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NEW CLOSEDNESS RESULTS AND ALGORITHMS FOR FINDING EFFICIENT SETS IN MULTIPLE OBJECTIVE MATHEMATICAL PROGRAMMING By ERJIANG SUN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 ACKNOWLEDGMENTS I would like to express my appreciation of the members of my entire supervisory committee, Dr. Harold P. Benson, Dr. S. Selcuk Erenguc, Dr. Gary J. Koehler, and Dr. Panos M. Pardalos. I am especially grateful to Dr. Benson, my committee chairman, for his invaluable guidance and tireless support throughout my years in the program. I would also like to thank Dr. S. Schaible for his support in my academic career in general. I would like to acknowledge all of the doctoral students, especially George Boger and Lawrence Nicholson, for their friendships. Finally, I would like to thank my entire family for the love and support they have provided me through the years. And I am grateful beyond words to my beloved wife, Hong Tang, and my daughter, Yantang Sun, who suffered and rejoiced with me through it all. TABLE OF CONTENTS page A CK N O W LED G M E N T S ................................................... .............. .... ............ ii A B STR A C T ........................................v............................. CHAPTERS 1 INTRODUCTION .................................... ............................ .. .............. Historical Perspective on M CDM .............................................. ..... ...... .......... M motivation for this R research ................................................................................ . Contents of this Research ................................................. .......................... 15 Organization of this Dissertation....................... ..... ............................. 17 2 LITERATURE SURVEY ................ ....................... ............................. 18 Approaches to Characterizing Efficient Solutions...................................................18 Closedness of the Efficient Set .................. ............... .... ......... ................ 22 Methods for Solving Multiple Objective Linear Programming Problems ................... 24 3 ON THE CLOSEDNESS OF THE EFFICIENT SET OF PROBLEM MOMP......... 31 D efinitions and N otation.................................................................................. ... 33 Quasiconcave and Strictly Quasiconcave VectorValued Functions .........................35 Parametric Representations of the Efficient Set .....................................................47 The Closedness of E(f, X) for General MOMP Problems .................................. 54 The Closedness of E(f, X) for Bicriteria Programming Problems .............................. 59 C including R em arks ................... ....... .... ......................................... ............. 62 4 FINDING THE SET OF ALL EFFICIENT EXTREME POINTS FOR PROBLEM MOLP IN THE OUTCOME SPACE ............... .............................................63 Decision SetBased Decomposition of the Weight Set W ....................................... 67 Outcome SetBased Decomposition of the Weight Set W ................... ................ 70 The Basic Weight Set Decomposition Algorithm ....................................................81 Tree Search Approach for STEP 3 of BWSDA.................................... .................. 89 Concave Programming Approach for STEP 3 of BWSDA............................. .......... ........ 97 D iscu ssion .............................................................. ....... 109 Concluding Remarks .................................................................................... 112 5 FINDING THE EFFICIENT OUTCOME SET OF PROBLEM MOLP .................. 114 P relim inaries .................................................................................. ............... 115 Theoretical Background ..................................................................... ............... 116 The Algorithm for Finding the Efficient Outcome Set ............................................ 133 Concluding Remarks .......................................................... 138 6 SUMMARY AND FUTURE RESEARCH ...................................................... 139 REFEREN CES......................... ................................................. ............... 144 BIOGRAPHICAL SKETCH .......................................................................... ....... 154 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NEW CLOSEDNESS RESULTS AND ALGORITHMS FOR FINDING EFFICIENT SETS IN MULTIPLE OBJECTIVE MATHEMATICAL PROGRAMMING By Erjiang Sun August 2000 Chairman: Dr. Harold P. Benson Major Department: Decision and Information Sciences A multiple objective mathematical programming (MOMP) problem involves simultaneously maximizing or minimizing several noncomparable objective functions over a nonempty feasible region. In order to help the decision maker find a most preferred solution to the underlying problem, researchers have shown that one can generally restrict one's attention to the subset of feasible solutions called the efficient set. One important and new (compared to traditional single objective mathematical programming) research area in multiple objective mathematical programming is to investigate the characteristics, properties and structure of the efficient set and the weakly efficient set. The characteristics and properties of the efficient set not only have important theoretical meaning but also provide the theoretical foundations upon which various algorithms are based. Another important research area is to develop algorithms that can be used to solve largescale multiple mathematical programming problems efficiently. In this dissertation, we first investigate the necessary and sufficient conditions for the efficient set of the problem (MOMP) to be closed. There are various theoretical, algorithmic, and practical reasons for investigating the necessary and sufficient conditions for the efficient set of the problem (MOMP) to be closed. However, only a small number of results of this type, limited to some special cases, have been developed. In this research, we present a necessary condition and several sufficient conditions for the efficient set of a general MOMP problem to be closed. Our approach relies in part upon generalizing the concepts of quasiconcavity and strict quasiconcavity for realvalued functions to vector valued functions. Our approach also relies upon some new characterizations of efficient solutions of problem (MOMP) that we developed in this research. The remainder of this research is focused on developing new outcome spacebased algorithms for potentially solving largescale multiple objective linear programming (MOLP) problems. In this dissertation, two new outcome spacebased weight set decomposition algorithms have been developed for solving the problem (MOLP) by using the vector maximization approach. One algorithm is devoted to finding the set of all efficient extreme points in the outcome space for problem MOLP. The other algorithm is devoted to finding the entire efficient outcome set of problem MOLP. To our knowledge, this algorithm is the first one capable of generating the entire efficient outcome set of a multiple objective linear programming problem directly. Both algorithms are based in part upon our new partition of the weight set. These two algorithms overcome the difficulties that face the traditional decision space based weight set decomposition algorithms. They may prove to be capable of solving largescale (MOLP) problems efficiently. CHAPTER 1 INTRODUCTION Decision making is the process of selecting a possible course of action from all of the available alternatives. In such processes, the decision maker or decision makers want to attain one or more than one objective or goal in selecting a preferred course of action while satisfying the constraints dictated by the environment. In the real world, many decision making processes involve a single decision maker (DM) who chooses among a countable or an uncountable set of alternatives by using two or more criteria. These processes are called multiple criteria decision making (MCDM) processes. When the values of the criteria are assumed to be known with certainty, the MCDM problem is called deterministic; otherwise it is called nondeterministic (or stochastic). Throughout this dissertation, we will limit our discussion to deterministic MCDM problems. 1.1. Historical Perspective on MCDM In this subsection, we will give a brief historical perspective on the development of MCDM. We will confine our discussion of solution techniques to the mathematical programming approaches. Our review is largely based upon the perspective required for understanding the chapters to follow. For deeper background and additional references in the field, the reader is referred to the books and survey papers by Cohon (1978), Stadler (1979), Hwang and Masud (1979), Zeleny (1982), Chankong and Haimes (1983), Evans (1984), Sawaragi et al. (1985), Yu (1985), Gal (1986), Steuer (1986), Shin and Ravindran (1991), Dyer et al. (1992), Korhonen et al. (1992), Pardalos et al. (1995), and references therein. Mathematically, one way to represent a multiple criteria decision making (MCDM) problem is by modeling it as a multiple objective mathematical programming problem. A multiple objective mathematical programming problem may be written MOMP: vmax f(x) = (fl(x), ... (x))T s.t. x eX, where x eR" is a vector of decision variables, fi, i = 1,2,...,p, are objective functions and X is the set of feasible decision alternatives. X is usually called the decision set of problem MOMP. Let Y= { yeRP y = f(x), X}. Y is usually called the outcome set of problem MOMP. When f(x) is a linear vectorvalued function and X is a convex polyhedron, the problem MOMP is called a linear vector maximization problem or a multiple objective linear programming (MOLP) problem. In problem MOMP, the goal is to simultaneously maximize all of the objective functions. However, with rare exceptions, the objective functions of problem MOMP conflict with one another. As a result, a solution that simultaneously maximizes all of the objective functions exists only in rare cases. Therefore, instead of trying to find a solution that maximizes all of the objective functions simultaneously, researchers and practitioners generally try to find a solution that maximizes the DM's preferences. In other words, "solving" problem MOMP means finding a most preferred solution of the DM. In order to find such a solution, some information about the DM's preference structure is required. The most complete way of expressing this information is by finding the DM's utility function, also called a value function, over the objective function space of the problem (see e.g. Keeney and Raiffa, 1976). Definition 1.1.1. A function u, which associates a real number u(f(x)) to each x in X, is said to be a utility function representing a particular decision maker's preference structure provided that (1) f(x1) ~ f(x2) if and only if u(f(x1)) = u(f(x2)) for all x', x2 in X; and (2) f(x) f(x2) if and only ifu(f(x')) > u(f(x2)) for all x', x2 in X, where f(x') f(x2) denotes that the decision maker is indifferent between outcomes f(x1) and f(x2), and f(x') > f(x2) denotes that the decision maker prefers f(x') to f(x2). Given a utility function u for the DM, the decision maker's most preferred solution is one that maximizes u over all feasible solutions; such a solution is also called a best compromise solution (in the sense that it is typically a compromise among the problem's various objective functions). Under the assumption that more of each objective function is preferred to less, a best compromise solution must also be an efficient solution, where an efficient solution is defined as follows. Definition 1.1.2. A point xOR" is called an efficient solution (or a nondominated solution) of MOMP if xo EX and there is no x eX such that f(x) > f(xo) and f(x) # f(x). The set of all efficient solutions of problem MOMP is usually called the efficient decision set of problem MOMP. If xo is an efficient solution of problem MOMP, then y = f(xo) is usually called an efficient outcome of problem MOMP. The set of all efficient outcomes of problem MOMP is called the efficient outcome set of problem MOMP. We will refer to both the efficient decision set and the efficient outcome set as the efficient set when the results apply to both sets. Closely related to the concept of an efficient solution is the concept of a weakly efficient solution. A weakly efficient solution may be defined as follows. Definition 1.1.3. A point xORn is called a weakly efficient solution (or a weakly nondominated solution) of MOMP if xo X and there is no x eX such that f(x) > fx). Similarly, we have the concepts of weakly efficient outcome, weakly efficient decision set, weakly efficient outcome set and weakly efficient set. The concept of efficiency has played a useful role in analyzing problem MOMP. In particular, it has been found to be very useful in maximizing the decision maker's utility function over X when the form of u is unknown. The notion of an efficient solution was first introduced by Pareto in 1896 (see Pareto, 1896). However, until the 1950's when Koopmans (1951) introduced the concept of an efficient vector, very few researchers were involved in the field of MCDM. An early formulation of the problem of vector maximization was due to Kuhn and Tucker (1951) and later to Karlin (1962). A direct extension of Koopmans' ideas also appears in a book by Charnes and Cooper (1961). In order to exclude certain efficient solutions that display an undesirable anomaly, several authors have developed several more sophisticated concepts of efficiency. Kuhn and Tucker (1951) were the first to propose a special concept of efficiency which they called proper efficiency. Geoffrion (1968) later refined Kuhn and Tucker's definition. The underlying domination cone utilized in both of these definitions was the ordinary nonpositive orthant. In 1974, Yu (1974) introduced a domination structure defined by a generalized convex cone. Soon after, Borwein (1977) proposed a definition of proper efficiency for the case when the domination cone is any nontrivial, closed convex cone. Benson (1979) and Henig (1982) later refined Borwein's definition. In a related vein, Hu (1990) subsequently introduced the concept of major efficiency when the domination structure is defined by a certain nonconvex cone. Much of the research in MCDM has occurred since 1970. In fact, Zeleny (1982) notes that the general area of multiple criteria decision making was the most rapidly growing area of operations research during the 1970's. Most of the MCDM research in the 1970's was focused on the theoretical foundations of multiple objective mathematical programming and the development of algorithms and procedures for solving some of these problems especially multiple objective linear programming problems and problems with discrete alternatives. Since a best compromise solution must be an efficient solution under mild assumptions usually satisfied in practice, one approach to solving MOMP is to find all or a part of the efficient set and present them to the DM for evaluation. This kind of approach is usually referred to as the vector maximization approach. During the 1970s, most researchers focused on developing and investigating algorithms that use this approach. As a result, many algorithms for determining the set of all efficient extreme points or the entire efficient decision set of a multiple objective linear program were developed in this period. We will give a literature review of the algorithms that use the vector maximization approach for solving multiple objective linear programming problems in Section 2.3. The reader is also referred to the review paper by Evans (1984) and to the book by Steuer (1986) for deeper reviews of the algorithms that were developed during this period for multiple objective linear programming. One important and new (compared to traditional single objective mathematical programming) research area in multiple objective mathematical programming is to investigate the characteristics, properties and structure of the efficient set and the weakly efficient set. The characteristics and properties of the efficient set not only have important theoretical meaning but also provide the theoretical foundations upon which various algorithms are based. For example, many algorithms are based upon the theoretical fact that an efficient solution can be characterized as an optimal solution to an appropriate single objective mathematical programming problem. In addition, the simplexbased methods for finding the entire efficient decision set of a multiple objective linear programming problem, for example, are usually based in part upon two theoretical facts. The first of these is the fact that the efficient decision set for such a problem is the union of its efficient faces. The second is the fact that the set of all efficient extreme points for such a problem is connected in the sense that one can move from one extreme point to any other without having to leave the edges of the efficient