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PAGE 1 1 A POLYHEDRAL STUDY OF INTEGER BILINEAR COVERING SETS By NITISH GARG A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE U NIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Nitish Garg PAGE 3 3 To my p arents and my sisters PAGE 4 4 ACKNOWLEDGMENTS I acknowledge my adviso r, Dr. Jean Philippe P. Richard for his immense help and encourag ement at every stage of my thesis. The frequent technical ses sions with him provided me a good insight of Operations R esearch and helped me to structure my work to its present state. I would also like to thank Dr. Yongpei Guan for serving on my committee. I am thankful to my parents and my sisters for their constan t motivation and support. I also take this opportunity to thank all my friends especially Navjeet Johal Karan Dadlani Ameya Bhosle and Onkar Bhende experience that I will cherish for many coming years PAGE 5 5 TABLE OF CON TENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 8 LIST OF ABBREVIATIONS ................................ ................................ ............................. 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 11 1.1 Background ................................ ................................ ................................ ....... 11 1.2 Mathematical Programming ................................ ................................ .............. 12 1.2.1 Optimization Problem ................................ ................................ .............. 12 1.2.2 Types of Optimization Models ................................ ................................ 17 1.2.2.1 Convex optimization models ................................ .......................... 19 1.2.2.2 Nonconvex o ptimization models ................................ .................... 22 2 BRANCH AND BOUND ALGORITHM FOR NONCONVEX PROBLEMS .............. 26 2.1 Branch and Bound ................................ ................................ ............................ 28 2.2 Building Convex Underestimator ................................ ................................ ...... 32 2.3 Factorable Rela xations ................................ ................................ ..................... 34 3 CONVEX RELAXATIONS FOR BILINEAR COVERING SETS .............................. 3 9 3.1 Motivation ................................ ................................ ................................ ......... 39 ................................ ................................ ................ 39 3.1.2 Staff Scheduling Prob lems ................................ ................................ ...... 42 3.2 Problem Description ................................ ................................ .......................... 44 3.3 Building Convex Hull for Different Cases ................................ .......................... 46 3.3.1 Case 0 ................................ ................................ ................................ ..... 46 3.3.1.1 Problem 1 ................................ ................................ ....................... 47 3.3.1.2 Problem 2 ................................ ................................ ....................... 48 3.3.1.3 Convex hull for n=2 ................................ ................................ ........ 49 3.3.1.4 Problem 3 ................................ ................................ ....................... 50 3.3.1.5 Convex hull for case 0 ................................ ................................ .... 51 3.3.2 Case 1 ................................ ................................ ................................ ..... 51 3.3.2.1 Problem 4 ................................ ................................ ....................... 52 3.3.2.2 Problem 5 ................................ ................................ ....................... 53 3.3.2.3 Convex hull for n = 3 ................................ ................................ ...... 53 PAGE 6 6 3.3.2.4 Convex hull for case 1 ................................ ................................ .... 54 3.4 Facet Defining Inequalities and Convex Hulls ................................ .................. 56 3.4.1 Facet Defining Inequality for Case 0 ................................ ....................... 57 3.4.2 Facet Defining Inequality for Case 1 ................................ ....................... 66 3.4.3 Convex Hull Proof for Case 0 ................................ ................................ .. 71 4 SUMMARY ................................ ................................ ................................ ............. 84 APPENDIX: MATLAB CODES ................................ ................................ ...................... 85 LIST OF REFERENCES ................................ ................................ ............................... 94 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 95 PAGE 7 7 LIST OF TABLES Table page 2 1 Sub problems obtained after breaking down u 1 =(x y) 2 z ................................ ..... 35 2 2 Sub problems obtained aft er breaking down u 2 =xy 2 ................................ ........... 37 3 1 Feasible points for x 1 y 1 +x 2 y 2 3 and u 1 =u 2 =3 ................................ ...................... 47 3 2 Convex Hull for x 1 y 1 +x 2 y 2 ................................ ................................ ............... 48 3 3 Convex Hull for x 1 y 1 +x 2 y 2 4 ................................ ................................ ............... 49 3 4 Convex Hull for x 1 y 1 +x 2 y 2 +x 3 y 3 4 ................................ ................................ ....... 51 3 5 Convex Hull for x 1 y 1 +x 2 y 2 +x 3 y 3 in case 1 ................................ ........................ 52 3 6 Convex Hul l for x 1 y 1 +x 2 y 2 +x 3 y 3 5 ................................ ................................ ....... 53 3 7 Convex Hull for x 1 y 1 +x 2 y 2 +x 3 y 3 ................................ ................................ ....... 54 PAGE 8 8 LIST OF FIGURES Figure page 1 1 Feasible region of Problem (1 2) with only sign restrictions. .............................. 15 1 2 Feasible region of Problem (1 2) with sign restrictions and production constraint for item 1. ................................ ................................ ........................... 15 1 3 Feasible region of Problem (1 2) with sign restrictions and production constraint for items 1 and 2. ................................ ................................ ............... 16 1 4 Infeasibility resulting fro m the addition of constraint x 1 +x 2 600. ......................... 16 1 5 P 1 P 2 P 3 are local minima and P 3 is a global minima. ................................ ....... 17 1 6 A) Convex set B) No n convex set C) Convex set ................................ ............... 18 1 7 Chord of a convex function ................................ ................................ ................. 19 2 1 Convex underestimator g(x) and convex envelope (x) of the f unction f(x) ....... 27 2 2 A) Set X B) Convex hull of X ................................ ................................ ............... 27 2 3 Convex relaxation of the sub problem u 4 =u 3 2 ................................ ..................... 36 2 4 Convex relaxation of the sub problem u 5 =y 2 ................................ ....................... 37 PAGE 9 9 LIST OF ABBREVIATION S BIP Binary Integer Program BP Bilinear Program IP Integer Program LP Linear Program OR Operations Rese arch QCQP Quadratically Constrained Quadratic Program QP Quadratic Program PAGE 10 10 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science A POLYHEDRAL STUDY OF INTEGER BILINEAR COVERING SETS By Nitish Garg August 2011 Chair: Jean Philippe P. Richard Major: Industrial and Systems Engineering We study the polyhedral structure of an integer bilinear covering set which appears in the formul ation of various practical problems including st aff scheduling. Starting from the analysis of specific instances we develop a polyhedral description of the convex hull of solution for special settings of the parameters and derive some facet defining inequ alities. PAGE 11 11 CHAPTER 1 I NTRODUCTION 1.1 Background The origins of Operations R esearch (OR) can be found before World War II, when a group of scientists in the British army started to research operation on the radar technology to improve the efficiency of e arly warning radar systems. During the war, the potential of the Operational Research Group was recognized and its contributions were extended far beyo nd problems related to radars [ 4 ] gap between war technologies and th eir practical use. Operations Research then became synonymous to the set of mathematical techniques developed for the efficient use of technologies and resources. After the end of the war, OR kept growin g because of two main factors [5 ] The first factor i n its progress was the industrial boom that followed the war. OR caught the eyes of many researchers because it had almost unlimited potential outside the military. A key example is the development of simplex algorithm in 1947 by George Dantzig, which is a popular algorithm to solve a very common fami ly of O R problem s called Linear Programs (we describe linear programming models, later in this chapter) The second factor that enabled the development of OR was the computer revolution, which led to the implem entation of computer codes to solve OR problems. A large computational effort is typically required to solve an OR problem, a feature that makes hand calculations often unpractical. Computers allowed the use of OR to spread to a large number of practitione rs and organizations. The basic steps in the OR approach include, (i) formulating the problem in a mathematical form that can be analyzed, and (ii) designing/using algorithms to obtain PAGE 12 12 solutions to the model. Classical OR models include : Mathematical Prog rams Stochastic Processes Models and Simulation Models. In this thesis, we focus on mathematical programs 1 2 Mathematical Programming Mathematical programs are composed of a set of unknown variables, whose values must be found so as to minimize/maximiz e a certain criterion ( objective function ). Those values must be selected in such a way that specified relations (called constraints ) between variables and problem parameters are satisfied. Combination of lem constraint s are said to be feasible There are typically many ways of formulating a practical problem as a mathematical program, not all of which are equivalent in terms of solution efficiency. Before introducing the type of mathematical models we will be studying, we first give a more precise definition of mathematical programs and the basic notions associated with their solution 1 2 1 Optimization Problem Mathematical programs (also called optimization problems ) are mathematical problems of the form: max/min (1 1 ) s.t. where, , are real valued functions, i.e. , PAGE 13 13 In (1 1 ), is called objective function while and are called inequali ty constraints and equality constraints respectively. The objective function allows us to measure the quality of a particular set of values for the variables whereas constraints define the requirements that variables must satisfy To further illustrate the concept of a mathematical program and describe notion related to its solution we next present a simple example. A company hires an OR analyst to maximize the revenue it gains from selling two products that we call item 1 and item 2. The unit revenue generated by the sale of these products is 4 and 6 respectively. The objective function of the problem is therefore where and are the deci sion variables representing number of items 1 and 2 sold respectively. Sign restrictions specify the nature of decision variables, whether they are nonnegative or can assume both positive and negative values. In our example, it is clear that because the number of sold items cannot be negative. If the company assumes that there is no restriction on the amount of these two products it can produce and if the demand for these two products is unlimited, then the problem will be unbounded. A problem is said to be u nbounded if there are feasible points for which the objective function takes arbitrarily large values (for a maximization problem) So far, the model we have built is unbounded because the revenue becomes arbitrarily large when and/or become arbitrarily large; s ee Figure 1 1 for a depiction of the feasible region of the problem, where the feasible region is defined as the set of values from and that satisfy all of the problem constraints (in our case, the sign restrictions). PAGE 14 14 Typically, unbounded problem s arise from issue s in modeling. In the situation that is occurring above, the analyst will realize that producing an unlimited amount of item 1 and item 2 is unlikely to respect physical production capacity in the plant and therefore will define new constraints to represent the problem better. In particular, the