decision set (Yu and Zeleny, 1975). For details of various simplexbased algorithms for finding the entire efficient decision set of problem MOLP, the reader is referred to Steuer (1986). During the 1970's, research on the characteristics, properties and structure of the efficient set and of the weakly efficient set was focused on developing various kinds of approaches for characterizing efficient solutions and weakly efficient solutions of problem MOMP. We will give a more detailed review of the characteristics, properties and structure of the efficient set of problem MOMP in Sections 2.12.2. The reader is also referred to the review paper by Gal (1986). Duality and stability theories were also substantially investigated by many authors during the 1970s. The reader is referred to the book by Sawaragi et al. (1985) and to the review paper by Gal (1986) for details. During the 1980's, emphasis shifted toward the implementation ofMCDM models and algorithms on computers. Decision support systems were developed to aid this implementation. With respect to the development of new algorithms, in part since the size of the efficient set can make it difficult to find a final best compromise solution, interactive procedures moved to center stage in the 1980s. This class of procedures relies on the progressive definition of the DM's preferences along with the exploration of the criterion space. The progressive definition takes place through a DManalyst or DMmachine dialogue at each iteration. A typical interactive algorithm can be characterized by the following procedure: (1) find a solution (preferably feasible and efficient); (2) interact with the DM to obtain his/her reaction/response to this solution; and (3) repeat the above steps until the current solution is "close enough" to a best compromise solution or until some other termination criterion is met. For details about the research of this type in this period, see the review papers by Korhonen et al. (1992) and Shin and Ravindran (1991) and the references therein. As we mentioned earlier, one important topic in multiple objective programming is the investigation of the characteristics and properties of the efficient set. During the 1980s, research in this topic was focused on investigating the connectedness of the efficient set and of the weakly efficient set. The connectedness of the weakly efficient set of problem MOMP was first studied by Warburton (1983). For further results on the connectedness of the weakly efficient set, the reader is referred to Luc (1989). For the connectedness of the efficient set, one open problem raised in this period was whether or not the efficient set of an mdimensional MOMP problem is connected when all objective functions are continuous and strictly quasiconcave, and the decision set is compact and convex. Schaible (1983) solved the problem in the case of two objective functions, and then Choo et al. (1985) obtained a partial solution in the case of three objective functions. Finally, Daniilidis, Hadjidis and Schaible (1997) completely solved the problem in the case of three objective functions. For higher dimensions, Hu and Sun (1993) solved the problem in the case where the efficient set is closed. Subsequently, Sun (1996) solved the problem in the case where one objective function is strongly quasiconcave. Finally, Benoist (1998) extended the results of Sun (1996) and solved the problem. Since the late 1980s, MCDM research has increasingly focused on how to support the real world decision making process. Traditionally, any problem in operations research is divided into two phases, formulation and solution. However, this scheme is improper in the real world. It is not fully realistic to assume that a DM is able to formulate a problem precisely prior to its solution. For the majority of decision making problems, the actual process is a multiplyrepeated, cyclic process of "formulationsolutionanalysiscorrection ...." (see Statnikov and Matusov, 1995). As a result, in recent years, much MCDM research has focused on developing systems that support the entire decision making process from problem structuring through solution and implementation (see, e.g. Korhonen and Wallenius, 1988). 1.2. Motivation for this Research The real world challenges us with many largescale multiple criteria decision making problems. It is known that the efficient decision set of problem MOMP is generally a complicated nonconvex set and grows rapidly as the size of the problem increases (see e.g. Benson, 1998a). As a result, even with the aid of recent decision support systems, the algorithms that find all or a substantial part of the efficient decision set for problem (MOMP) in the decision space are usually not able to usefully solve largescale problems. For example, consider some results reported by Benson. In Benson (1998a), the ADBASE algorithm of Steuer (1989) was used to find the set of all efficient extreme points in the decision space of some randomlygenerated multiple objective linear programming problems with four objective functions. Benson found that when n=30 and X is described by 25 linear inequalities, the average number of efficient extreme points in X in a set often randomlygenerated problems was 7245.90. When n was increased to 50 and X was described by 50 linear inequalities, this average jumped to 83,780.60 points. With n=60 and with 50 linear inequalities describing X, each of the ten problems that were randomlygenerated in this study exceeded the solution capacity of ADBASE, indicating that the number of efficient extreme points in the decision space in each of these problems exceeded 200,000. These results indicate that, for a largescale MCDM problem, we either cannot generate all or a substantial part of the efficient decision set, or the generated set will probably overwhelm the DM. It is largely because of this that interactive procedures became popular in the 1980's. The interactive approach is actually a learning process. The DM, in fact, takes part in the solution process of an interactive procedure. Because of these and other advantages, it seems reasonable to think that interactive approaches would be very useful in practice. However, Korhonen et al. (1992) pointed out that, in actuality, only a handful of interactive multiple objective procedures have been applied and implemented in practice. One problem with interactive approaches is that nearly all of them require consistent responses from the DM to be successful. For largescale problems, it may becomes very difficult for the DM to respond consistently. This is because, for a large scale problem, it may take numerous iterations to finally reach a solution that satisfies the DM. Furthermore, each iteration may take much more time for a largescale problem than for a smallscale problem. Thus, the assumption that the DM can provide consistent responses usually draws severe criticism. Some researchers, such as French (1984), feel that the value of the interactive methods is significantly diminished due to this drawback. Another problem with the interactive approach is that only a few points in the feasible set or the efficient set are generally explored. This is in part because the learning process is tedious and the DM tends to get tired after a relatively small number of interactions. As a result, the DM may not be provided with sufficient information about the efficient set to make a useful choice of a best compromise solution. For largescale problems, this situation may become worse because of the larger size of the efficient set. The discussions above show that it is still very important to develop effective algorithms that can solve largescale problems. Some ideas and algorithms have been proposed, in particular, for potentially solving largescale multiple objective linear programming problems by the vector maximization approach. One of these ideas involves analyzing multiple objective linear programming problems in the outcome space (see e.g. Dauer, 1987; Benson, 1995a). Three reasons for analyzing multiple objective linear programming problems in the outcome space, rather than in the decision space, were summarized in Benson (1998c). First, the dimension p of the outcome space is typically much smaller than the dimension n of the decision space. As a result, the efficient outcome set is invariably much smaller and has a much simpler structure than the efficient decision set; see, e.g. Benson (1995a, 1997, 1998ac), Dauer (1987, 1993), Dauer and Saleh (1990, 1992) and Dauer and Liu (1990). Generating all or parts of the efficient outcome set is therefore expected, in general, to be less computationally demanding than generating all or portions of the efficient decision set. Furthermore, it also follows that the DM will be less likely to be confused or overwhelmed if all or portions of the efficient outcome set are presented to him or her than if all or portions of the efficient decision set are presented. Second, the DM generally prefers searching for a most preferred solution by examining the efficient outcome set rather than the efficient decision set. This has been shown by empirical research. Third, it is well known that frequently the objective functions map many points in the efficient decision set onto either a single outcome or onto essentiallyequivalent outcomes in the efficient outcome set. Thus, generating points directly from the efficient outcome set avoids risking redundant calculations of points in the efficient decision set. To our knowledge, the first algorithm capable of generating the set of all efficient extreme points in the outcome space for general multiple objective linear programming problems was proposed by Benson (1998a). This algorithm uses outer approximation techniques from global optimization (Horst and Tuy, 1993). It works by generating a finite number of polyhedra that each approximates an "efficiencyequivalent" polyhedron for the problem. The algorithm is finite, works in the outcome space, and can be implemented relatively easily by using univariate search techniques, linear programming, and some global optimization techniques. The most computationallydemanding task in the algorithm calls for determining the set of extreme points of each new polyhedron that is created by adding a linear inequality cut at each step of the algorithm. This is accomplished with the aid of outer approximation. As a byproduct of the algorithm, the entire weakly efficient outcome set of problem MOLP is also generated (see Benson, 1998a, 1998c). A newer outcome spacebased method for multiple objective linear programming, called a hybrid approach, has also been proposed by Benson (1998c). This approach adapts two global optimization, decision setbased methods to outcome space. These methods are a special simplicial partitioning technique of Ban (see Ban, 1983; Tuy and Horst, 1988), proposed originally for solving concave minimization problems, and a general outer approximation method that has been used very frequently to help solve a variety of global optimization problems (see Horst and Tuy, 1993 and Horst and Pardalos, 1995). In particular, the simplicial partitioning technique is systematically integrated into the outer approximation scheme in outcome space to determine all efficient extreme points in the outcome set of the problem in a finite number of iterations. The two algorithms of Benson have many potential advantages (see Benson, 1998a, 1998c). Because of their advantages, these two algorithms may prove to be suitable for solving largescale multiple objective linear programming problems. However, since the two algorithms of Benson, to our knowledge, are the only two outcome space based algorithms that are capable of generating the set of all efficient extreme points of a general MOLP problem, and since we don't know how well these two algorithms will work, we need to develop and explore more new outcome space based algorithms. In particular, because different kinds of problems may need different approaches, the algorithms of Benson (1998a, 1998c) may work well on some problems but not on others. As a result, we may need alternative algorithms for efficiently finding the efficient outcome set of problem MOLP in some cases. This also calls for more outcome space based algorithms for generating the set of efficient extreme points of problem MOLP. The rationale for generating the set of all efficient extreme points is based on the assumption that the best efficient extreme point is an acceptable approximation of a best compromise solution. However, this assumption may not true. As a result, we may need to generate the entire efficient set in some cases. The algorithms in Benson (1998a and 1998c) can generate the entire weakly efficient set in the outcome space. Since the work of generating the entire efficient set in the outcome space from the weakly efficient set in the outcome space is generally nontrivial, we need algorithms that can directly generate the entire efficient set in the outcome space. As we know, there are various kinds of algorithms for finding the set of efficient extreme points in the decision space or the entire efficient decision set of problem MOLP. All of these algorithms are based in part on some kind of modified simplex method. The idea of using simplex techniques was also adapted by some authors to develop simplelike procedure in the outcome space (see e.g. Dauer and Saleh, 1990; Dauer and Liu, 1990). However, all these simplelike procedures have to deal with certain unfortunate difficulties, such as tedious bookkeeping, backtracking, and degeneracy of extreme points. To overcome these problems, we would like to develop nonsimplexlike algorithms. Motivated by all of the above issues, part of this research will focus on developing new outcome space based, nonsimplexlike algorithms for solving problem MOLP. As we have mentioned earlier, one important research area in multiple objective mathematical programming is to investigate the characteristics, properties and structure of the efficient set. Among the properties of the efficient set, closedness is of interest. One reason is that some algorithms are based in part upon this property (see e.g. Kornbluth and Steuer, 1981). However, up to now, there are only a few results about the closedness of the efficient set for some special cases. These results are scattered or implied in several papers; see e.g. Yu and Zeleny (1975), Schaible (1983) and Choo and Atkins (1983). We will give a review on these results in Section 2.2. The reason that only a few results on closedness are reported in the literature is probably that the efficient set is generally not closed. Even when all of the objective functions are continuous and strictly qusaiconcave and the feasible set is compact, the efficient set need not be closed (see Choo and Atkins, 1983). Thus, an interesting and important question is whether or not there are general conditions under which the efficient set of problem MOMP is closed. Motivated by this, we will focus part of this research on investigating necessary and sufficient conditions for the efficient decision set of problem MOMP to be closed. 1.3. Contents of this Research In this research, we will present two new outcome spacebased algorithms that may be potentially useful for solving largescale multiple objective linear programming problems. One algorithm is devoted to finding the set of all efficient extreme points in the outcome space for problem MOLP. Since this algorithm is based on the decomposition of the weight set. We will call it Basic Weight Set Decomposition Algorithms (BWSDA). The algorithm BWSDA works in the following way: At each iteration, the algorithm will first either find a weight vector which will lead to an unexplored efficient extreme point in the outcome space or conclude that all points in YE r Yex have been generated. If a new weight vector is found at some iteration, the algorithm will in the next iteration call for solving at most (p+l) linear programs in order to find an unexplored efficient extreme point in the outcome space. Two different kinds of approaches will be developed for finding, if it exists, a new weight vector in each iteration that leads to an unexplored extreme point in the outcome space. One approach uses a tree search method. The other calls for solving a special concave minimization problem over a polyhedron. These two different approaches yield two versions of the Basic Weight Set Decomposition Algorithm, Weight Set Decomposition Algorithm I (WSDA I) and Weight Set Decomposition Algorithm I (WSDA II). The other algorithm is devoted to finding the entire efficient outcome set of problem MOLP. To our knowledge, this algorithm is the first one capable of generating the entire efficient outcome set of a multiple objective linear programming problem. The remainder of this research is focused on investigating necessary and sufficient conditions for the efficient set of problem MOMP to be closed. We first introduce a new definition of strict quasiconcavity for a vectorvalued function. This definition extends the definition of strict quasiconcavity for a realvalued function to vectorvalued functions. We will show that the efficient decision set of a general MOMP problem is closed when the vectorvalued function formed by the objective functions is strictly quasiconcave under our new definition and the feasible region is compact and convex. For the special case of the biobjective programming problem, we will extend the results obtained by Schaible (1983). In Schaible (1983), it is shown that the efficient decision set for a biobjective programming problem is closed when the two objective functions are continuous and strictly quasiconcave and the feasible region is compact and convex. In this dissertation, for the biobjective programming problems, we will extend Shaible's results to cases where the feasible region need not be convex and the objective functions need not be strictly quasiconcave. 