analyst may study the production plant and realize that no more than 200 units of item 1 can be produced due to limited capacity. If so, the feasible region will be affected; see Figure 1 2 for a graphical depiction of the updated feasible region. After studying production capacity for item 2, the analyst might find out that product ion is limited to 300 units. If so, the feasible region will be affected; see Figure 1 3 for a graphical depiction of the updated feasible region. After these modifications are performed, the analyst will have defined the problem, gathered data and genera ted the following mathematical pro g ram : max (1 2 ) s.t. Model (1 2 ) is an instantiation of model (1 1 ) if one sets , and The next step in the OR approach is to find a solution to model (1 2 ). In the case of a maximization problem w e say that a feasible solution is an optimal so lution to (1 1 ) if for all feasible solutions in the feasible region. The value of the objective function at an optimal solution is called optimal value i.e. the optimal value is PAGE 15 15 Fo r our example, it is easy to verify from Figure 1 3 that and hence the optimal value for the problem is Now suppose that the company has committed to put 600 units of its product in t he market. To accommodate this new restriction, one would have to add the constraint to the previous model. The introduction of this constraint makes the model infeasible, as there is not a single vector that satisfies all constraints and sign restrictions; see Figure 1 4 for a graphical illustration. Figure 1 1. Feasible region of Problem (1 2 ) with only sign restrictions. Figure 1 2 Feasible region of Problem (1 2 ) with sign restrictions and production constraint for item 1. Feasible region x 1 x 1 x 2 Feasible region x 1 x 2 PAGE 16 16 Figure 1 3. Feasible region of Problem (1 2 ) with sign restrictions and production constraint for items 1 and 2. Figure 1 4. Infeasibility resulting from the addition of constraint x 1 +x 2 600 Feasible region x 1 x 2 x 2 x 1 x 1 x 2 x 1 + x 2 x 1 x 2 PAGE 17 17 The notion of optimal solution we have presented before is that of a global optimum Global optimum. In a maximization problem, we say that a point is globally optimum if the objective function value at i s greater or equal to the objective function of any other feasible point where is the feasible region of the problem. Another notion is regularly used in opti mization that is less restrictive about the qualit y of solution s produced. Local optimum. In a maximization problem, we say that a point is locally optimum if the objective function value at is greater or equal to the objective function of any other feasible p oint in a neighborhood of where is the feasible region of the problem. It is easy to verify that Global optima are always local optima Local optima are not necessarily global (Figure 1 5) and A problem might have multiple local and global optima. Figure 1 5. P 1 P 2 P 3 are local minima and P 3 is a global minima Typically problems that have local optima that are not global are harder to solve since local solution techniques can be fooled to believe that a local solution is global. The situation is simple r in the presence of convexity, a concept that we introduce next 1 2 2 Types of Optimization Models Not all optimization problems are equally simple to solve. The notion of convexity presented next, help delineate those problems that are easy to solve with current P 1 P 2 P 3 PAGE 18 18 optimization techniques from those that are not. To introduce this concept, we first introduce the following definitions. Affine s ets. A set is affine if the line joining any two points in the set completely lies in Mathematically, we say that is affine if for all and for every Affine f unctions. A function is affine if it is a sum of a linear function and a constant, i.e., where and Convex s ets. A set is convex if the line segment joining any two points in the set also lies in Mathematically, we say that is convex if for all and for every We illustrate the concept of convex set in Figure 1 6. A B C Figure 1 6 A) Convex set B) Non convex set C) Convex set It is simple to verify that the intersection of two conv ex sets is convex but the union of two convex sets is not necessarily convex. The concept of convexity can also be defined for functions as we present next Convex f unctions. Let be a real valued function wh ere is a convex set. The function is said to be convex if the chord joining and for any two points is greater or equal to function for any point PAGE 19 19 on the line segment joining and Mathematically, we say that is convex if for all and for all The notion of convex function is illustrated in Figure 1 7. We say that a function is concave if is convex. In other words, we say that function is concave, if for all and for all The concept s of convex function and convex set are related in several ways. First a function is convex if and only if its epigraph is a convex set, where the epigraph of is defined as Second, if is convex and then the sub level set of of level defined as is a convex set. Figure 1 7. Chord of a convex function 1 2 2 1 Convex optimization m odels We define convex optimization problems to be optimization problems of the form : min (1 3 ) s.t. (x 2 f(x 2 )) f(x) (x 1 f(x 1 )) PAGE 20 20 where, is a convex real valued function, i.e. , are convex function s and are affine functions Because the sublevel sets of convex function are convex sets, hyperplanes are convex and intersection s of convex sets are convex, it is easy to verify that the feasible region of a convex program is a c onvex set. A fundamental property of convex optimization models is that any local optimum will also be a global opt imum (for a proof, refer to [3 ]), whereas nonconvex optimization problems can have many locally optimal solutions that are not global optima An important consequence of this result is that local search algorithms that converge to a locally optimal solution always produce a globally optimal solution when applied to convex problems whereas this is not always the case for nonconvex problems. Conv ex optimization problems can be solved efficiently in practice and therefore are key models in OR. Among convex problems, several subclasses have received particular attention. 1 2 2 1 .a Linear programming m odels Linear programming problems are convex pro blems in which the objective function and constraints are affine functions. L inear P rogram s (LP) are convex optimization problem s since it is c lear from their definition that affine functions are convex LPs are usually written in the following standard fo rm in which the only i sign restrictions : PAGE 21 21 min (1 4 ) s.t. In (1 4 ), is a row vector of dimension is a column vector of dimension is matrix of dimension is a column vector of dimension and requires that each component of the column vector is greater or equal to 0. Generally, LP problems arising from application are not in standard form. However, they can be converted into that form by adding slack variables, by subtra cting surplus variables, converting unrestricted variables in to the form For more details on such transformations, we refer to [8 ] We mention that the example problem we discussed in the previous section is an LP. Linear Programs are important since they can be solved very efficiently in practice. In particular, variants of the simplex algorithm [12] and interior point methods [ 3 ] have been implemented in commercial solution software that can nowadays be used to solve very large in stances. 1 2 2 1 .b Quadratic programming m odels Quadratic programming problems are optimization problems in which the objective function is quadratic and const raints are affine functions. A quadratic program (QP) can be expressed in the form: min (1 5 ) s.t. where, , , , and PAGE 22 22 Not all QP s are convex. We describe next conditions under which they are. To this end, we define to be the set of symmetric positive semidefinite matrices: Proposition 1 2 2 1 Problem (1 5 ) is convex if In Problem (1 5 ), if quadratic inequality constraints are added the problem becomes known as quadratically constrained quadratic program (QCQP). A QCQP can be expressed in the form: min (1 6 ) s .t. where, , , and QCQPs are convex if for There are various methods to solve convex QCQPs W e refer interested readers to [ 3 ] for a description. 1 2 2 2 Nonconvex optimization m odels Although c onvex optimization problems can be solved efficiently in theory and practice, the situation for nonconvex optimization problems is more difficult. Entire families of nonconvex problems can be shown to be hard to solve (under reasonable complexity assumptio ns) such as integer programs which we discuss in the next section. However, some nonconvex problems can be readily converted into convex optimization problem, therefore maki ng them easy to solve. Consider for instance min (1 7 ) s .t. PAGE 23 23 an example that i s presented in [3 ]. It can be observed that is not affine and is not convex. However, (1 7 ) can be readily conv erted into the following convex optimization problem : min (1 8 ) s.t. Problem (1 8 ) satisfies all the requirements o f a convex optimization problem. T he set s of optimal solutions to (1 7 ) and (1 8 ) are clearly identical. We next describe two families of nonconvex problems that are hard to solve in theory (and also in practice). 1 2 2 2 .a Integer programming m odels Integer programs (IP s ) are among the most challenging classes of nonconvex optimization models and have wide practical applicability because in many practical applications decision variables can only assumes integers values Pure integer p rogramming. A pure IP is a linear program in which all th e variables are required to take integers values. Pure IPs can be written as : min (1 9 ) s.t. In (1 9 ), is a row vector of dimension is a column vector of dimension is matrix of dimension is a column vector of dimension and is the set PAGE 24 24 of non negative integers. Among IPs, those in which all the variables are restricted to take the values 0 or 1 are called binary integer program (BIP ). BIPs are particularly important, since they arise in the modeling of virtually a ll combinatorial problems Mixed integer p rogramming. In mixed IP s some of the variables in the optimization problem can only assume integer valu es. Mixed IPs can be written as: min (1 1 0) s.t. where In (1 1 0), is a row vector of dimension is a column vector o f dimension is matrix of dimension is a column vector of dimension and is the set of indices of the problem variables. Over the years, many algorithms h ave been developed to solve IPs. B ranch and bound is one such an algorithm which we are going to discuss in a subsequent part of the thesis. 1 2 2 2 .b Bilinear programming m odels Another family of nonconvex p rograms is that of bilinear programs (BP) in which the objective/constraint functions are bilinear. Bilinear functions are functions of the form: (1 1 1) where, and PAGE 25 25 Bilinear functions are a particular form of quadratic functions. In fact, defining one can write (1 1 1) as from which it can be verified that bilinear funct ions are typically not convex s ince is typically not positive semi definite. A fundamental property of bilinear functions is that they become linear if anyone of the vector or is assigned a particular value In this thesis, we wil l study relaxation techniques for a particular family of BP s BP came into existence through problems called bimatrix game s Various solution methods have been proposed for solvin g BP s both locally and globally. A gener al discussion regarding BP s and their solvi ng techniques can be found in [6 ]. In particular, a branch and bound algorithm for BP s was dev eloped by Al Khayyal and Falk [1 ] We emphasize this particular reference, since will discuss general branch and bound concepts in the following chapters of this thesis. PAGE 26 26 CHAPTER 2 BRANCH AND BOUND ALGORITHM FOR NONCONVEX PROBLEMS We mentioned in the previous chapter that finding global solution for nonconvex optimization problem s is difficult in general. To circumvent this di fficulty, we seek to construct good convex approximation s of nonconvex problems that will allow us to exploit the fact that convex optimization problems are easier to solve. This idea motivates the introduction of the notion of convex relaxation Convex r elaxation Let where and be an optimization problem. We say that is a relaxation of if (i) for all and (ii) Further, we say that is a convex relaxation of if is a convex optimization pr oblem. Assuming that and have optimal solutions, it is easy to see that It is also easy to verify that is unbounded if is unbounded and that is infeasible if is infeasible. Given an real valued objective function that is nonconvex, where is a convex set, w e define to be a convex underestimator of over i f is convex over and, for all We define the convex envelope of over to be the pointwise supremum of all convex underestimators of