1.4. Organization of This Dissertation The dissertation is organized in the following way. In Chapter 2, we give a literature review of common approaches for characterizing efficient solutions, a literature review of studies on closedness of the efficient set of problem MOMP, and a literature review of the algorithms for solving MOLP problems. In Chapter 3, we will study some new approaches for characterizing efficient solutions and some necessary and sufficient conditions for the efficient set of problem MOMP to be closed. New definitions for the quasiconcavity and strict quasiconcavity of a vectorvalued function are also given in this chapter. In Chapter 4, we will give two new algorithms for finding the set of all efficient extreme points of a multiple objective linear programming problem in the outcome space. In Chapter 5, we will give a new algorithm for finding the entire efficient set of a multiple objective linear programming problem in the outcome space. Finally, in Chapter 6, we will discuss some ideas for further research. CHAPTER 2 LITERATURE SURVEY In this chapter, we will present a literature review of some important approaches to characterizing efficient solutions, a literature review of the closedness of the efficient set and a literature review of the algorithms for solving multiple objective linear programming problems. 2.1. Approaches to Characterizing Efficient Solutions In order to operationalize the concept of an efficient solution, a common idea is to relate it to a familiar concept. The most common strategy of this type is to characterize efficient solutions in terms of optimal solutions of appropriate scalar optimization problems. There are many ways of forming appropriate scalar problem for accomplishing this. Among them are two kinds of commonly used scalar optimization problems. These are "the weighting problem" and "the kthobjective e constraint problem Since our theoretical results and algorithms are based on these two common approaches, we will give a brief review on these two kinds of scalar optimization problems and their relationships with the MOMP problem. For other approaches to characterizing efficient solutions in terms of appropriate scalar optimization problems, the reader is referred to the book by Chankong and Haimes (1983) and to references therein. For example, a strategy used by Sun (1996) is to characterize efficient solutions of an MOMP problem in RP in term of efficient solutions of some particular MOMP problems in RP'. One advantage of this strategy is that we can use the induction method with this approach to obtain some theoretical results (see Sun, 1996). Let W = (wl, ..., WpT I wj > 0 for allj = 1,...,p }, WO = { w = (wi, ... W)T I wj > 0 for all j = 1,...,p }. We will refer to both W and W as the weight set associated with the MOMP. The weighting problem can be defined as follows: p P(w): max w Jfj(x) j3i s.t. x e X, where w e W is a nonnegative weight. The weighting problem P(w) has been used to analyze the efficient solutions since the vector maximization problem was first formulated by Kuhn and Tucker in 1951. Here we will state three major results which show relationships between P(w) and MOMP in the cases that fi, ..., fp are general realvalued functions, concave functions and linear functions, respectively. In the following, Result 1 is due to the work of Geoffrion (1968), Zadeh (1963) and Yu (1974). Result 2 is due to Karlin (1959). Some similar results can be found in DaCunha and Polak (1967), and Yu (1974). Result 3 is due to Zeleny (1974), Yu and Zeleny (1975) and Isermann (1976). Result 1. x* is an efficient solution of MOMP if there exists w e W such that x* is an optimal solution of P(w) and either one of the following two conditions holds: (i) wj > 0 for allj = ,...,p; (ii) x* is the unique optimal solution of P(w). For a given weight vector wo, if we solve P(wo) and obtain an optimal solution x', we can claim from Result 1 that x* is an efficient solution of MOMP if either x* is the unique optimal solution of P(wo) or w e WO. Thus, Result 1 implies that we can find some efficient solutions of (MOMP) by solving P(w) for some properly chosen value of w. Result 2. Assume that X is a convex set and fj, j=l,...,p, are concave on X. If x* is an efficient solution of (MOMP), then there exists w e W such that x* is an optimal solution of P(w). The above result implies that under the convexity assumptions on f and X, we can find all efficient solutions by solving P(w) for all w e W. Since Result 2 does not guarantee the efficiency of an optimal solution of P(w) when w e W, we need to verify whether an optimal solution ofP(w) is efficient. Result 3. Assume that X is a convex polyhedron and fj, j=l,...,p, are linear functions. Then, x* is an efficient solution of (MOMP) if and only if there exists a weight vector w E Wo such that x* is an optimal solution ofP(w). The above Result 3 implies that the set of all efficient solutions of a multiple objective linear programming problem can be found by solving P(w) for all w e W. Many theoretical results and algorithms in the literature on multiple objective mathematical programming are based upon the above three results. Our algorithms in Chapter 4 and Chapter 5 are also based upon Result 3. In Chapter 4, we will decompose the weight set Wo into a number of subsets. Based upon this decomposition, we will then present two new algorithms for finding the set of all efficient extreme points and an algorithm for finding the set of all efficient points for problem MOLP in the outcome space. The kthobjective econstraint problem was introduced by Haimes (1970) and Olagundoye (1971), and it can be defined as follows: Pk(e): max fk(X) s.t. fj(x) j, j= 1,...,p, j k, x X, where e = (ei,..., sk1, Ek+i,..., Ep)T. For a given x*, we will use Pk(e*) to represent the problem Pk(e), where s j = e j* = f,(x*), j # k.. The theoretical equivalence between Pk(e) problems and a general MOMP problem was established by Haimes et al. in 1971 and, then, was extended by Chankong and Haimes in their book in 1983. Some other results for general MOMP problems were also obtained by Lin (1976, 1977). Their main results are stated in the following. Result 4. Let x* be an optimal solution of Pk(8*) with 8 = fj(x'), j # k.. Then x* is an efficient solution of MOMP (i) if and only if x* is an optimal solution of Pk(8*) for every k = 1,...,p; (ii) if and only if the optimal value of Pk(e) is strictly less than fk(x*) for any 8 2 ; (iii) if x* is the unique optimal solution of Pk(8*) for some k. The sufficient condition for efficiency in Result 4 (i) requires Pk(e) to be solved for all k before a conclusion can be drawn. The sufficient condition in Result 4 (ii) requires one to compare fk(x') with the optimal values of all problems Pk(s), e > s*. The sufficient condition in Result 4 (iii) requires x* to be the unique optimal solution of Pk(e*). Under some generalized convexity assumptions, Sun (1996) obtained the following result. Result 5. Let X c R" be a nonempty compact and convex set, and let fi, i = 1,...,p be continuous and quasiconcave on X. If fk is strongly quasiconcave on X, then x* is an efficient solution of problem MOMP if and only if there is e e {(e,,..., Ek1, Sk+1,..., p)TI si = fi(y) for all i # k, y e X} such that x* is an optimal solution of Pk(g*). In this research, we will give a different sufficient condition which does not have these requirements for an optimal solution of Pk(E) to be efficient. Some necessary and sufficient conditions for the efficient set to be closed will be obtained based upon this sufficient condition. 2.2. Closedness of the Efficient Set One important research area in the theory of multiple objective mathematical programming is to investigate the topological properties of the efficient set and of the weakly efficient set. Among these properties, closedness is of particular interest. The closedness of the efficient set and of the weakly efficient set is both theoretically and algorithmically important. Following are some reasons for this. (1) The closedness of the efficient set guarantees the existence of at least one optimal solution for the problem of optimizing a continuous function over the efficient set of a bounded MOMP problem. (2) An algorithm for finding all weakly efficient extreme points of a multiple objective linear fractional programming problem that was proposed by Kornbluth and Steuer (1981), is based in part on the facts that the weakly efficient set is both closed and connected. (3) The closedness of the efficient set in biobjective strictly quasiconcave programming has been used by Daniilidis et al. (1997) to prove the connectedness of the efficient set in threecriteria quasiconvex programming. It is well known that the weakly efficient set is closed when the objective functions are continuous and the feasible set is closed (Choo and Atkins, 1983). However the efficient set is not closed, in general, even if the objective functions are continuous and strictly qusaiconcave and the feasible set is compact (Choo and Atkins, 1983; Steuer, 1986). Up to now, only a few results exist concerning the closedness of the efficient set for some special cases. Yu and Zeleny (1975), for instance, showed that the efficient set of a multiple objective linear programming with a compact feasible region can be represented by the union of its maximal efficient faces. Since the number of maximal efficient faces is finite and each maximal face is closed, the efficient set of a multiple objective linear programming problem with a compact feasible region is closed. Later, Schaible (1983) showed that the efficient set of a biobjective programming problem is closed when the objective functions are continuous and strictly quasiconcave and the feasible region is compact and convex. As a result, it is important to investigate necessary and sufficient conditions for the efficient set of a general MNOMP problem to be closed. 2.3. Methods for Solving Multiple Objective Linear Programming Problems In the literature, the methods for solving a MOLP problem are usually classified into three categories on the basis of the time at which the DM needs to articulate his/her preference structure over the set of feasible alternatives (see Hwang, 1979, and Evans, 1984). The first category consists of techniques that require prior articulation of the DM's preferences. This means that the preference information of the DM is given to the analyst (who is responsible for solution of the MOLP) before the analyst actually solves the problem. The information may be either (1) cardinal information, or (2) mixed (ordinal and cardinal) information. The major methods using the approach of cardinal information are utility function methods. In all of the utility function methods, the MOMP (or MOLP) is converted to max U(f)= U(ft,...,fp), s.t. x e X, where U(f) is the utility function of the DM over the multiple objectives. Thus these methods require that U(f) be known prior to solving the MOMP (or MOLP). The literature on utility function methods and on problems of determining U(f) is discussed and reviewed in Farquhar (1977), Huber (1974), Keeney and Raiffa (1976), and Dyer and Sarin (1979). The major advantage of utility function methods is that if U(f) has been correctly assessed and used, it will ensure the most satisfactory solution to the DM. The major difficulty with the utility function methods is that the DM is required to articulate preference judgements in an information vacuum. The methods using the approach of mixed (ordinal and cardinal) information are called goal programming (Lee, 1973), and lexicographic ordering (Keeney and Raiffa, 1976). Goal programming was originally proposed by Charnes and Cooper (1961) for a linear model. The method requires the DM to set goals that he/she wishes to attain for each objective function. A preferred solution is then defined as the one that minimizes the deviations from the set goals. In the most common formulation of goal programming, the DM, in addition to setting goals for the objective functions, must also be able to give an ordinal ranking of the objectives. For recent developments in goal programming, we refer the reader to the book edited by Tamiz (1996). The lexicographic method requires the objective functions be ranked in order of importance by the DM. The preferred solution obtained by this method is the one that maximizes the objectives starting with the most important and proceeding according to the order of the importance of the objectives. Since the obtained solution is very sensitive to the ranking of the objectives given by the DM, the analyst should exercise caution in applying this method when some objectives are of nearly equal importance. The second category of methods for solving problem MOLP consists of methods that require the progressive articulation of preferences by the DM. This class of methods, generally referred to as interactive methods, relies on the progressive definition of the DM's preferences along with the exploration of the criterion space. The progressive definition takes place through a DManalyst or DMmachine dialogue at each iteration. As explained by Hwang and Masud (1979), and Shin and Ravindran (1991), the advantages of the interactive methods are (1) there is no need for 'a priori' preference information; (2) they each yield a learning process for the DM from which the DM eventually understands the behavior of the system; (3) only local preference information is needed; and (4) since the DM is part of the solution process, the solution obtained has a better prospect of being implemented. Again, as explained by Hwang and Masud (1979), and Shin and Ravindran (1991), the disadvantages are (1) solutions depend upon the accuracy of the local preference information that the DM can give; (2) for many methods there is no guarantee that the preferred solution can be obtained within a finite number of interactive cycles; (3) much more effort is required of the DM than in the methods in the previous category; (4) nearly all interactive methods require consistent responses from the DM to be successful. For surveys of interactive algorithms, we refer to Steuer (1986) and Shin and Ravindran (1991). The third category of solution methods for problem MOLP consists of methods that require posteriori articulation of preferences. Each of these methods, generally referred to as vector maximization approaches, typically generates all or a substantial part of the efficient set. The efficient set or the subset of the efficient set that was generated is then presented to the DM for evaluation. Since our algorithms lie in this category, we will give a more detailed review of the algorithms in this category. Algorithms in this category can be divided into two subclasses based upon the solutions they seek: (a) those which concentrate on finding a representative subset of the efficient set; (b) those which concentrate on finding the entire efficient set. Algorithms in this category can also be classified as either decisionspacebased or outcomespacebased. A decisionspacebased algorithm works in the decision space and concentrates on finding all or part of the set of the efficient solutions. An outcomespace based algorithm works in the outcome space and concentrates on finding all or part of the set of efficient objective function values. Most current algorithms are decisionspacebased algorithms. The algorithms in subclass (a) include those of Evans and Steuer (1973a, b), Zeleny (1974), Steuer (1976a, b), Ecker and Kouada (1978), Armand and Malivert (1991), Benson and Sayin (1997), Benson (1998a, c), among others. The algorithms in Benson and Sayin (1997) and Benson (1998a, c) are outcomespacebased algorithms. The algorithms in subclass (b) include those of Yu and Zeleny (1975), Isermann (1977), Gal (1977), Ecker et al. (1980), Armand and Malivert (1991), Armand (1993), among others. All of these algorithms are decisionspacebased algorithms. The most frequently generated representative subset of the efficient decision set for the algorithms in subclass (a) is the set of all efficient extreme points in the decision space. The rationale for generating the set of all efficient extreme points is based upon the assumption that the best efficient extreme point is an acceptable approximation of a best compromise solution. Steuer (1976a) notes that algorithms for finding all efficient extreme points in the decision space generally consist of three phases. Phases I and II consist of finding an initial extreme point and an initial efficient