over In other words, the convex envelope of over is the largest convex function underestimating An illustration of these concepts is given in Figure 2 1 PAGE 27 27 Figure 2 1. C onvex underestimator g(x) and convex envelope (x) of the funct ion f(x) It is clear that if is an optimization problem where is a convex set and is a nonconvex function, then is a convex relaxation of where is the convex envelope of over Given a set we define the convex hull of to be the smallest convex set containing An illustration of this concept is given in Figure 2 2 A B Figure 2 2. A) Set X B) C onvex hull of X PAGE 28 28 It is clear t hat if is an optimization problem where is a non convex set and is a convex function, then is a convex relaxation of In general, given a feasible set defined by constraints for it is difficult to find Ho wever it is easily established that is a convex relaxation of where is the envelope of over Convex relaxations p lay an important role in solving many nonconvex optimization problems. In the next section, we describe how convex relaxati ons can be combined with divide and conquer principle s to give rise to branch and bound algorithms for the global soluti on of nonconvex problems. 2 1 Branch and Bound Branch and bound (BB) is one of the most widely used methods to solve nonconvex optimization problem to globally optimality In this method, the initial problem is relaxed into a convex problem that is solved. If the optimal solution of the relaxation is also optimal for the initial problem, the process is stopped. Otherwise the problem is divided into parts (branching) so that stronger relaxations can be built over the pieces that hopefully have better bounds (bounding). To illustrate the approach, w e consider the following problem : min (2 1 ) s.t. PAGE 29 29 Because this problem has a concave ob jective function and a polyhedral feasible region, it is clear that a n optimal solution can be found at an extreme point of the feasible region of (2 1 ). It follows then by inspection that is an optimal solution to the problem. Howe ver, we will ignore this to illustrate how the branch and bound algorithm would solve th is problem. A possible implementation of the branch and bound algorithm would work as follows. Step 0. First, the feasible region is relaxed to which is chosen to be the convex hull of This set is a simplex that contains the feasible region For more information about simplices, we refer to [6 ]. Since the objective function of (2 1 ) is concave, t he affine function coinciding with at the vertices of is the convex envelope of over ; see definition of concave f unction in the previous chapter. In particular, the parameters of can be obtained by solving the system , which equates the values of and at the vertices of We obtain : and so we conclude that is a convex underestimator of i.e. Now, a lower bound to (2 1 ) can be found by solving the following linear program min (2 2 ) s.t. PAGE 30 30 The optimal value of (2 2 ) is and is achieved at It follows that Now, observe that the two points and belong to the feasible region. Since the feasible region is convex, the segment between them, call it als o belongs to Optimizing over yield which is achieved at the point Since is feasible w ith value we conclude that Combining these observations, we can say that Because the gap between upper and lower bound is large, we w ill subdivide the problem into smaller pieces S tep 1. In this step, we partition into two other simplices and After partitioning we check that each one of these intersects the feasible region, i.e. or If not then it is unnecessary to investigate the corresponding simplex further since it cannot contain the optimal solution (as it does not even contain a feasible solution) In classical branch and bound term inology, we prune the corresponding branch because we know that a solution to the actual problem cannot lie in that branch. In our example, both and intersect and so, they bot h must be considered. We then must construct a convex relaxation of (2 1 ) over and PAGE 31 31 Proceed ing as before, we compute and We calculate the coefficients of these envelopes as explained in the previous step to obtain and We now can solve convex relaxation s of (2 1 ) over and respecti vely by solving linear programs. We ob tain the lower bounds achieved at and achieved at Therefore, we conclude that Looking for feasible solutions in and we obtain that is achieved at and is achieved at where, and This leads to an overall upper bound of achieved at Because we will partition the problem further. Step 2. In the next iteration, we check first whether any of the previous branch es can be pruned on the basis that or In fact, in such a situation, the best possible solution in the branch is inferior to one already discover ed in the tree and therefore can no t be optimal. This step is one of the fundamental features of branch and bound algorithms that allows entire portion s of the feasible region to remain unexplored on the basis that they cannot possibly contain an optimal so lution. In our case, the best solution found so far only has value and therefore no pruning occur at this stage We must therefore partition into and and to p artition into and so as to define PAGE 32 32 , Following the same procedure as mentioned in Step 1 to find and we obtain that, is achieved at and Since we stop the algorithm with the conclusion that is a globally optimal solution to (2 1 ). There are many implementation details that are crucial in applying branch and bound algorithm successfully, including how the feasible region is divided, how relaxations are obtained, and in what o rder nodes are evaluated. C onvergence of the algorithm to an optimal solution is not obvious but can be guaranteed under some conditions. We r efer the interested reader to [6 ] for a discussion of these important questions 2 2 Buil ding Convex Underestimator The above example illustrates the fact that, in order for branch and bound to be effective, one needs to be able to construct convex relaxations of nonconvex problems easily. It is also clear that such relaxations must improve as the problem gets subdivided. Next, we describe classical results on the derivation of convex underestimator s and convex envelope s for bilinear problems. Consider the problem min (2 3 ) s.t. where, PAGE 33 33 Let underestimate the function over where is any convex set in domain of the function Then it can be argued that (2 4 ) where denote s the convex underestimator of over Now, it requires designing a procedure to build L et be a hyper rectangle that contains the feasible region, and let be such that This type of partitioning is used in the branch and bound algorithm for BP developed by Al Khayyal an d Falk [1 ] We seek to develop relaxations of the terms over We can safely say that : (2 5 ) and (2 6 ) Multiplying ( 2 5 ) and ( 2 6 ) gives, In other words, underestimates over Similarly, (2 7 ) and (2 8 ) Multiplying (2 7 ) and (2 8 ) gives In other words, also underestimates over PAGE 34 34 Since the pointwise supremum of convex underestimators of a function als o underestimates the function, w e conclude that (2 9 ) is a convex underestimator of over It can be shown that (2 9 ) forms the convex envelope of over We refer to Al Khayyal and Falk [1 ] for a proof. 