extreme point, respectively. Phase III involves finding all remaining efficient extreme points. Phases I and II are easy to implement and only require classical linear programming procedures. Phase III is where the various algorithms differ in their approaches. Steuer also notes that there are three classes of approaches for Phase III among those algorithms for finding all efficient extreme points in the decision space. The three classes are the decomposition of parametric space approach; the adjacent efficient basis approach; the adjacent efficient extreme point approach. We will give a detailed discussion of the decomposition of the parametric space approach later in Chapter 4. The adjacent efficient basis approach involves pivoting among all efficient bases. The adjacent efficient extreme point approach involves the pursuit of all efficient edges emanating from each efficient extreme point. All of these algorithms use modified simplex method approaches. One general difficulty of these algorithms concerns the handling of degeneracy. It is will known that several bases may correspond to a single degenerate vertex. Since all of these methods use basic feasible solutions to characterize the extreme points of the feasible region, this makes the determination of all efficient extreme points more complicated when degeneracy is present. To deal with degeneracy, some special pivoting rules are used in Armand and Malivert (1991). It is well known that the efficient decision set of a MOLP problem can be decomposed into the union of its maximal efficient faces in the decision space (Yu and Zeleny, 1975). Each efficient face is completely determined by the efficient extreme points and extreme rays that lie in it. Thus, the algorithms for finding the entire efficient decision set in subclass (b) usually work in the following way. First, all efficient extreme points and extreme rays are generated. Next, some identification and bookkeeping techniques are usually used to generate all maximal efficient faces. All of these algorithms are also based upon some type of modified simplex method approach and must deal with the degeneracy problem. Apart from the computational burden of employing these decisionspacebased methods and the complications induced by the presence of degeneracy, there are a number of other problems related to these algorithms. One problem is that the efficient decision set is generally a complicated nonconvex set that grows rapidly as the size of the problem increases. Thus, generating the set in its entirely is possible only in certain special cases. Another problem is that the size and nature of the generated set in the decision space can easily overwhelm the DM. It has been noted (Dauer, 1987, 1990, and Benson, 1995a) that since the number of objective functions is usually much less than the number of variables in problem MOMP, the efficient outcome set has a much simpler structure than the efficient decision set. Recently, some researchers have begun to develop tools for analyzing problem MOLP in the outcome space, rather than in the decision space of the problem. To our knowledge, the first algorithm that is capable of generating the set of all efficient extreme points in the outcome space for a general MOLP problem was proposed by Benson in 1998 (Benson, 1998a). This algorithm uses outer approximation techniques from global optimization (Horst and Tuy, 1993). It works by generating a finite number of polyhedra that each approximates an "efficiencyequivalent" outcome polyhedron for the problem. The most computationallydemanding task in the algorithm calls for determining the set of extreme points of each new polyhedron that is created by adding a linear inequality cut at each step of the algorithm. This is accomplished with the aid of outer approximation. A newer outcomespacebased method for multiple objective linear programming, called a hybrid approach, has also been proposed by Benson (1998c). This approach adapts two global optimization, decision setbased methods to the outcome space. These methods are a special simplicial partitioning technique of Ban (see Ban, 1983, and Tuy and Horst, 1988), proposed originally for solving concave minimization problems, and a general outer approximation method that has been used very frequently to help solve a variety of global optimization problems (see Horst and Tuy, 1993, and Horst and Pardalos, 1995). In particular, the simplicial partitioning technique is systematically integrated into the outer approximation scheme in outcome space to determine all efficient extreme points in the outcome set of the problem in a finite number of iterations. To our knowledge, there are no direct algorithms for finding the entire efficient outcome set. The algorithms in Benson (1998 a, 1998c), however, can be used to find the entire weakly efficient outcome set. As discussed in Section 1.2, it is important to develop new outcomespacebased algorithms for finding the set of all efficient extreme points in the outcome space, and to develop algorithms for finding the entire efficient outcome set directly. CHAPTER 3 ON THE CLOSEDNESS OF THE EFFICIENT SET OF PROBLEM MOMP We restate problem MOMP as follows: MOMP: vmax f(x) = (fi(x), ... f,(x)), s.t. x X, where p 2 2, x e R" is a vector of decision variables, fi, i = 1,2,...,p are objective functions and X is the set of feasible decision alternatives. Since problem MOMP has p objective functions, we call it a pdimensional multiple objective mathematical programming problem (or a pdimensional MOMP problem). The vectorvalued function f: X>RP is called a pdimensional vectorvalued function. The efficient decision set and weakly efficient decision set of problem MOMP are denoted by E(f, X) and Ew(f, X), respectively. The efficient outcome set and weakly efficient outcome set of problem MOMP are denoted by E(f(X), R ) and Ew(f(X), R ), respectively. One important research area in the theory of multiple objective mathematical programming is to investigate the topological properties of the efficient set and of the weakly efficient set. Among these properties closedness is of interest. The closedness of the efficient set and of the weakly efficient set are both theoretically and algorithmically important (c.f Section 1.2 and Section 2.2). It is well known that the weakly efficient decision set Ew(f, X) is closed when the objective functions are continuous over X and the feasible set is closed (Choo and Atkins, 1983). However the efficient decision set is not closed in general, even if the objective functions are continuous and strictly qusaiconcave on X and the feasible set is compact (Choo and Atkins, 1983; Steuer, 1986). As mentioned earlier, the only closedness results for the efficient set that exist in the literature concern certain special cases (see, e.g., Section 2.2). Thus the question as to what the necessary conditions are for the efficient set of a general MOMP problem to be closed remains open. The question as to whether or not we can find any sufficient conditions for the efficient set of a general MOMP problem to be closed also remains open. Motivated, in part, by answering these questions, we focus this chapter on finding necessary and sufficient conditions for the efficient decision set E(f, X) of problem MOMP to be closed. We will first introduce a new definition of strict quasiconcavity for a vector valued function. This definition extends the definition of strict quasiconcavity for a real valued function to vectorvalued functions. It is shown that a vectorvalued function is strictly quasiconcave on a compact convex set X if each of its component functions is strongly quasiconcave or if each of its component functions is linear and X is a polytope. We also show that the reverse of the above relations may not be true. Under our new definition, it is shown that the efficient decision set E(f, X) of problem MOMP is closed when the vectorvalued function f(x) is strictly quasiconcave and the feasible region X is compact and convex. Some other sufficient conditions are also obtained. By applying some of the new results to bicriteria mathematical programming problems, we obtain a new result on the closedness of E(f, X), which extends a result obtained by Schaible (1983). In Schaible (1983), it is shown that the efficient decision set for a bicriteria programming problem is closed when the two objective functions are continuous and strictly quasiconcave and the feasible region is compact and convex. Here, for the bicriteria programming problem, we extend Shaible's result to cases where the feasible region need not be convex and the objective functions need not be strictly quasiconcave. A necessary condition for E(f, X) to be closed will also be given in this chapter. This chapter is organized in the follow way. In Section 3.1, we introduce some notation and review the definitions of upper semicontinuous and lower semicontinuous pointtoset maps. In Section 3.2, we introduce and study new definitions for a vector valued function to be quasiconcave and strictly quasiconcave. Some new results on characterizing the efficient set of a pdimensional MOMP problem in terms of the optimal solution sets of some singlevalued optimization problems and in terms of the efficient sets of some (p1)dimensional MOMP problems are given in Section 3.3. In Section 3.4, we present some necessary and sufficient conditions for the efficient set of a general MOMP problem to be closed. The results on the closedness of a bicriteria programming problem are given in Section 3.5. Finally, some concluding remarks are given in Section 3.6. 3.1. Definitions and Notation Suppose that X c R" is a nonempty and compact set, and that fi, i = 1, 2, ..., p, are continuous functions on X. We define fi = min {fi(x) I x e X}, (3.1) f, = max {fi(x) I x X}, (3.2) f(i)(x) = (fi(x),..., fl.i(x), fi+(x),..., f,(X))T, (3.3) and i,)(X) = { fi(x) x e X}, (3.4) for all i = 1, 2,...,p. Notice that f(;)(x) is a (p1)dimensional vectorvalued function on X if p > 2, and f(i)(X) is a nonempty set in R'. Let i {1, 2, ...,p}. For any t e [ fi,f], and e e fi)(X), we define xi(t) = {x eX fi(x) t}, (3.5) and X()( ) = {x E X I f(i)(x)> }. (3.6) Let y = (y',...,ym)T, z = (z1,...,Zm)T be any two vectors in R" with m 2 1. We define min{ y, z,} min {y,z}= .... (3.7) min { Y, Zm}) Let F be a pointtoset map from a set U c Rk to subsets of a set V c Rm, and let U' c U. Then, the union of the sets F(u) for u e U' is denoted by u{F(u)l u e U'), and the intersection of the sets F(u) for u e U' is denoted by n{F(u)l u e U'}. Now, we review the concepts of upper semicontinuous and lower semicontinuous pointtoset maps. Definition 3.1.1. Let F be a pointtoset map from a set U c Rk to subsets of a set V c R". Then, F is said to be (a) upper semicontinuous at a point u* e U, if {uk c U, uk > u* (when k oc) and vk e F(uk) with vk v* (when k * oc) imply that v* e F(u*); (b) lower semicontinuous at a point u* e U, if {uk} c U, uk > u* (when k oc) and v* e F(u*) imply the existence of an integer M and a sequence of points {vk}c V such that vk e F(Uk) for all k 2 M and vk > v* when k > oc. 3.2. Quasiconcave and Strictly Quasiconcave VectorValued Functions The definitions of quasiconcavity, strict quasiconcavity, and strong quasiconcavity for a realvalued function can be stated as follows. Definition 3.2.1. Let S c R" be convex, and let g be a realvalued function defined on S. Then, g is said to be (a) quasiconcave on S, if g(Xx' + (1I)x2) > min {g(x'), g(x2)} for any x', 2 S, (0, 1); (b) strictly quasiconcave on S, if g(Xx' + (1X)x2) > min (g(x'), g(x2)} for any x', x2 e S, g(x') g(x2), and X e (0, 1); (c) strongly quasiconcave on S, if g(Xx' + (1X)x2) > min (g(x'), g(x2)} for any x, x2 e S, x' # x2, and X e (0, 1). We now review some properties that we will need concerning the quasiconcavity and strict quasiconcavity of realvalued functions. For an excellent review of various properties of quasiconcave and strictly quasiconcave functions, we refer the reader to the book by Avriel et al. (1988). The following property is well known. It can be proved directly from the definition. Property 3.2.1. Let g be a realvalued function defined on the convex set S c R". Then, g is quasiconcave on S if and only if the level set L(a) = {x e SI g(x) ac} is convex for any scalar a e R' such that L(a) # 0. The following properties will be useful in this chapter. Property 3.2.2 (Karamardian, 1967). Let g be a continuous realvalued function defined on the convex set S c R". If g is strictly quasiconcave on S, then it is also quasiconcave on S. Property 3.2.3 (Elkin, 1968, Martos, 1969). Let g be a continuous quasiconcave realvalued function defined on the convex set S c R". Then, g is strictly quasiconcave on S if and only if every local maximum of g in S is also a global maximum of g on S. Property 3.2.4 (Zang and Avriel, 1975). Let g be a realvalued function defined on the convex set S c R". Then, L(a) is a lower semicontinuous pointtoset map on G = (a L(a) 0)}ifand only if every local maximum ofg in S is also a global maximum on S. From Properties 3.2.23.2.4, we immediately have the following result concerning strictly quasiconcave functions. Property 3.2.5. Let g be a continuous realvalued function defined on the convex set S c_ R". Then, g is strictly quasiconcave if and only if g is quasiconcave on S and L(a) is a lower semicontinuous pointtoset map on G = {a I L(a) #0}. Properties 3.2.1 and 3.2.5 imply the following alternative definitions for quasiconcave and strictly quasiconcave functions. Definition 3.2.2. Let S c R" be convex, and let g be a continuous realvalued function defined on S. Then, g is said to be (a) quasiconcave on S, if the level set L(a) = (x e SI g(x) > a} is convex for any a e R1 such that L(a) # 0; (b) strictly quasiconcave on S, ifg is quasiconcave on S and L(a) = {x e S g(x) > a} is a lower semicontinuous pointtoset map on G = { I L(a) 0}). We now extend Definition 3.2.2 to vectorvalued functions. To aid in this extension, we define a vectorvalued function to be continuous on a set if each of its component functions is continuous on the set. Definition 3.2.3. Let f(x) = (fi(x),...,f(x))T be a pdimensional continuous vector valued function defined on the convex set X c R". Then, f is said to be (a) quasiconcave on X, if the level set M(y) = {x e XI f(x) > y) is convex for any y E RP such that M(y) # 0; (b) strictly quasiconcave on X, if f is quasiconcave on X and the pointtoset map M(y) = {x e X f(x) 2 y} is lower semicontinuous on G' = {y e RP  M(y) # 0}. We now present some properties of quasiconcave vectorvalued functions and strictly quasiconcave vectorvalued functions. Some of these properties will be used later in this chapter in the proofs of our results on closedness. In addition to this, these properties also show us the relationships among various classes of generalized convex functions and linear functions. Theorem 3.2.1. Let f(x) = (fi(x), ..., fp(x))' be a pdimensional continuous and quasiconcave vectorvalued function defined on the convex set X c R". Assume that {i1,...,ik) C {1, 2,...,p}, where 1 < k < p. Then, f'(x) = (f, (x),...,fi (x))'r is continuous and quasiconcave on X. Proof. We only need to prove the case where k = p1. Without loss of generality, we assume that {il,...,ik} = { 1,..., p}. Choose y' E RP' such that {x e XI f'(x) > y' } 0. Let y, z e {x e XI f'(x) > y' }. Since X is convex, we have Xy + (1X)z X for any 0 < X < 1. Since y, z e {x e XI f'(x) 2 y' }, we have f'(y) > y' and f'(z) 2 y'. It follows that min {f'(y), f'(z)) > y'. Let y = min {f(y), f(z)}. Then, y, z {x e XI f(x) 2 y }. Since f is quasiconcave, we have that {x E XI f(x) > y ) is convex. It follows that y + (1X)z e{x e X f(x)2 y } for any 0 < < 1. That is f(ky + (lX)z) y = min {f(y), f(z)}. This implies that f'(Xy + (IX)z) 2 min {f'(y), f'(z)} > y' for any 0 < X < 1. As a result, the set {x e X1 f'(x) 2 y' ) is convex. By Definition 3.2.3, f' is quasiconcave on X. O Theorem 3.2.2. Let f(x) = (fi(x), ..., f,(x))T be a pdimensional continuous vector valued function defined on the convex set X c R". Then, the vectorvalued function f is quasiconcave on X if and only if each of its component functions f1, i = 1, 2, ...,p, is quasiconcave on X. Proof If f is quasiconcave on X, then it follows from Theorem 3.2.1 that every component function of f is quasiconcave on X. Now, we assume that every component function fi is quasiconcave on X for i e (1,...,p). To prove fis quasiconcave on X, we need to prove that the level set M(y)) = {x e X f(x) > } is convex for any y in R' with M(y ) f 0. Let y be any given point in RP such that M(y *) a 0, and let x, y e M(y *). Then, f(x) > y and f(y) 2 y *. Since fi is quasiconcave on X for each i =1,...,p, we have that Sf, I(Ax + (1 2)y) f(2 Ax + (1 A)y)= ...... fP (A2x+(1A)y)) min {f, (x), f, (y)}) = min {f(x), f(y)} y * for any 0 < A< 1. This implies Ax + (1A)y e M(y*) for any 0 < A< 1. Therefore, M(y *) is a convex set. D Theorem 3.2.3. Let f(x) = (fi(x), ..., fp(x))' be a pdimensional, continuous strictly quasiconcave vectorvalued function defined on the compact, convex set X c R". Assume that {i1,...,ik} C {1, 2,...,p), where 1 < k < p. Then f'(x)= (f, (x),..., f" (x))T is strictly quasiconcave on X. Proof We only need to prove the case where k = p1. Without loss of generality, we assume that {il,...,ik) = {1,..., p }. Since f is continuous and strictly quasiconcave on X, we have that it is also quasiconcave on X. It follows from Theorem 3.2.1 that f'(x) = (f,(x), ..., f1 (x))T is quasiconcave on X. Therefore, to complete the proof, we need only to prove that the pointtoset map M'(y) = {x e XI f'(x) > } is lower semicontinuous on G' = {y e RP'I M'(y) 0 0}. Towards this end, let y e G', {y k} c G', and y k + y when k > oc, and let x* E M'(y *). By definition, to prove that M'(y) is lower semicontinuous at y *, we need to find an integer K and a sequence {xk}such that xk e M'(y k) for k > K and xk  x* when k Since f is continuous on the compact set X, fp(x) has minimum value on X. Let S* = min {fp(x) I x X}, and let t* =, Then, fp(x*) 2 f *. This, together with x* e M'(y *), yields f(x*) 2 t*. Therefore, x* e M(t*) = { x e XI f(x) 2 t*}. Let tk = Y #*) for k = 1, 2,.... We have tk > t* when k  oc. Since y k e G' (k = 1, 2,...), it follows that there exists yk e X such that f,(yk) k for all k = 1, 2,.... Since f = min {fp(x) I x E X}, we have fp(yk) > f *. Therefore, f(yk) S = tk for k = 1,2.... It follows that tk e G = {t e RP M(t) 0} for any k. As a result, since fis continuous on the compact set X, we can show that t* e G. Therefore, t* e G, and we have found a vector x* e M(t*) and a sequence of vectors {tk} c G such that tk  t* when k  oc. Since f is strictly quasiconcave on X, the pointtoset map M(t) = { x e X1 f(x) > t} is lower semicontinuous at t* e G = {t e RPI M(t) # 0}. Therefore, there exist an integer K and a sequence of vectors {xk) such that xk e M(tk) for k > K and xk x* when k + oc. That is, there exist an integer K and a sequence {xk) such that xk M'(yk) for k > K and xk + x* when k > oc. This implies that M'(y) is lower semicontinuous at y*. By the choice of y*, this implies that M'(y) is lower semicontinuous on G'. Consequently, the proof is complete. D Before going further, we need to prove a lemma. Consider the system Du = v * uTy*=0 (3.8) u>0, where D is a n x q matrix, v* e R" and y* e R' are two given vectors, and y* > 0. Let I = {il yi* # 0, i = 1,...,q}. Let ui = 0 for i e I, and let u' be the vector whose elements are ui, i S{ 1,...,q}\I. Then, the above system is equivalent to the linear system {D'u' = v * u'> 0, (3.9) where D' is a submatrix of D. If the system (3.8) is consistent, then, from linear programming theory, there is a nonsingular submatrix of D', denoted by B, and a subvector of v*, denoted by v', such that u* = Bv') is a solution of (3.8). Notice that Ilv'I < Iv*I. Therefore, lu* = IB'v'I < IB 0 'II IIv'll ,1B'Il IIv*ll, where IB''II = max {1B'vll I Ilvj = 1). Notice that since B is a nonsingular submatrix of D', it is also a nonsingular submatrix of D. Let M = max {(lB'iI I B is a nonsingular submatrix of D}. Then, Ilu*ll MIv*ll. Consequently, we have proven the following lemma, which will be used in the proof of Theorem 3.2.4 (c). Lemma 3.2.1. Let D be an n x q matrix, let v* = Rn and y* e Rq be two given vectors with y* > 0, and let M = max {JB'111 I B is a nonsingular submatrix of D}. If system (3.8) is consistent, then it has a solution u* such that Ilu*ll < M1\'*1I Theorem 3.2.4. Let f(x) = (fi(x), ..., fp(x))T be a pdimensional continuous vector valued function defined on the nonempty compact, convex set X c R". (a) If the vectorvalued function f is strictly quasiconcave on X, then the real valued function fi is strictly quasiconcave on X for each i = 1,..., p. (b) If the realvalued functions f,, i = 1,...,p are strongly quasiconcave on X, then the vectorvalued function f is strictly quasiconcave on X. (c) If the realvalued functions fi, i = 1,...,p are linear functions and X is a polytope, then the vectorvalued function f is strictly quasiconcave on X. Proof (a) The result follows immediately from Theorem 3.2.3, Definition 3.2.2, Definition 3.2.3, and Property 3.2.5. (b) Since f2, i = 1,...,p are strongly quasiconcave on X, they are also quasiconcave on X. It follows from Theorem 3.2.2 that the vectorvalued function f is quasiconcave on X. So, by the definition of strict quasiconcavity we only need to prove that the pointto set map y > M(y) = {x e XI f(x) 2 y } is lower semicontinuous at any given point y in RP such that M(y *) a 0. Towards this end, let y e RP satisfy M(y *) # 0 Let {y k} c G'= {y I M(y) # 0 }, yk ' when k > oc, and x* e M(y *). Since X is a compact set, M(y k) is a nonempty compact set for each k. Therefore, we may choose xk e M(y k) for each k such that Ixk x* =min ({lx x* I x EM(y k). Assume that y* is a cluster point of the sequence {xk} and y* # x*. Then, there is a subsequence of {xk} convergent to y*. Without loss of generality, we assume that xk > y* when k oc. Then, Ilxk x*il y* x* when k + oc. So, there is a positive integer K1 such that for all k 2 K1, III xk x*l ly* x*ll I < 0.5 y* x*II. Therefore, SXk x*1 >2 0.5 IIy* x*ll (3.10) when k > Ki. For each k, since xk E M(Y k), f(xk) > y k. By the continuity of f, this implies that f(y*) > y *. Since x* e M(y *), we also have that f(x*) y *. Since fi, i =1, 2, ..., p, are strongly quasiconcave and y* # x*, it follows that f(Ax* + (1A)y*) > min {f(x*), f(y*)}> y for any A such that 0 < 2< 1. By choosing A = 0.6, we obtain that f(0.6x* + 0.4y*) > y *. Since 7k y when k > oc, there exists a positive integer K2 such that f(0.6x* + 0.4y*) > y k when k > K2. That implies that 0.6x* + 0.4y* e M(yk) when k > K2. Therefore, ii (0.6x* + 0.4y* ) x* II min {Ix x*ll I x e M( k)} = Ixk x*ll for all k > K2. This implies that 0.4 Iy* x*ll > Ixk x*l (3.11) for all k > K2. From (3.10) and (3.11), it follows that 0.4 Ily* x*l > 11 xk x* > 0.5 liy* x*ll for all k > max {K1, K2). This is a contradiction. Therefore y* = x* must hold. By the choice of y*, we have proven that xk  x* when k + oc. Thus, M(y) is lower semicontinuous at y *. (c) Since f is a linear vectorvalued function and X is a polytope, we may assume that f(x) = Cx and X = (x e R"I Ax < b, x > 0}, where C is a p x n matrix, A is a m x n matrix and b e Rm. Similarly to (b), we only need to prove the pointtoset map y + M(y) = {x e XI Cx > y } is lower semicontinuous at any given point y in RP with M(y *) 0. Suppose that y e RP and M(y *) 0. Suppose that {y k} c G' = {y I M(y) 0), yk + y when k > c, and x* E M( *). Since X is a compact set, M(yk) is a nonempty compact set. Therefore, we may choose xk e M(y k) for each k such that xk is an optimal solution of the problem (Q) given by min (1/2)Ix x*112 s.t. Cx > yk, Ax < b, x 0. Problem (Q) is a convex quadratic programming problem. By using the Karush KuhnTucker conditions, we see that xk is an optimal solution if and only if xk is a feasible solution and there exists a vector u e Rp+" such that xk x* + [CT, A', I]u = 0, (3.12) Cxk yk cXk k (u)T bAxk =0, (3.13) xk u 0, (3.14) where I is the n x n identity matrix. In Lemma 3.2.1, let D = [CT, AT, I], v* = x* xk, y* = [(Cxk 7k)T, (b Ak)T, (xk)T]T, and M = max {IB'1I I B is a nonsingular submatrix of D). By Lemma 3.2.1, for each k = 1, 2,..., there exists a vector uk that satisfies (3.12) (3.14) such that Ilukll < M Ixk x*ll. Since (Xk} c X, and X is compact, {xk) has convergent subsequence. Without loss of generality, we may assume that (xk} is a subsequence convergent to a vector z*. Since xk e M(y k) for each k, z* e M(y *). Notice that M is a fixed number and Ilukil < MIIxk  x*ll for each k. Therefore, {uk) is bounded. This implies that {uk} has at least one cluster point. Let u* be a cluster point of {uk). Then, from (3.12)(3.14), letting k + ac, we obtain z + [CT, A, I]u* = 0, ICz*y* (u*)T bAz* = 0, z* u* > O. This implies that z* is an optimal solution of the problem min (1/2)ixx x*1 s.t. x e M(y *). Since x* e M(y *), the above problem has the unique optimal solution x*. Therefore, z* = x*. Summarizing, we have found a sequence {xk) such that xk e M(y k) for each k and xk > x*. Thus, M(y) is lower semicontinuous at y *. By the choice of y *, the proof is complete. D Remark 3.2.1. It is well known that a linear realvalued function need not be strongly quasiconcave and a strongly quasiconcave realvalued function need not be linear. Therefore, the converses of Theorem 3.2.4 (b) and Theorem 3.2.4 (c) are not true. Let f= (fi,...,f) be a pdimensional continuous vectorvalued function defined on the compact, convex set X. Then, from the above results, we have the following relationships shown in Figure 3.1. Figure 3.1 3.3. Parametric Representations of the Efficient Set In this section, we will present some new results that characterize the efficient solutions of a pdimensional MOMP problem. These characterizations are in terms of the optimal solutions of appropriate parametric scalar optimization problems or in terms of efficient solutions of appropriate (p1)dimensional parametric MOMP problems. First, we present a new result that characterizes the efficient solutions of a MOMP problem in term of the optimal solutions of appropriate parametric scalar optimization problems. As we mentioned in Section 2.1, one of the commonly used scalar optimization problems is the "ithobjective, econstraint problem", which can be defined as Pi(e): max fi(x) s.t. x e X()( 6), where i e { 1, 2,...,p}and e e Rp'. Let us define the set of all optimal solutions of problem Pi(s) by X>(iI). From Theorem 4.1 of Chankong and Haimes (1983), we know that x* e E(f, X) if and only if x* is an optimal solution of Pi(s*) for every i = 1, ...,p, where 6* = f(i)(x*). The following theorem extends this result. Theorem 3.3.1. Let X c R" be nonempty, and let f = (f,...,fp)T be a pdimensional vectorvalued function defined on X. Then, x* e E(f, X) if and only if for every i = 1, ..., p, there is E' e f(i)(X) such that x* is an optimal solution of Pi(e'), i.e. p E(f, X)= niU ,{ o) sf(,)(x)}. i=l Proof. By Theorem 4.1 of Chankong and Haimes (1983), P E(f, X)g iU nte) i e c f.)(x)) i=1 P Now, we prove the reverse inclusion. Let x* e flUj (i>() 16o f,(X)}. i=1 Suppose to the contrary that x* o E(f, X). Then, there is x e X such that f(x) > f(x*), and f(x) : f(x*). (3.15) Without loss of generality, suppose that fi(x) > fi(x*). Now, we claim that x* e U (l) () fE(,)(X))}. Otherwise, suppose x* e Xo(*() ( for some e *e f() (X). Since x* e X1) (e*),we have f(l)(x*) *. (3.16) By (3.15) and (3.16), f(i)(x) > f(i)(x*) > E*. This together with x e X yields x e X(1)(e*). By our assumption that fi(x) > fi(x*), we have x* o Xj,(e*). This contradicts our assumption that x* e X() (E*). Thus, x* U { X(1 (e) lee f(,)(X)}. This contradicts P x* e n U{ f lc f. (x)}. Therefore, U ,)(e) 1e f(,)(X)}c E(f, X). D i=1 Next we present a new result that characterizes the efficient solutions of an MOMP problem with p objective functions in terms of efficient solutions of some particular MOMP problems with p1 objective functions. Sun (1996) introduced the following (p1)dimensional parametric MOMP problem P(i)(t): vmax f(i)(x) s.t. x e Xi(t), where i { 1, 2,...,p), t e R1. The efficient decision set and the efficient outcome set of problem P(i)(t) are denoted by E(f(i), Xi(t)) and E(f(i)( Xi(t)), RP~"), respectively. When p = 2, problem P(i)(t) is a scalar optimization problem. For convenience, we still use E(f(i), Xi(t)) and E(f(i)( Xi(t)), RP') to present the optimal solution set and optimal value set, respectively, when p = 2. Sun (1996) proved the following result. Theorem 3.3.2. Let X c R" be a nonempty compact convex set, and let f = (fi,...,f)T be a pdimensional continuous and quasiconcave vectorvalued function defined on X. If there isj e { 1,...,p} such that fj is strongly quasiconcave on X, then x* e E(f, X) if and only if, for any given i e (l,...,p}\ {j}, there is t E [fi,f,] such that x* is an efficient solution of P()(t), i.e. E(f, X)= { E(fi), Xi(t))l t [f,,f,]) for any given i { 1,...,p)\ {j}. In the following, we present a new result that characterizes an efficient solution for a general MOMP problem. Theorem 3.3.3. Let X c R" be a nonempty compact set, and let f= (fi,...,P)T be a pdimensional continuous vectorvalued function defined on the compact set X. Then, x* e E(f, X) if and only if for every i = 1, ..., p there is ti e [fi,f,] such that x* is an efficient solution ofP(i)(ti), i.e. P E(f, X)= U{E(f ,), X,(t)) t [f ,,f ]}. 1=1 Proof We first prove that P E(f, X) c U{E(f(O), X, (t)) t [fi,f 1)}. (3.17) 1=I Let x* e E(f, X). Suppose to the contrary that P x* nu{E(f(o, Xi(t))I t [fi, ]}. Then, there is i e {1,...,p} such that x* e U{E(fo),Xi(t)) t e [fi,f ]}. That is, for any t [f, f ], x* e E(f(i), Xi(t)). Take t* = fi(x*). Since x* t E(fv), Xi(t*)), there is x e Xi(t*) such that f(i)(x) > f(o)(x*), f<1)(x) # f()(x*), and fi(x) > t* = fi(x*). Thus, we have found a x e X such that f(x) > f(x*) and f(x) f(x*). This contradicts x* e E(f, X). Thus, (3.17) holds. Now, we prove that P nU{(E(f(,,, Xi(t)) t e [f,,f, ]} c E(f, X). (3.18) Let x* E nU.{E(f,), Xi(t))I t c [.f, ]}. Suppose to the contrary that x* z E(f, i=1 X). Then there exists a vector x e X such that f(x) > f(x*) and f(x) # f(x*). (3.19) This implies that there exists an i 6 (1, 2,...,p} such that f(i)(x) 2 f(i)(x*), and f(o)(x) f(i)(x*). Without loss of generality, suppose that f(l)(x) > f(l)(x*), and f(i)(x) : f()(x*). We claim that x* 0 U(E(fo), X,(t)) tE[f,,f,]}. (3.20) To verify (3.20), suppose that x* e E(f(l), Xi(t*)) for some t* e [f,,f,]}. Since x* e XI(t*), fi(x*) > t*. (3.21) By (3.19) and (3.21), fi(x) > fi(x*) > t*. This together with x e X yields x e XI(t*). Since f()(x) > f(l)(x*), and f()(x) # f(o)(x*), this implies that x* 0 E(f(), XI(t*)). This contradicts our assumption that x* E E(fi), Xi(t*)). Thus (3.20) must hold. The fact that (3.20) is true contradicts the assumption that x* e Uj {E(f(,, X,(t))1 t e [ f, f ]}. Therefore, (3.18) is true, and the proof is complete. I i=1 Next, we present a new result that characterizes the efficient solutions of a p dimensional MOMP problem in terms of the optimal solutions of problem Pi(e), where e is an efficient outcome of the (p1 )dimensional MOMP problem P(1)(t) for some t e [f,, fi ]. Theorem 3.3.4. Let X c R" be a nonempty compact set, and let f = (fi,...,fp,) be a pdimensional continuous vectorvalued function defined on X. Assume that i { 1,..., p}. Then, x* e E(f, X) if and only if there is an efficient outcome e of problem P(1)(t) for some t e [f,,, ] such that x* is an optimal solution of problem Pi(6 ), i.e. E(f, X)=u { X(() (e) e e F} (3.22) where F = u { E(f(,)( Xi(t)), R ') It e [fit f ]}. Proof. We first prove that E(f, X) c u { X(,() e F}. (3.23) Suppose that x* E E(f, X). Set t = fi(x*) and e = f(i(x*). Then x* e X)(e) and Sf(i)(Xi(t)). Suppose that e s E(f)( Xi(t)), RP'1). Then there exists x e Xi(t) such that 52 f(i)(x) > e = f(i)(x*) and f()(x) ; f(i)(x*). (3.24) Since x e Xi(t), it follows that fi(x) > t = fi(x*). (3.25) From (3.24) and (3.25), we obtain that f(x) > f(x*) and f(x) : f(x*). This contradicts that x* e E(f, X). Therefore e e E(f(i)( Xi(t)), RP~ ) c F. Now suppose that x* X^) ,(). Then there exists a vector x' e X) (e) such that fi(x') > fi(x*). (3.26) Since x' e X),,(), it follows that f(<)(x') 2> = fi)(x*). (3.27) By (3.26) and (3.27), we have that f(x') 2 f(x*) and f(x') # f(x*). This contradicts that x* e E(f, X). Hence, x* e X ,) (). Since 6 F and x* e X5()(), (3.23) holds. We now demonstrate the inclusion opposite to (3.23). Let t e [f 1], E E(f()(Xi(t)), R6'), and x* e Xi)(e). Since E E f such that f(i)(x) = e. Thus, x e XO)(e). This together with x* XO)(e)and x e Xi(t) yields that fi(x*) 2 fi(x) > t. (3.28) Suppose that x* v E(f, X). Then there exists a vector x' E X such that f(x') > f(x*) and f(x') ; f(x*). (3.29) By (3.28) and (3.29), since x* e X,)(E), we have that fi(x') > fi(x*) > t, f()(x') 2 f()(x*) > E (3.30) Since e e E(fi)(Xi(t)), RP'), it follows from (3.30) that f(i)(x') = f)(x*)= E. (3.31) Thus, by (3.29), fi(x') > fi(x*). From (3.31), since x' e X, this contradicts that x* e X() (). Therefore, x* e E(f, X), and we have proved the opposite inclusion to (3.23). D Theorem 3.3.5. Let X c R" be a nonempty compact set, and let f = (f,.. .,f)T be a pdimensional continuous vectorvalued function defined on X. Assume that i { 1,..., p}. If there exist a t* E [f,, f ] and an e* e f(j)(X) such that x* e X(,) (*)and c* e E(f(i)(Xi(t*)), RP '), then fi)(x*) = * Proof. Since e* e E(f(i)(Xi(t*)), RP A), there exists an x e Xi(t*) such that f()(x) = 6*. Therefore, x e X(,)(e*). Since x* e X(,) (*)and x e Xi(t*), this implies that fi(x*) > f(x) 2 t*. Therefore, x* Xi(t*). Since x* e X, this implies that f(i)(x*) e f,)(Xi(t*)). Suppose that f(i)(x*) ;e *. It follows from x* E X(1) (*)that f(i)(x*) > * Thus, we have found a vector f(i)(x*) e f(i)(Xi(t*)) such that fi)(x*) e and f(i)(x*) > e * This contradicts that e E(f(i)(Xi(t*)), R'). Consequently, ii)(x*) = s *. D 3.4. The Closedness of E(f, X) for General MOMP Problems We now are in a position to focus on our main topic in this chapter: Conditions for the closedness of the efficient solution set E(f,X) of problem MOMP. We first present a necessary condition for E(f, X) to be closed. Theorem 3.4.1. Let X c R" be a nonempty compact set, and let f = (fl,...,f,)' be a pdimensional continuous vectorvalued function defined on X. If E(f,X) is closed, then for any ie { 