2 3 Factorable Relaxations For more general functions, finding convex envelope s is typically a difficult problem. However, several techniques have been proposed to derive convex underestimators We d escribe one such technique next that was initially proposed by G. P. McCormick [9] This technique applies to factorable functions that can be obtained recursively, using sums and products of a family of univariate base functions whose convex envelopes are known. The family of base functions can con tain polynomial, exponential or trigonometric functions. The technique proceeds by breaking down the initial function into its constituent terms and then approximating these terms individually. The technique can be used to obtain convex underestimators of objective funct ions but also can be used to construct convex relaxations of nonconvex constraints as is a convex relaxation of whenever is a convex underestimator of To illus trate the ideas behind the factorable relaxation technique we consider the following constraint: (2 1 0) PAGE 35 35 where, it is known otherwise that variables belong to the unit hypercube, i.e. , and O ur goal is to find a good convex underestimator of over as will provide a convex relaxation to At fi rst glance, it may appear difficult to devi se such a convex underestimator For this reason, we will start breaking down the function into components until we obtain terms that are sufficiently simple to approximate. To break down we introduce two new variables and as (2 1 1) (2 1 2) Still, it is difficult to relax these terms and hence we will break them down further by introducing ne w variables and as , Each of these terms is then relaxed as summarized in Table 2 1. Table 2 1. Sub problems obtained after breaking do wn Sub problem Convex relaxation This constraint is c onvex and As shown in the Figure 2 3 the feasible region defined by this constraint is not convex over It can be observed from the figure that provides a convex relaxation of the feasible set defined by this constraint. As discu ssed earlier in this chapter, w e can use standard techniq ue s to find a convex relaxation to this expression ; see convex envelope for bilinear problems which yields , and PAGE 36 36 Figure 2 3 Convex relaxation of the su b problem A convex relaxation for the set if feasible solutions to equation for can therefore be written as: (2 1 3) (2 1 4) (2 1 5) (2 1 6) (2 1 7) (2 1 8) (2 1 9) For we proceed similarly and introduce new variable as Each of these terms is then relaxed as summarized in Table 2 2. (u 3 ) 2 u 3 PAGE 37 37 Table 2 2. Sub problems obtained after breaking down Sub problem Convex relaxation As shown in Figure 2 4 the feasible region defined by this constraint is not convex over It can be observed from the figure that provides a convex relaxation of the feasible set defined by this constraint. Similarly to the previous case, a convex relaxation is given by , and Figure 2 4 Convex relaxation of the sub problem A convex relaxation for the set of feasible solution to equation for can therefore be written as: (2 2 0) (2 2 1) (2 2 2) (2 2 3) (2 2 4) y 2 1 y PAGE 38 38 (2 2 5) together with , (2 2 6) Combining (2 1 3) (2 1 9) and (2 2 0) (2 2 6 ) with (2 2 7 ) provide s a convex relaxation for the set of feasible solution to constraint (2 1 0). The ideas presented here can be applied general factorable functions. We refer to Tawarmalani and Sahinidis [11] for a more detailed descript ion. Factorable relaxations are used in commercial global optimization software such as Baron and Lindoglobal. PAGE 39 39 CHAPTER 3 CONVEX RELAXATIONS F OR BILINEAR COVERING SETS 3 1 Motivation As discussed in the previous chapters, building strong convex relaxat ions is key in solving many complex nonconvex problems to global optimality. We next demonstrate through two examples that multilinear covering constraints of the form (3 1 ) occur in practical problems. This will be the primary mo tivation for our later polyhedral studies. 3 1 1 The problem of multiplying two matrices of dimension is a fundamental problem in linear algebra. When multiplying matrix with we create a new matrix whose components are obtained through the formula for and A straightforward applicatio n of the formula shows that it is possible to compute the product of two matrices using multiplications and additions, PAGE 40 40 resulting in a algorithm. There are, howe ver, algorithms for matrix multiplication with running time of where [10] is one such algorithm. The fundamental idea in S when consider ing ma trices and it is possible to compute their product using only 7 multiplications and 18 additions/subtractio ns. This can be done as follows. First we compute : = = = = ` = = = Then, we combine th ese terms as follows : = PAGE 41 41 = = = Because matrices can be b lock multiplied, a recursive application of the above equation s yields a algorithm for matrix multiplication. An important question is that of determi ning whether such rule can be obtained systematically This question can be answer ed positively as schemes to multiply two matrices using only 7 multiplications corresponds to feasible solution s to a mixed integer nonlinear problem with constraints: (3 2 ) where, and In the above model, the variable represents the coefficient of in represents the coefficient of in represents the coefficient of in and and and More precisely a feasible solution to Problem (3 2 ) yields a way of computing with the 7 multiplications as = with the following additions, = Similarly, one could determine if there exists a way of multiplying matrices PAGE 42 42 using 23 multiplications by determining if the foll owing system is feasible or not: (3 3 ) where, and an d A f easible solution to Problem (3 2 ) would