1,...,p}, the pointtoset map s + X(, )() is upper semicontinuous on F = v {E(f(i)(Xi(t)), RP') It [f,,f, ]}. Proof Let {( k} c F, e e F, E k  xk E X ()(ek), and Xk+ x. To complete the proof, we need to show that x e X(,)(E). From Theorem 3.3.4, for each k, we have that xk e E(f, X). Since E(f, X) is closed and xk x, this implies that x e E(f, X). Furthermore, from Theorem 3.3.5, for each k we have that f(i)(xk) = k. Since f is continuous, this implies that f(i)(x) = e. Therefore, x e X,)(E). Suppose, to the contrary, that x a X^()(s). Then, there exists x' e X that fi(x') > fi(x). Since x' e X(j)(e) and f()(x) = e, fi)(x') f(i)(x). Because fi(x') > f(x), this implies that f(x') 2 f(x) and f,(x') > fi(x). This contradicts xe E(f,X), so that the proof is complete. O By adding an additional condition to the necessary condition in Theorem 3.4.1, we obtain the sufficient conditions given in the next result for E(f, X) to be closed. Theorem 3.4.2. Let X c R" be a nonempty compact set, and let f = (fl,... ,f)T be a pdimensional continuous vectorvalued function defined on X. If there is an i {1,...,p}such that the pointtoset map e + Xj,)() is upper semicontinuous on F = u (E(f(i)( X(t)), R ')j t [f ,,f]}, and such that the pointtoset map t + E(f(i(Xi(t)), RP') is upper semicontinuous on [fi, f ], then E(f, X) is closed. Proof. Let {x") be a sequence of vectors in E(f, X) such that x"  x* as n oc. We need to show that x* e E(f, X). For each n, since x" e E(f, X), it follows from Theorem 3.3.4 that if we choose i as stated in the theorem, then there exist t" e [f,, ,] and e" e E(f(i)(Xi(t")), RP') such that x" e X0)(e"). Since {tn} and { e n) are bounded, it follows that they have convergent subsequences. Without loss of generality, assume that {t") and { e } are two convergent subsequences with t" > t* and e" > e *. Since the pointtoset map t > E(f(i)(Xi(t)), R"') is upper semicontinuous on [fi, f ], it follows that S* e E(f()(Xi(t*)), RP ') c F. (3.32) Since the pointtoset map e X^(,)(e) is upper semicontinuous on F, it also follows that x* E X((,)(e*) (3.33) By (3.32), (3.33) and Theorem 3.3.4, x* E E(f, X). D One may notice that, in practice, it is generally quite difficult to verify the conditions in Theorem 3.4.2 for a given MOMP problem. This may suggest that Theorem 3.4.2 be of little practical use. However, as we will see in the next section, Theorem 3.4.2 can be used to develop some other more practical sufficient conditions. In the following, we will present some sufficient conditions for E(f, X) to be closed under some generalized convexity assumptions on f Theorem 3.4.3. Let X c R" be a nonempty, compact convex set, and let f = (f,...,fp)T be a pdimensional, continuous, vectorvalued function defined on X. If the vectorvalued function f is strictly quasiconcave on X, then E(f, X) is closed. Proof By Theorem 3.3.1, we have that E(f, X)= Ufile>) lEf()o(X)). i=1 To prove that E(f, X) is closed, we need only to prove that U({X(i)(c) seff((X)}is closed for every i = 1,...,p. Let i e {1, 2,...,p), {xk} c U(j{X(i)()1 ef(,)(X)} and xk + x* when k oc. Then, there exist 8k E f(i,(X) such that xk Xk ( ) for k = 1, 2,.... Since { ek} f(i)(X), and f(,(X) is bounded, { ek } has at least one convergent subsequence. Without loss of generality, assume that { k} is an arbitrary convergent subsequence, and that sk > *. Since f(,(X) is closed and { sk } f(,)(X), E e f(i)(X). For each k, since xk e X(,)(.k), we know that xk E X()(ck), i.e. f(,)(xk)> _k. By continuity of f(i), this implies that f(i)(x*) > c *, i.e. x* e X()(6*). Suppose that x* e X0((e*). Then there exists y* e X(,(e*) such that f,(y*) > f,(x*). (3.34) Since fis strictly quasiconcave on X, it follows from Theorem 3.2.3 that f(; is also strictly quasiconcave on X. By Definition 3.2.3, this implies that X(,(e) is a lower semicontinuous map on { e X, )() # 0). Therefore, since y* e X()(e*), X(,(e) is lower semicontinuous at *. Hence, there exists {ykk such that yk e X()(ek) for k large enough, and such that yk > y* when k  oo. (3.35) It follows from (3.34), (3.35) and xk > x* that fi(yk) > f,(k) when k large enough. This contradicts that xk e X (, (k)for all k = 1, 2,... Therefore, x* e X(i(e*). Consequently, E(f, X) is closed. O Theorem 3.4.4. Let X c R" be a nonempty compact convex set, and let f = (f,,...,f)T be a pdimensional, continuous, vectorvalued function defined on X. If each function fi, i = 1,...,p, is strongly quasiconcave on X, then E(f, X) is closed. Proof This result follows immediately from Theorem 3.2.4 (b) and Theorem 3.4.3. D A different result related to Theorem 3.4.4 is given in Theorem 4.1.11(i) of Luc (1989). Theorem 3.4.5. Let X c R" be a nonempty compact convex set, and let f = (fi,...,fp)T be a pdimensional continuous quasiconcave vectorvalued function defined on X. If there exists at least one i{ 1, ..., p} such that fi is strongly quasiconcave on X and the optimal solution set map X,)(e) of problem Pi(6) is upper semicontinuous on f()(X), then E(f, X) is closed. Proof Let (xk}c E(f, X) and xk  x*. Choose i as in the statement of the theorem. Since f is continuous and quasiconcave on X, and fi is strongly quasiconcave on 58 X, Theorem 3.1 of Sun (1996) implies that for each k there exists a vector 8k e f(,)(X) such that xk E X1)(Ek). Since { e }c f()(X), and f(,)(X) is bounded, { Ek } has at least one convergent subsequence. Without loss of generality, assume that { k is a convergent subsequence and 6k ) as k co. Then e* e f(,(X), since f(j(X) is closed. Since the pointtoset map e > X5()(E) is upper semicontinuous on f(i)(X), we have that x* e X()(e*). Since f is continuous and quasiconcave on X, and f, is strongly quasiconcave on X, from Theorem 3.1 of Sun (1996), x* e E(f,X). Consequently, E(f,X) is closed. D Theorem 3.4.6. Let X c R" be a nonempty compact convex set, and let f = (fi,...,fp)T be a pdimensional continuous quasiconcave vectorvalued function defined on X. If there exists at least one is { l,...,p} such that fi is strongly quasiconcave on X and the pointtoset map e  X,)(e) is lower semicontinuous on f(i)(X), then E(f, X) is closed. Proof By Theorem 3.4.5, we only need to prove that the pointtoset map from e to the optimal solution set X(l)(c) of problem Pi(E) is upper semicontinuous on fi)(X). Let { k } fi)(X), 6 f(i)(X), 6k ) *, k e X()(ek), xk ) x*. Since, for each k, xk e X()(6k), we have xk e X,)(e6k), i.e. f()(xk)2 6ek By continuity of f on X, this implies that f(,)(x*) 2 6*, i.e. x* E X(,(e*). Suppose that x* X^() (6*). Then there exists a point y* e X,,(6*) such that f (y*) > f (x*). (3.36) Since X(,)() is lower semicontinuous at 6 *, there exists a sequence {yk} such that for k large enough, yk e X(, (Ek), and such that yk + y* as k + oo. (3.37) It follows from (3.36), (3.37) and xk _ x* that there exists an integer k of sufficient magnitude such that f(yk) > f,(xk). Since yk e XO)(ek) for k sufficiently large, this contradicts that xk E I,(ek) for all k = 1, 2,.... Therefore, x* e X^((e*). Hence, the pointtoset map e > XR()(E) is upper semicontinuous on f(i)(X). ] 3.5. The Closedness of E(f, X) for Bicriteria Programming Problems In this section we consider the closedness of E(f, X) for the special case when p=2. Theorem 3.5.1. Let X c R" be a nonempty compact set, and let f = (fi, f2)T be a twodimensional continuous vectorvalued function defined on X. If every local maximum solution offi in X is also a global maximum solution offi on X for i = 1, 2, then E(f, X) is closed. Proof Since every local maximum solution of f; (i = 1, 2) in X is also a global maximum solution off, on X, it follows from Theorem 3.3 of Zang and Avriel (1975) that the pointtoset map t > Xi(t) is lower semicontinuous on [f, f, ], and the pointtoset map 6 + X2( ) is lower semicontinuous on [f, f2 ]. To establish the theorem, we will use these two results and Theorem 3.4.2. Thus, it is sufficient to show that the pointtoset map e . X())() is upper semicontinuous on [f2 f, 2], and that the pointtoset map t + E(f2(XI(t)), R ) is upper semicontinuous on [fI, f, ]. We first prove that the pointtoset map t + E(f2(XI(t)), R') is upper semicontinuous on [ f,, ]. Suppose that t* [f,, f, ], that (tk} c I[f,, satisfies tk t* as k oo, that yk e E(f2(Xi(tk)), R ) for each k, and that yk y* as k oo. Since yk e f2(Xi(tk)), it follows that there exists xk e Xi(tk) such that yk= f2(xk) and f(xk) > tk. (3.38) Since {xk} is bounded, it follows that it has at least one convergent subsequence. Without loss of generality, assume that {xk} is a convergent subsequence and that xk + x* e X for some x*. Then, by continuity off, y* = f2(x*) and fi(x*) 2 t*. (3.39) Hence, y* e f2(Xl(t*)). Suppose that y* o E(f2(Xi(t*)), R ). Then there exists a vector x' e XI(t*) such that f2(x') > y* = f2(x*). (3.40) It follows from the lower semicontinuity of the pointtoset map t  XI(t) at t* e [ f, Tf ] that there exist {zk} c X such that zk eXi(tk) for k large enough, and zk when k oo. (3.41) By (3.38), (3.40), (3.41) and xk  x*, we have that f2(zk) > f2(xk) = yk (3.42) when k large enough. Notice further that zk e Xi(tk) for each k large enough. This together with (3.42) contradicts that yk e E(f2(Xl(tk)), R ) for each k. Therefore, y* e E(f2(XI(t*)), R ). Hence, the pointtoset map t  E(f2(XI(t)), R') is upper semicontinuous on [f,, f, ]. Similarly, we can also prove that the pointtoset map e + X5(^0(e) is upper semicontinuous on [f2 f2 ]. Consequently, we have that E(f, X) is closed. O Remark 3.5.1. In Schaible (1983), Schaible proved that the efficient solution set E(f, X) is closed if X is compact and convex and fi is continuous and strictly quasiconcave on X for each i = 1, 2. However, we do not require any convexity assumptions on X in Theorem 3.4.6. Therefore, Theorem 3.5.1 extends the result of Schaible (1983) to cases where f need not possess any type of generalized concavity or convexity properties and X need not be convex. Remark 3.5.2. It is well known that every local maximum solution of a continuous and strictly quasiconcave function on a convex compact set is a global maximum solution. However, we can easily find examples to show that a continuous function may not be strictly quasiconcave even though each of its local maximum solutions is also a global maximum solution. Therefore, Theorem 3.5.1 extends the result of Schaible (1983) to the cases where f; may not be strictly quasiconcave for any i = 1, 2. 3.6. Concluding Remarks From the literature, it is known that the efficient decision set E(f, X) of problem MOMP is generally not closed when p > 3, even if every component of f is strictly quasiconcave on X and X is a compact, convex set (see, e.g., Choo and Atkins (1983) and Steuer (1986)). It is also well known that the efficient decision set of a multiple objective linear programming problem is closed. This leaves open the question of whether or not, when X is nonempty, compact and convex, there exist classes of vectorvalued, nonlinear functions f whose components are not all strictly quasiconcave on X for which it is guaranteed that E(f, X) is closed. In this chapter, we have answered the question posed above, and several other questions concerning the closedness of E(f, X). One of the main tools that we used to accomplish this was to introduce the new notions of quasiconcavity and strict quasiconcavity for vectorvalued functions. These notions are direct extensions of the definitions of quasiconcavity and strict quasiconcavity for realvalued functions. Using these two new definitions and some other results, we showed that E(f, X) is closed when f is a continuous, strictly quasiconcave vectorvalued function over the nonempty, compact convex set X. We also showed that if each component of f is continuous and strongly quasiconcave on the nonempty, compact convex set X, then E(f, X) is closed. We went on to show several other necessary, sufficient, and necessary and sufficient conditions for E(f, X) to be closed. For instance, in the bicriteria case, we extended a result of Schaible (1983). Our new result gives a sufficient condition for E(f, X) to be closed in cases where f is continuous and X is nonempty and compact. CHAPTER 4 FINDING THE SET OF ALL EFFICIENT EXTREME POINTS FOR PROBLEM MOLP IN THE OUTCOME SPACE A multiple objective linear programming problem (MOLP) can be written as follows: MOLP: vmax Cx s.t. Ax < b x >0, where C e RPX", A e Rm"n, b e Rm. Then, the decision set X for problem MOLP is X = {x e R" Ax and the outcome set Y for problem MOLP is Y= {Cx x e X). Throughout this chapter we will assume that X is a compact set. It follows that Y is also a compact set. In order to make the chapter more selfcontained, we will restate some concepts for problem MOLP. A point x0 E R" is called an efficient solution for problem (MOLP) when x0 e X and there exists no point x e X such that Cx > Cx0 and Cx # Cxo. Similarly, a point yO e RP is called an efficient outcome for problem (MOLP) when yO e Y and there exists no point y e Y such that y 2 yo and y # yo. The set of all efficient solutions and the set of all efficient outcomes for (MOLP) are called the efficient decision set and the efficient outcome set, respectively, for problem MOLP, and are denoted by XE and YE, respectively. The set of all extreme points of X and the set of all extreme points of Y are denoted by Xex and Yex, respectively. We will also call a point in XE r) Xx an efficient extreme point in decision space, and a point in YE n Yex an efficient extreme point in outcome space. During approximately the past thirty years, various algorithms have been developed for finding the set XE n Xex, or the entire set XE. Although these algorithms are quite different in certain ways, they all utilize some modified version of the simplex method. One general difficulty with these methods is that, because of their complexity and the complexity of XE and XE r X.x, they often encounter CPU time and computer storage limitations. Most of the computer time is consumed by efficiency tests that evaluate the nonbasic variables at each basis (see, p. 245 of Steuer (1986)). The computer storage limitations arise due to the sheer size of XE and XE n Xex (Benson, 1998a). Another general difficulty concerns the handling of degeneracy (see Section 2.3). Apart from these difficulties, there are some other problems related to these methods. For example, these methods frequently generate a set so large that it overwhelms the DM (see e.g. Benson, 1998a). Because of these difficulties, these methods have achieved only limited success in practice. Recently, some researchers have begun to turn their attention to investigating tools and methods for generating all or part of the efficient outcome set YE, (see e.g. Benson, 1995a, 1997, 1998a, 1998b, and 1998c; Benson and Sayin, 1997; Dauer, 1987, 1993; Dauer and Liu, 1990; Dauer and Saleh, 1990; Dauer and Gallagher, 1996). This is in part because the dimension of the outcome space is usually much smaller than the dimension of the decision space. Furthermore, the efficient outcome set generally has a much simpler structure than the efficient decision set. Thus, generating all or part of the efficient set in the outcome space is expected to be more practical than doing so in the decision space. In this chapter, we will present a weight set decomposition algorithm for generating the set YE n Yex. The approach of decomposing the weight set was originally developed by Gal and Nedoma to deal with multiparametric linear programming problems (see Gal and Nedoma, 1972). Later, this approach was adapted by Zeleny for use in attempting to generate the set XE r Xex (see Zeleny, 1974). The weight set decomposition approach involves "decomposing" the weight set W = { w = (w, ... wp)1 I wj > 0 for all j = 1,...,p } into a finite number of subsets. In Zeleny (1974), the weight set W0 is decomposed into a finite number of subsets associated with the different efficient bases in the decision space of problem MOLP (the definition of an efficient basis will be given later in Section 4.1). We call this decomposition the decision setbased decomposition of the weight set W. In this chapter, we will decompose the weight set W into a finite number of subsets associated with the different efficient extreme points in the outcome space of problem MOLP, rather than the different efficient bases in the decision space of problem MOLP. We call this decomposition the outcome setbased decomposition of the weight set W. Unlike the decision setbased decomposition, our outcome setbased decomposition will establish a onetoone correspondence between the efficient extreme points in the outcome set and subsets of W. Based upon this decomposition, we will then present a new algorithm, called Basic Weight Set Decomposition Algorithm, (BWSDA), for generating the set YE r Yex. The algorithm BWSDA works in the following way: At each iteration, the algorithm will first either find a weight vector which will lead to an unexplored efficient extreme point in the outcome space or conclude that all points in YE r Yex have been generated. If a new weight vector is found at some iteration, the algorithm will in the next iteration call for solving at most (p+l) linear programs in order to find an unexplored efficient extreme point in the outcome space. Two different kinds of approaches will be developed for finding, if it exists, a new weight vector in each iteration that leads to an unexplored extreme point in the outcome space. One approach uses a tree search method. The other calls for solving a special concave minimization problem over a polyhedron. These two different approaches yield two versions of the Basic Weight Set Decomposition Algorithm, Weight Set Decomposition AlgorithmI (WSDAI) and Weight Set Decomposition AlgorithmII (WSDAII). This chapter is organized in the following way. In Section 4.1, we will review the decision setbased decomposition of W developed by Zeleny (1974). Our outcome set based decomposition of W0 will be given in Section 4.2. The new algorithms for generating the set YE r Yex will be given in Section 4.3. In Section 4.4, we will compare the new algorithms with Zeleny's decision setbased algorithm (see Zeleny, 1974) and with Benson's outcome setbased algorithms (see Benson, 1998a, c). Section 4.5 gives some conclusions. 4.1. Decision SetBased Decomposition of the Weight Set W As stated in Section 2.1, one common strategy for dealing with problem MOMP is to characterize efficient solutions of problem MOMP in terms of optimal solutions of the weighting problem P(w). For problem MOLP, we use LP(w) to denote the corresponding weighting problem P(w), i.e. LP(w) is given by LP(w): max wTCx s.t. Ax< b x>0 in the case of problem MOLP. From Theorem 2.6 in Yu and Zeleny (1975), a vector x0 e R" is an efficient solution of problem MOLP if and only if there exists a vector w e W such that xo is an optimal solution for problem LP(wo). We can therefore, in theory, find the set of all efficient solutions for problem MOLP by solving problem LP(w) for a properly chosen set of weights in W. In order to introduce the concept of an efficient basis for problem MOLP and to show relationships between efficient bases for problem MOLP and subsets of W, we first transform LP(w) to its standard form. By adding slack variables, LP(w) is equivalent to the following problem: LP(w)': max wT x s.t. A b, ), , where s eRm, C= (C, Opx.), and A=(A, I,.,). Let B = Bmxm be a nonsingular submatrix of A, and let B' denote the remaining submatrix. B is called a basis of problem LP(w)'. If the basic solution determined by B is also an optimal solution, then B is called an optimal basis of problem LP(w)'. If there is a w in WO such that B is an optimal basis of problem LP(w)', then it is called an efficient basis of MOLP. Let C, be the submatrix of C related to B, and let CB, the remaining submatrix. The simplex tableau T of LP(w)' related to the basis B is of the form =I B'B' B'b) 0 wTZ w'z' where Z = C, B'B' C8. and z = C, B'b. Let x(B) denote the basic solution determined by B. If Bb > 0, then x(B) is a basic feasible solution. In this case, the point determined by x(B) is an extreme point of X. If additionally wTZ > 0, then the basic feasible solution is an optimal solution for problem LP(w)'. If we further have w > 0, then the basic feasible solution is an efficient extreme point of problem MOLP in the decision space and the basis B is an efficient basis. Let W = { w = (wi, ..., w)T I wj > 0 for all j = 1,...,p }. For any given efficient basis B for problem LP(w)', consider the polyhedral cone W(B) defined in Zeleny (1974) by W(B) = {w wTZ >0}. From Theorem 3.6 in Yu and Zeleny (1975), there exists a finite number of efficient bases, Bi,..., Bq, such that W Wc UW(B,). (4.1) 1=] As a result, W can be decomposed into a finite number of subsets W r W(Bi), i=l,...,q such that for each x' e XE r Xe there exists at least one W(Bi), i { ,...,q}, such that x' = x(Bi) and x' is an optimal solution of problem LP(w)' for any w in W(Bi) We will call the decomposition of WO which is based upon (4.1) the decision set based decomposition of W. The weight set W can also be decomposed into subsets associated with different efficient extreme points in the decision space. To see this, for a given point x in X, let W(x) be defined by W(x)= {weRPI WrCx > wrCx' for all x' in X}. (4.2) It can be shown that We UW(x). (4.3) xeX r'XE A similar result can be found on p.157 of Chankong and Haimes (1983). As a result of (4.3), the weight set W can be decomposed into a union of subsets WO n W(x), x e XE n Xex. Zeleny (1974) discussed the algorithmic possibilities and strategies for locating XE n Xex by using decision setbased decompositions of W based upon (4.1) or (4.3). However, as noted by Zeleny (1973, 1974), there are some difficulties implementing algorithms that are based upon either of these decompositions. Two of the major difficulties are as follows. The first difficulty pertains to algorithms that might decompose WO using (4.1). In this approach, a difficulty arise due to the fact that the onetoone correspondence between an efficient basis B and the subset W(B) may be destroyed when degeneracy occurs. When x is a nondegenerate efficient extreme point, there is only one efficient basis B associated with x. In this case, it can be shown that W(x)=W(B) When x is a degenerate efficient extreme point, two or more efficient bases can correspond to x. Suppose that B1,...,Bi are the distinct efficient bases associated with x. Then, it can be shown that W(x)= UW(B). 1=1 In this case, there is a possibility that each W(B;) is a strict subset of W(x), and that W(Bi) # W(Bj) for any given two different bases Bi and Bj associated with x. This implies that in methods that use a decision spacebased decomposition of W based upon (4.1), all efficient extreme points may be discovered before Wo is fully decomposed. The second difficulty pertains to approaches that might decompose Wo based upon (4.3). This difficulty is due to the fact that there need not be a onetoone correspondence between the efficient extreme points x and the sets W(x). Given two different efficient extreme points x' and x1, there is a possibility that W(x') = W(xJ). This would imply that in some cases, we may fully decompose Wo via a decision setbased decomposition method based upon (4.3), yet all efficient extreme points may not be found thereby. 4.2. Outcome SetBased Decomposition of the Weight Set W In this section, we will present an outcome setbased decomposition of W. We will show that the outcome setbased decomposition approach overcomes the difficulties of the decision setbased decomposition approaches. Some useful properties of the outcome setbased decomposition are also given. In the following, we will first introduce some definitions and notation. All definitions from convex analysis are standard and may be found, for instance, in Rockafellar (1970). Let y denote a point in Y. Define W(y) by W(y) = { weRPI wTy > wTy' for all y' in Y ). (4.4) The set W(y) is called the normal cone to Y at y (see Rockafellar (1970) for example). It is easy to show that x is an optimal solution to problem LP(w) if and only if we W(y), where y = Cx. For a convex set S in RP, the intersection of all convex cones containing S is also a cone and is called the convex cone generated by S. We denote the convex cone generated by a convex set S by cone (S). For any convex set U in Rk, the interior and the relative interior of U will be denoted int U and ri U, respectively. Proposition 4.2.1. For a point y in Y, let Y{y} = (y'yl y'e Y}. Then, cone (Y {y}) is a polyhedral cone containing the orign and cone (Y{y}) = {k(y'y)l y'e Y, k >0}. Proof. Since Y is a polyhedron and yeY, Y{y} is a polyhedron containing the origin. It follows from Corollary 19.7.1 of Rockafellar (1970) that cone (Y{y}) is a polyhedral cone containing the origin. Furthermore, by Corollary 2.6.3 of Rockafellar (1970) we have cone (Y{y}) = {k(y'y)l y'e Y, k > 0}. 3 For a cone K in RP, the cone {z'e RP cone of K and is denoted by K. Proposition 4.2.2. For each y e Y, the cones cone (Y(y}) and W(y) are polyhedral sets and are polar to one another. Proof The results are immediate from Theorem 2.1 of Varaiya (1967), Theorem 14.1 of Rockafellar (1970), and Corollary 19.2.2 of Rockafellar (1970). O Given a nonempty convex set K in RP, let O0K be defined by 0'K = yeRPly + K c K}. The set 0'K is called the recession cone of K. From Theorem 8.1 of Rockafellar (1970), the recession cone 0+K is a closed cone containing the origin. If K is a closed convex cone, it is easy to verify that 0*K = K. Recall that the lineality of a nonempty convex set K is the dimension of the set (0OK) n O'K. Proposition 4.2.3. Let y e Yex. Then, the lineality of cone (Y{y)) equals zero. Proof Since y e Yex, it is easy to show that cone (Y{y}) is a pointed cone. That is, ( cone (Y{y})) n cone (Y{y})= {0}. Therefore, the lineality of cone (Y{y}) equals zero. l Since Y is a polyhedron, it has a finite number of extreme points. Therefore, Y has at most a finite number of efficient extreme points. Since Y is compact, the problem of minimizing w y over Y has at least one extreme solution for any w>0. Therefore, Y has at least one efficient extreme point. Without loss of generality, assume that YE Yex = {y1,.. .,yq}, where q > 1 is an integer. Notice that the efficient outcome set of problem MOLP is identical to the efficient decision set of the problem vmax y, s.t. yeY, where Y is the outcome set of problem MOLP. By applying (4.3) to the above problem, we obtain the following result. Theorem 4.2.1. W c W(y'). Theorem 4.2.1 implies that W can be decomposed into the union of its subsets W n W(y'), i=l,...,q. We will call this decomposition of W the outcome setbased decomposition. In the following, we will present some properties of W(y'). Then we will show that the decomposition of WO given above establishes a onetoone correspondence between the efficient extreme points y', i = 1, 2,..., q, ofY and the subsets WO W(yi), i =1,...,q, of W. Theorem 4.2.2. For each i = 1,2,...,q, Won int W(y') # 0. Proof. Choose i E {1, 2, ..., q}. By Propositions 4.2.1 and 4.2.2, cone (Y{y')) is a polyhedron in RP containing the origin, and cone (Y{y'}) and W(y') are polar to each other. Thus, from Corollary 14.6.1 of Rockafellar (1970), the dimension of W(y') equals pk, where k is the lineality of cone (Y{y'}). Since y' e Yex, by Proposition 4.2.3, the lineality of cone (Y{y'}) equals zero. Therefore, the dimension of W(y') equals p. This implies that the relative interior of W(y') equals its interior, i.e. ri (W(y') )= int (W(y')). (4.5) Since y' e YE, WrW(y') 0. Notice that WO is an open set. Consequently, by Corollary 6.3.2 of Rockafellar (1970), it follows that W n ri (W(y') # 0. (4.6) From (4.5) and (4.6), we have W n int (W(y')) ; 0. D Remark 4.2.1. For each i = 1, 2, ..., q, since int (Wo 'W(y')) = Wor int W(y'), it follows from Theorem 4.2.2 that W n W(yi) has a nonempty interior. Based upon this result, the outcome spacebased decomposition method will subdivide the weight set W into a union of subsets of W with nonempty interiors. We will use Example 4.2.1 later to illustrate this property. The example will also show that this property does not hold for the decision spacebased decompositions. Theorem 4.2.3. Let w e Wor int W(y') for some i e{1,...,q}. Then y' is the unique optimal solution to the linear program max {wTy yeY}. Proof. Let ek, k = 1, ..., p, denote the vector in RP whose kth component equals one and whose other components equal zero. Since W(y') is a polyhedron, it is a convex set. Since we int W(y'), this implies that there exists a scalar M sufficiently large so that for each k = 1,...,p, w + k ek int W(y'), M where Wk is the kth component of w. Notice that Z(w + i) pw + ( )w = (p + )w. (4.7) k= M M M Setwk =w+ w ek, k= 1,...,p, and set t = 1/(p+ ). Then M M t > 0 and, by (4.7), p w = twk. (4.8) k=1 Since weint W(y'), yi is an optimal solution to max {wTyl yeY}. Suppose that there exists a vector y eY such that wTy = wTyi. By (4.8), we have P P wTyi= :t(wk) y' and wTy = Lt(wk)Ty. (4.9) k=1 k=l Since wkeint W(yi) for all k = 1,2,...,p, we have (wk)Tyi > (wk)Ty for each y in Y and for all k=l,...,p. Since y eY, this implies that (wk)Tyi > (wk)T for each k=l,...,p. It follows from t > 0 that t(Wk)Tyi > t(wk)T (4.10) for each k=l,...,p. We now claim that for each k = 1, 2, ..., p, (wk)Tyi = (k)Ty. Suppose, to the contrary, that there exists some koe { 1,2,...,p} such that (wko )Tyi > (wko)Ty Then, t(w ko )Tyi > t( W k )Ty. (4.11) From (4.9)(4.11), we see that P P wTyi= y> t(wk > k )T y=WT k=l k=l which contradicts the assumption that wr = w'yi. We have thus proved the claim that for each k = 1, 2, ..., p, (wk)Tyi =(w)T y . By the claim proven in the previous paragraph and the definitions of wk, k=l,...,p, (w+ wk k )Tyiw+ W ek)T M M for each k = 1, ..., p. Since w y = wTyi, this implies that Wk k)Ti k (k(412) (ek e)Tyi W()Te (4.12) M M for each k = 1,...,p. Since w e Wo, we know that wk > 0, k = l,...,p. Thus, by (4.12), since M > 0, (ek)Ty' = (ek)T for each k = 1,...,p. As a result, yi= y. Consequently, we have proven that, ifwTyi = wTy and y eY, then y' = y. Therefore, y' is the unique optimal solution to max {wTyI yeY}. E Remark 4.2.2. Let y' be an efficient extreme point of problem MOLP in the outcome space. From Theorem 3.1 in Benson (1982), there exists a weight vector w in W such that y1 is the unique optimal solution to the linear program max {wTyl yeY}. Furthermore, from Theorems 4.2.2 and 4.2.3, we see that y' is the unique optimal solution to the linear program max {wyl yeY} for any we Won int W(y'), where WV int W(y')>0. This observation slightly extends the result in Theorem 3.1 of Benson (1982). Theorem 4.2.4. Suppose that i, j { 1, 2, ..., q} and that y' i yJ. Then, W n int W(y') n W(yi) = 0. Proof Suppose that WO n int W(y') n W(y') # 0. Then, we may choose a point w in W n int W(y') W(yj). Since w e WO int W(y'), it follows from Theorem 4.2.3 that y' is the unique optimal solution to the linear program max {wTy yeY}. Since w e W(y'), yJ is an optimal solution to the linear program max {wryl yeY}. Therefore, yi = y. This contradicts y1i yj. Thus, W n int W(y') r W(y') = D Remark 4.2.3. Theorem 4.2.4 implies that there exists an outcome spacebased decomposition of WO that consists of a union of subsets of W with no interior points in common. Later, Example 4.2.1 will illustrate this property. Theorem 4.2.5. Suppose that i, je {1, 2,...,q} and that yi y'. Then, WO W(y') WO n W(y'). Proof. The result follows immediately from Theorem 4.2.2 and Theorem 4.2.4. D Remark 4.2.4. Theorem 4.2.5 implies that there exists a onetoone correspondence between the efficient extreme points in the outcome space and the subsets of the corresponding weight set W. Theorem 4.2.6. Let H be a nonempty face of Y, and let y, y be two points in ri H. Then, W(y)= W( ). Proof. Since H is a nonempty face of Y, it is a convex set. In fact, from Rockafellar (1970), since Y is a polyhedron, H is a nonempty polyhedral set. By Theorem 6.4 of Rockafellar (1970), since y, y e ri H, we may choose t1, t2 > 1.0 such that (1ti) + tx e H, (lt2) y + t2 E H. By the definition of W( y), for each w in RP, weW(y) iff wT(z ) <0, forallz e Y, iff wT(z 9)+wT(y)< 0, forall z Y, (4.13) By choosing z = (1ti) 9 + t1 y in (4.13), we obtain wT[t](y 9)] + wT(9y)< 0. This implies that (ti 1) wT(5 y)<0. Since tj > 1, we see that wT(y y) 0. (4.14) Similarly, by choosing z = (1t2) y + t2 9 in (4.13), we see that wT(9Y))< 0. (4.15) Together (4.14) and (4.15) yield w T( 9)=0, so that wT = WTy. By definition of W(y), this implies that weW(y) if and only if weW(9). O A set S' is said to be a proper subset of S if S' is a nonempty subset of S and S'S. Theorem 4.2.7. Let I be a proper subset of{1, 2,..., q}. Then, Wo0 (UW(y')) is ieI a proper subset of W. Proof. Since I is a proper subset of {1, 2,..., q}, we may choose some j {1, 2,..., q} such that jel. By Theorem 4.2.2, we can choose a point we W n int W(y'). Since j I, we have y' # y' for each i e I. It follows from Theorem 4.2.4 that w z W(y') for each i e I. Therefore, we WO (U W(y')). Consequently, Wo0 (U W(y')) is a proper subset of iEl JeI Remark 4.2.5. Theorem 4.2.7 will be used later to prove that the Basic Weight Set Decomposition Algorithm for finding all efficient extreme points in the outcome space is finite and valid. The following example is a slight modification of Example 22 on p. 185 of Steuer (1986). It shows that the outcome spacebased decomposition establishes a onetoone correspondence between the efficient extreme points y', i = 1, 2,..., q, and the subsets W nW(y'), i = 1, 2, ..., q, of W. It also shows that, for each i { 1, 2, ..., q}, W nW(y') has a nonempty interior, and that, for any i : j and i, j e {1, 2, ..., q), WO rW(y') and W nW(y') have no interior points in common. Example 4.2.1. Consider the MOLP problem max x+ + 2x2 = y 2x1 = Y2 s.t. 2x, + x3 6 xi + 3x2 6 xI + X2 < 2 x3 2 X1, x2, X3 2 0. The decision set X (see Figure 4.1) has eight extreme points. These are x' = (0, 2, 0)T, x2 = (0, 2, 2)T, x3 = (3, 3, 0)T, X4 = (2, 8/3, 2)T, x5 = (0, 0, 0)T, x6 = (0, 0, 2)T, x7 = (2, 0, 2)T and x8 = (3, 0, 0)T. Here, x' and x2 are each degenerate extreme points, and x3, x4, ...,x8 are nondegenerate extreme points. It can be shown that in this problem Xex r XE = {x', X2, X3, X4}. The outcome set Y (see Figure 4.2) has four extreme points. These are y' = (4, O)T, y3 = (9, 6)T, y8 = (3, 6)T and y5 = (0, )T. In the outcome space, the set of efficient extreme points is given by Yex n YE = {y3, y3}. In Figure 4.2, for each i = 1, 2, ..., 8, y' is the outcome point corresponding to x'. Figure 4.1. Decision Set X Figure 4.2. Outcome Set Y It can be shown that W(x') = W(x2) = (weR21 w1 > 0, 5w1 6w2 < 0), W(x3) = {weR2l w1 >0, 5w1 6w2 > 0), and W(x4) = {weR2I w > 0, 5w, 6w2 = 0}. It is obvious the correspondence between x' and W0 r W(x'), i = 1, 2, ..., 8, is not a onetoone correspondence. Notice that int W(x4) n W = 0 and that int (Wo n W(x1)) = int (W n W(x2)). For y' and y3, it is not difficult to show that W(y') = { weR2 W 12 0, 5w, + 6w2 2 0}, and W(y) = { weR2I w1 > 0, 5wi + 6W2 < 0}. It is obvious that the correspondence between y' and W n W(y'), i = 1, 3, is a onetoone correspondence. Notice also from the above observations that while int W(x4) r Wo = 0, int W(y') n Wo 0 and int W(y3) r W # 0. Furthermore, notice that Wo = (Wo nW(y')) u (Wo nW(y3)), and that the subsets (Wo n int W(y')) and (W r int W(y3)) of Wo have no points in common. 