yield an algorithm for multiplying matrix with 23 multiplications instead of 27. Such a feasible solution, in tu rn would lead to a algorithm for matrix multiplication. A feasible solution to Problem (3 3 ) can be found in [ 7 ] T he constraints of the above models involve multilinear function s It is simple to reformulate them using variables to transform (3 2 ) and (3 3 ) into a problem conta ining only bilinear constraints, some of which can be relaxed to form (3 1 ) 3 1 2 Staff Scheduling Problems Traditional staff scheduling models also can be formulated as bilinear inte ger programs if the shifts are part of the decision variables. To clarify the idea, consider a problem in which a call center has to ensure that at least operators are on duty during time slot Operators mus t be assigned to one of the shifts (whose characteristics must be decided) in a way that minimizes the number of operators used. Shifts are characterized only by the ir start time. We assume that shifts are required to contain a sequ ence of consecutive time slots of wo rk followed by 1 time slot of rest, followed by time slots of work. To model such problem, we introduce variables as PAGE 43 43 if time slot is part of shift otherwise. We introduce as if shift starts at time otherwise, and let to represent the number of operators assigned to shift T he problem can then be formulated as : min s.t. (1) (2) (3) , (4) (5) In the above model constraint (1) requires that s ufficient ly many operators are present during each time period, constraint (2) imposes that o perators work for time periods, c onstraints (3) and (4) state that the shift requires work in the period and and rest in period after the beginning of the shift while constraint (5) requires that t he shift has a sing le starting period In model (1) (5), we replace with when ever PAGE 44 44 In the above model, we see that the constraints that require that sufficient ly many operators are available at any poi nt in time are bilinear and are of the form (3 1 ) We also observe that they contain only integer variables. 3 2 Problem Description As illustrated before in this chapter constraints of the form : (3 4 ) where, , , appear in the formulation of several prac tical problems. It would therefore be useful to derive convexification procedures for them. Current convexification procedures applied to this problem would (i) relax integrality and (ii) build convex relaxation s by finding concave overestimators of the pr oduct This result s in a double source of weakness as (i) it can be seen that integrality makes the convex hull of feasible solutions to (3 4 ) polyhedral while the convex hull of the continuous relaxation is not and (ii) the relaxat ion where is a concave overestimator of do es not yield the convex hull of feasible solutions to (3 4 ). Therefore, we wish to derive expression s for the convex hull of set s defined by (3 4 ) PAGE 45 45 We define a polyhedron as the set of feasible solutions to a finite number of inequalities and equalities: ( 3 5 ) where , and We say that is bounded if is contained in a ball of finite radius. A bounded polyhedron is said to be polytope Because the feasible sets of integer bilinear sets are composed of a finit e number of points, the following result can be easily established using Minkowski Weyl Theorem [13] We denote the feasible region of Problem (3 4 ) by and the convex hull of by P roposition 3 2 1 : The convex hull of the feasible solutions to Problem (3 4 ) is a polytope, i.e., is a polytope. As a first step in the derivation of such convex hull s we use MATLAB to generate the c onvex hulls of sample problems w here the values of and are fixed. To this end we use the MATLAB function K = convhulln(X) where and s are the feasible solutions (vectors in dimensions) of Problem (3 4 ). We restricted ourselves to small values of , and s so that the function K = convhulln(X) work s relatively fast. During the study of the results pro duced by MATLAB we found that the structure of the convex hull is strongly influenced by the position of in the sequence Therefore we divide the study of the problem into different cases depending upon posi tion of in the sequence. In particular, we define case to be the PAGE 46 46 case where and We mention that, in case 0 every individual term can satisfy the right hand side requirement in Inequality (3 4 ) by itself (without the use of any other pair of variables). In case 1 however, the term must always be used in conjunction with at least another. Finally in case n none o f the variable pairs can be used without also selecting at least another pair. We further categorize c onvex hulls into two categories: specific convex hull and generic convex hull. We call convex hulls generic, when their structure remained unchanged for all values of in a particular case. For example, suppose that the convex hull is found for n = 2 in case 0 If the convex hull can be found for n = 3 in case 0 using the same method, then we say that case 0 ha s a generic convex hull. In our studies, we will mainly focus on those cases for which we could identify a generic structure for the convex hull. In the following se ctions, we will see that case 0 and case 1 have generic convex hulls. 3 3 Building Convex Hull for Different Cases In this section, we use our MATLAB codes to derive convex hull s of different instances of case 0 and case 1 and we use these examples to conjecture facet defining inequalities and convex hull description s for certain integer bilin ear covering sets. In Section 3 3 1 we consider case 0 while in Section 3 3 2 we consider case 1 3 3 1 Case 0 In this case, we have (3 6 ) PAGE 47 47 where , and with 3 3 1 1 Problem 1 In Problem ( 3 6 ), we set and to obtain : ( 3 7 ) where , , and To use MATLAB, we also need to choose values for and For we list all feasible solution s to the problem and find the convex hull using MATLAB function K = convhulln(x) For this instance, we present all feasible solutions in Table 3 1. These points were obtained using a code we wrote MATLAB The code is presented in Appendix Table 3 1. F easible points for and 1 3 0 0 1 3 1 0 1 3 0 1 1 2 1 1 1 3 1 1 1 3 0 2 1 1 1 2 1 2 1 2 1 3 1 2 1 3 0 3 0 0 1 3 1 0 1 3 0 1 1 3 1 1 1 3 0 2 1 3 1 2 1 3 0 3 1 3 1 3 1 3 Observe that, in Table 3 1, n ot all points are necessary to build the convex hull as some of the points are convex combination of other points. In particular, (1,3,0,1) is a PAGE 48 48 convex combination of (1,3,0,0) and (1,3,0,2) and therefore could be