4.3. The Basic Weight Set Decomposition Algorithm The algorithm BWSDA is based on the decomposition of the weight set Wo. At each iteration, the algorithm will first either find a weight vector which will lead to an unexplored efficient extreme point in the outcome space or conclude that all points in Ye r YE have been found. Two questions arise. One is how to find a weight vector which will lead us to an unexplored point in Yex n YE, or to determine that such a weight vector does not exist. We will address this question later. The other question is how to find a point in Yex rn YE, given a weight vector in Wo. The following two results will answer the latter question. Theorem 4.3.1. Assume that w is a weight vector in Wo. If x* is the unique optimal solution to problem LP(w), then y* = Cx* is an efficient extreme point of problem MOLP in the outcome space RP. Proof. Since x* is the unique optimal solution to problem LP(w), y* is the unique optimal solution to the problem max {wTyl yeY}. By linear programming theory, this implies that y* is an extreme point of Y. Since weWo, it follows from Geoffrion (1968) that x* is an efficient solution of problem MOLP. Therefore, y* is an efficient outcome of problem MOLP. Consequently, y* is an efficient extreme point of problem MOLP in the outcome space R'. O Theorem 4.3.2. Assume that w is a weight vector in W0 and that x* is an optimal solution to problem LP(w). If y' is an extreme point of the set Y n {(y RPI wTy = wTCx*}, then y' is also an efficient extreme point of problem MOLP in the outcome space RP. Proof. Notice that as in the proof of Theorem 4.3.1, it follows that x* is an efficient solution of problem MOLP and y' is an efficient outcome of problem MOLP. Thus, we only need to prove that y' is an extreme point of Y. Suppose, to the contrary, that y' is not an extreme point of Y. Then, there exist y', y2 Y, y'#y2 such that y' = ty' + (1t)y2 for some t such that 0 < t < 1. Choose such a t. Then wTy' = twTy + (1t)wTy2. (4.16) We claim that wTy = wTy2. To show this claim, suppose that w y # wy2 Without loss of generality, suppose that w'y' < wTy2. It then follows from (4.16) that wTy'< twTy2 + (1t)wTy2 =wT2. (4.17) 83 Since y' is in Y n {y e RPI wTy = wTCx*}, wTy' = wTCx*. Because y2eY, we may choose a vector x2eX such that y2 = Cx2. As a result, by (4.17), it follows that wTCx* < wTCx2. But since x2EX, this contradicts that x* is an optimal solution to problem LP(w), so that the claim must hold. By (4.16) and the claim proven in the paragraph above, wTCx* = wry = w'yI = w y2 Therefore, y', y2e Y n {y e Rp' wTy = WTCx*}. This contradicts that y' is an extreme point of Y {y e RPI wTy = wTCx*}. Hence, our assumption that y' is not an extreme point of Y must be false. O Basic Weight Set Decomposition Algorithm (BWSDA) STEP 1. Set W' = W. Choose any point w' e W'. Find any optimal extreme point solution x' to problem LP(w'). Set k := 1, EX := 0, EYo := 0. STEP 2. If xk is the unique optimal solution to problem LP(wk), then set xk = xk and yk = Cxk. Otherwise, find any extreme point yk of the set Y({yeRPj (wk)Ty (wk)TC k }, and any extreme point xk of X such that yk = Cxk. Set EXk = EXkl u{Xk}, EYk = EYkl u{yk} and Wk+1 = Wk\W(yk). STEP 3. Find any point wk+1 in Wk+l. If such a point wk+' does not exist, stop: Yex r YE = EYk. Otherwise, go to STEP 4. STEP 4. Find any optimal extreme point solution xk+' to problem LP(wk+1). Set k k+1 and go to STEP 2. Notice that the feasible region of problem LP(w) is a nonempty compact polyhedral set. Thus, for any point w in RP, problem LP(w) has at least one optimal extreme point solution. Thus, STEP 1 and STEP 4 are well defined. In STEP 2, if xk is the unique optimal solution to problem LP(wk), then it follows from Theorem 4.3.1 that yk = Cxk is an efficient extreme point of problem MOLP in the outcome space RP. Otherwise, the algorithm calls for finding an extreme point of the compact polyhedral set Ynr{yRP' (wk)Ty = (wk)TCk }. By the theory of linear programming, such an extreme point yk always exists. Therefore, STEP 2 is well defined. Furthermore, by Theorem 4.3.2, yk is an efficient extreme point of problem MOLP in the outcome space RP. In STEP 3, if Wk+' = 0, then the algorithm stops. Otherwise, the algorithm calls for finding any point in Wk+'. Therefore, STEP 3 is also well defined. The following result shows that the algorithm is finite and valid. Theorem 4.3.3. The Basic Weight Set Decomposition Algorithm terminates at iteration q 2 1, and EYq = Yex n YE, where q is the number of points in Yex. YE. Proof Since Y is nonempty and compact, problem MOLP has at least one efficient extreme point in the outcome space. Thus, q > 1. Since X and Y are nonempty and compact, we can always find the point y' called for in STEP 2 of the first iteration of algorithm BWSDA. Suppose that k > 1 and EYk = {y1, ..., yk}. It follows from Theorems 4.3.1 and 4.3.2 that, for each 1 I i < k, y' is an efficient extreme point of problem MOLP in the outcome space. Thus, EYk is a set of efficient extreme points of problem MOLP in the outcome space. Suppose that EYk # Yx r YE. In Theorem 4.2.7, let I = {1, 2,...,k}. k Then, we have that Wn(U W(y')) is a proper subset of WO. From this, there is a point w i=i k in WO such that w UW(y'), i.e., such that w eWk+. Therefore, when EYk e Yx r YE i=1 the algorithm can always find a point wk+' E Wk+1 in STEP 3 of iteration k, and the algorithm will continue at least to STEP 2 of iteration k+l. In STEP 2 of iteration k+l, the algorithm will then find a new point yk+1 e Yex r YE such that wk+'e W(yk+1). Since k wk+l UiJW(y), wk+1 eW(y') for any I < i k. Therefore, yk' # y' for all 1 i < k. This i=l implies that when EYk # Yx r) YE, the algorithm will continue to the next iteration where it will find an unexplored point in Yex r YE. Since the number of points in Yex n YE is q, the algorithm will eventually stop in iteration q with EY' = Yex n YE. [ Now, we address the implementation of BWSDA. In STEP 1 or STEP 4, we need to find any optimal extreme point solution, denoted by xk, to the linear programming problem LP(wk) for some k 2 1. This can be accomplished, for instance, by using the simplex method of linear programming. In STEP 2 of BWSDA, for any given k, it may be necessary to find an extreme point yk of the set Yrn{yRP (wk)Ty = (wk)TC xk ), and any extreme point xk of X such that yk = Cxk. We explain now one method for accomplishing this. By definition of Y, Yn~{yeRP ( (k)T = (wk)TC k }= (Cx Ax 5 b, x > 0, (wk)TCx = (wk)TC k ). Let D = [C, Opxm], A= A Imxm =(wk)C 01xm ~b b [(wk )T C k and Z= {z Az= b, z 0), where zT = (x1, sT) e R"n. Then Yn(yeR'I (wk)Ty = (k)TCk } = DZ {Dz z eZ}. Since xk is an extreme point of {x e R"n Ax < b, x > 01, it is also an extreme point of {x e R" Ax < b, x > 0, (wk)Cx = (wTCXk It follows that ((x)r,(Sk)T) is an extreme point of Z, where k = b Axk. We need to find an extreme point yk of DZ and an extreme point xk Of X such that yk = Cxk. Starting with z0 = ((Xk)T, (k )T), we can find such points yk and xk by using an algorithm in Benson and Sun (2000). Using this approach, we obtain the following procedure. Procedure for finding yk and xk in STEP 2 of BWSDA Subprocedure I: Stepl. Find any optimal extreme point solution, denoted by z = to the following linear program LPDi: LPDi: min where D1 denotes the first row of D. If z' is the unique optimal solution to LPDI, then set xk = X', y = C i and stop. Otherwise, let vl denote the optimal value of LPD1, set i=2, and go to Step i. Step i (i = 2, 3,...,p). Find any optimal extreme point solution, denoted by z = to the following linear program LPD;: LPDi: min s.t. zE Z, where Dt denotes the tth row of D, and vt is the optimal value for LPD,. If z' is the unique optimal solution to LPDi, or if i = p, then set xk = R', yk = Ci'and stop. Otherwise, let vi denote the optimal value of LPDi, set i = i+1, and go to Step i. From Benson and Sun (2000), the point yk obtained by the above procedure is an extreme point of DZ = Yn {yeRP' (wk)Ty = (wk)TCk }. Notice that Subprocedure I involves solving at most p linear programming problems. The following result shows that the point xk obtained by the procedure is an extreme point of X. Theorem 4.3.4. The point xk obtained by Subprocedure I is an extreme point of X. Proof. It is obvious whenever the procedure stops at some Step i, i > 1, that xk eX. Suppose, to the contrary, that x is not an extreme point of X. Then, there exist x', x" in X such that x' # x" and xk = a x' + ( o) x" (4.18) for some a such that 0 < a < 1. Since z' = e Z and k = i', it follows that (wk)TCxk = (w)TC k This implies that xk is an optimal solution to problem LP(wk), since xk is an optimal solution to this problem. This together with (4.18) yields that x' and x" are also optimal solutions to problem LP(wk). Therefore, (wk)TC x' = (wk)TC xk (wk)TC X" = (wk)TC k Notice that x' and x" are in X. Thus, x' and x" are distinct points in {x e R"I Ax < b, x > 0, (wk)TCx = (Wk)TC k }. Let s' = b Ax', s" = b Ax", z' = and z" = VJ. We then s' S" have that z' and z" are two distinct points in Z such that z= az' + (1 c)z". (4.19) Therefore, Dz' = aDz' + (1 a)Dz". From this equation, since z' is an optimal solution to problem LPDi, we obtain that for each t= 1, 2,....,i, vt = With t =1, since z', z" e Z and 0 < a < 1, (4.20) implies that z' and z" are distinct optimal solutions to problem LPDI. As a result, z' and z" are feasible solutions to problem LPD2. By setting t = 2 in (4.20), since 0 < a < 1, this implies together with (4.20) that z' and z" are distinct optimal solutions to problem LPD2. As a result, z' and z" are feasible solutions to problem LPD3. With t = 3 in (4.20), this implies in a similar manner that z' and z" are distinct optimal solutions to problem LPD3. By continuing in this fashion, we see that z' and z" are distinct feasible solutions to problem LPDi. This together with (4.19) yields that z' is not an extreme point of the feasible region of problem LPDi. This contradicts that z' is an optimal extreme solution of problem LPDi. [ In STEP 3, we need to find, if it exists, a point wk+1 e Wk+', where Wk+' is usually a nonclosed, nonconvex cone. We will present two approaches that either find a point wk+I Wk+l or show that no such point exists. These two approaches will yield two versions of the Basic Weight Set Decomposition Algorithm, WSDAI and WSDAII. 4.3.1. Tree Search Approach for STEP 3 of BWSDA k Let k e {1, 2, ..., q}. Notice that Wk+1 = W0 \ UW(y'). To introduce the first 1=1 approach, we will first present some necessary and sufficient conditions for w 1 W(y'), where = {1, 2,...,q}. Suppose that x e X, y Y, and y = Cx. Notice that W(y) = { weRPI wTy > wTy' for all y' in Y } = { weRPI wTCx > wTCx' for all x' in X ). Therefore, w e W(y) if and only if x is an optimal solution to problem LP(w). By duality theory of linear programming, this implies that w e W(y) if and only if there exists a point u e R" such that CTw ATu < 0, (4.21) uT(A b) = 0, (4.22) xT(CTw ATu) = 0, (4.23) u 0. (4.24) Let Ai be the ith row of A for i = 1, 2,... ,m, and let E; be the ith row of the nxn identity matrix for i = 1, 2, ...,n. For each given pair (x', y'), i = 1, 2,...,q, where x' is an extreme point of X and y' = Cx', let ID(x') = {j{ 1,2,...,m} Ajx'= bj}. For each i = 1, 2, ..., q, let A' be the matrix whose rows are Aj, jID(x'). For each i = 1, 2, ..., q, if ID(x') = 0, let A' equal the scalar 0. For each i = 1, 2, ..., q, let Io(xi) = { je{1,2,...,n} xj'= 0), and let E' be the matrix whose rows are El, leIO(x'). For each i = 1, 2, ..., q, if Il(x') = 0, let E' equal the scalar 0. Let i e {1, 2, ..., k). From (4.21)(4.24), we know that w e W(y') if and only if there exist a vector u' > 0 and a vector v' > 0 such that CTw (Ai)Tui + (E)Tvi = 0. (4.25) Therefore, w v W(y') if and only if the above linear system (4.25) of n equations has no solution (u', v') > 0. By Farkas' Lemma, this implies that w e W(y') if and only if there exists a point d' e R" such that wTCdi > 0, A'd' < 0, E'd' > 0. Let D(i) = {d' R"I A'd' 0, E'd' > 0}, and let CD(i) be the image of D(i) under C. We immediately obtain the following result. Theorem 4.3.5. Let i e {1, 2, ..., q}. For a given w e RP, w z W(y') if and only if there exists d' e D(i) such that w'Cd' > 0. The following theorem gives a way to generate D(i) and CD(i) for each i = 1, 2, ..., q. For any set T c R", let cone T = {at I a > 0, t e T). Theorem 4.3.6. For each i = 1, 2,...,q, we have (1) D(i) = cone (X{x'}); (2) CD(i) = cone (Y{y'}). Proof. (1). (a) First, we will show D(i) c cone (X{x'}) for any given i { 1, 2, ..., q). Let i E {1, 2,..., q), and let d E D(i). Then, A'd < 0, E'd > 0. Let t be any positive number. It follows immediately that A'(td) < 0, E'(td) > 0. Therefore, A'(td + x' x') 0 and E'(td + x' x') > 0. This implies that A'(td + x') < A'x' = b', (4.26) E'(td + x') 2 E'x' = 0, (4.27) where b' is the vector whose components are bj, j eID(x'). For each j ID(x'), we have Ajx' < bj. Thus, for t sufficiently small, we have Aj(td + x') < bj (4.28) for allj e ID(x'). By (4.26) and (4.28), A(td + x') < b, (4.29) for t sufficiently small. Similarly, notice that, for any j z I(xi), xj > 0. Thus, when t is sufficiently small, we have tdj + xj' > 0, (4.30) for all j e Io(x'). By (4.27) and (4.30), we have td + x' > 0, (4.31) for t sufficiently small. From (4.29) and (4.31), we may choose a t > 0 such that td + x' e X. Then, td e X (x'}. Since cone(X{x')) is a cone and t > 0, this implies that d E cone(X{x'}). We have thus proven that D(i) g cone (X(x')). (b) We now will prove the opposite inclusion. Let d e cone (X{x'}). Then, we may choose x e X and t > 0 such that d = t(xx'). By the definitions of A' and b', since x e X, this implies that A'd = t(A'x A'x') = t(A'x  b') < 0. Furthermore, since x e X, this also implies by definition of E' that E'd = t(E'x  E'x') = tE'x 2 0. Therefore, d e D(i). We have thus proved that cone (X{x'}) c D(i). By (a) and (b), D(i) = cone (X{x'}). (2). Let i { 1, 2,..., q}. It is obvious that C(cone (X{x'})) = cone (Y{y'}). This implies from the first part of this theorem that CD(i) = cone (Y{y'}). D Let i { 1, 2, ..., q}. Since X is a polyhedral convex set and x' is an extreme point of X, cone(X{x')) is a polyhedral convex cone generated by the edges of X emanating from x'. It follows that D(i) is a polyhedral convex cone generated by the edges of X emanating from x'. Since X is compact, all edges of X emanating from x' can be exactly determined by determining all extreme points of X adjacent to x'. Let Sat(x') denote the set of all extreme points adjacent to x' in X. Then, D(i) can be generated by determining all points in the set St(x') {x'}. The set Sat(x') can be generated by using the simplex method. Similarly, CD(i) is a polyhedral convex cone generated by the edges of Y emanating from y'. If we let Sat(y') denote the set of all extreme points adjacent to y' in Y, then CD(i) can be generated by determining all points in Sat(y') {y}. The set Sat(y') can be generated by using the algorithm in Benson and Sun (2000). Theorem 4.3.7. For each i = 1, 2,..., q, w e W(y') if and only if there exists a point Kx e Sat(x') {xi) such that wTC K > 0. Proof Without loss of generality, assume that for some positive integer li that St(x') = {x1", x1,...., x1 }. Since D(i) can be generated by all points in Sat(x') {x1}, we have D(i)= { a,(x''x') a, > 0, t= 1, 2, ..., li). t=1 By Theorem 4.3.5, w 4 W(y') if and only if there exists a point d' e D(i) such that wTCdi > 0. i, Suppose that d = Ya, (x' x'), where a, 2> 0, t = 1, 2,..., li. Then, t=1 11 w'Cdi = CatwrC(x'' x'). If wTCd' > 0, then from the above equation, there exists at least one j e { 1, 2,..., li} such that wTC(x1i xi) > 0. Set K' = x'j xi. Then wTC > 0. On the other hand, if X' e S,t(x') {xi} and wTC J > 0, then by Theorem 4.3.5, since KI e D(i), w o W(y'). D k Remark 4.3.1. Let k e {1, 2, ..., q}. From Theorem 4.3.7, w v UJW(y')if and 1=1 only if there are points x' e Sat(x') {x'), i = 1, 2,...,k, such that wTC > 0 for all i = 1, k 2, ...,k. Notice that Wk"' = Wo \ UW(y'). So, w e Wk+1 if and only if w > 0 and there are points S(x) i= 2,...,k, such that >forall= 1 2, ...,k. are points K' e St(x') {x'}, i = 1, 2,...,k, such that wTCK' > 0 for all i = 1, 2, ...,k. 
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