omitted from the list without altering the convex hull. However, computing convex hulls with these unnecessary points does not compromise the validity of our approach. The convex hull of feasib le points of Table 3 1 generated using MATLAB, is presented in Table 3 2 Table 3 2. Convex Hull for Convex hull (for restricted problem) Convex hull (for original problem) Repeating the computation for various values of and we conjecture the convex hull for the original problem ( 3 7 ), presented in the right side of Table 3 2. Intuitively, this works because the convex hull of Problem (3 4 ) has two kinds of inequalities: lower bounding and upper bounding. Lower bounding inequalities do not change as long as upper bounding inequalities do not interact with them Limit case s are encountered when an upper bounding inequalit y and a lower bounding inequality of the convex hull meet at a vertex 3 3 1 2 Problem 2 We consider now a case larger right hand side value In particular, we set and in Problem ( 3 6 ) to obtain : ( 3 8 ) where , , and PAGE 49 49 To conjecture the convex hull of this problem, we apply the same procedure as before. We first restr ict so that we can find the convex hul l of the restricted problem easily After finding it, we replace the upper bounds on and with and respectively The convex hull of feasible points generated using MATLAB, is presented in Table 3 3 Table 3 3 Convex Hull for Convex hull (for restricted problem) Convex hull (for original problem) In the right side of the table, we present our conjecture for the convex hull when and are general. From the results of Table 3 2 and Table 3 3 we next infer the general form for the convex hull of case 0 when 3 3 1 3 Convex h ull for n=2 We compare the inequalities obtained in Problem 1 and 2. Inequalities and are similar, and the difference can be ea sily attributed to We therefore infer that, is a required inequality for any value of We will verify this claim later in this chapter. Second we observe that inequalities and are similar and have an obvious dependence on that can be generalized to Similarly, inequalities and can be generalize d to Regarding this second and third inequalities, it can be PAGE 50 50 argued that if anyone of them is a valid inequality then the other one is also valid, because of symmetry. Fourth similar inequalities are and which can be extended in terms of to Hence, we conjecture that the convex hull of Problem (3 4 ) in case 0 when is given by : , We nex t seek to generalize this conjecture to situation whe re We do so by considering an example first. 3 3 1 4 Problem 3 In Problem ( 3 6 ), we set and to obtain : ( 3 9 ) where , , and Similarly to Problem 1, w e obtain t he convex hull of feasible points by first generating a list of all feasible solutions through a MATLAB code and then computing its convex hull using K=convhulln( X). Results are presented in Table 3 4 for the case where and a conjecture for the convex hull is presented on the right side of table when PAGE 51 51 Table 3 4 Convex Hull for Convex hull (f or restricted problem) Convex hull (for original problem) , 3 3 1 5 Convex hull for c ase 0 The result s of Table 3 4 combined with those of Table 3 2 and Table 3 3 suggest the following form for : where ( 3 1 0 ) ( 3 1 1 ) ( 3 1 2 ) for ( 3 1 3 ) for ( 3 1 4 ) 3 3 2 Case 1 We now focus on developing a conjecture for the convex hull of integer bilinear covering set in case 1 In this case, we consider the set of feasible solution s to ( 3 1 5 ) where , and when we assume PAGE 52 52 here that 3 3 2 1 Problem 4 In Problem ( 3 6 ), we set and to obtain : ( 3 1 6 ) where , , and Similarly to Section 3 3 1 1 we will build the convex hull for Problem ( 3 1 6 ). W e consider two cases. First, we set and S econd we set and After building the convex hull s of these two instances, we will conjecture a convex hull description for the general problem. Results are presented in Table 3 5 Table 3 5 Convex Hull for i n case 1 Convex hull ( and ) Convex hull ( and ) Convex hull (for original problem) , Comparing the inequalities, we present, in the right column a conjecture for the convex hull description of feasible solutions to Problem ( 3 1 6 ) We now examine the dependence of this convex hull on the right hand side value PAGE 53 53 3 3 2 2 Problem 5 In Problem ( 3 6 ), we now set and to obtain ( 3 1 7 ) where , , and Similarly to Section 3 3 2 1 we consider two sets of values for the bounds The first choice is obtained by setting and The second is obtained by setting and Results are presented in Table 3 6. Table 3 6 Convex Hull for Convex hull ( and ) Convex hull ( and ) Convex hull (for original problem) , Com paring the inequalities, we present, in the right column a conjecture for the convex hull description of feasible solutions to Problem ( 3 1 7 ) 3 3 2 3 Convex h ull for n = 3 We are now ready to conjecture a polyhedral description of the convex hul l of inte ger bilinear covering set s in case 1 Comparing inequali ties with similar structure for Problem s 4 and 5, we conjecture the following description of the convex hull of case 1 PAGE 54 54 for n = 3. Table 3 7 Convex Hull for Convex hull ( ) Convex hull ( ) Convex hull ( ) (1) ( 2 ) (3 ) (4 ) ( 5 ) (6 ) (7 ) (8 ) (9) (10) , , , (11) 3 3 2 4 Convex h ull for case 1 We nex t discuss each of the inequalities in Table 3 7 so a s to see how they can be generalized to problem s with pairs of variables. First, we observe the inequality ( 3 1 8 ) is always part of the description; see (7) As we discussed before alone cannot meet the right hand side of P roblem ( 3 1 5 ): o ne of the other has to be active. This argument holds for any value of Hence, we conj ecture that the inequality ( 3 1 8 ) is part of the convex hull de scription for any value of in case 1 Second, inequalities (1), (2), (3), (4), (5), and (6) suggest that ( 3 1 9 ) PAGE 55 55 where, , and where are part of the convex hull description. Hence, we conj ecture that the family of inequalit ies ( 3 1 9 ) is part of the description of for any value of in case 1 To generalize (8) and (9), we write ( 3 2 0 ) where, Inequality ( 3 2 0 ) can be interpreted in light of Problem ( 3 6 ) as an application of case 0 over variables when are set to their maximum values, with the exception of inequalities based on and since the cases and are already handled in ( 3 1 8 ) and ( 3 1 9 ). This argume nt also holds true for any value of in case 1 As a result of the above discussion we conjecture that case 1 has a generic convex hull of the form: ( 3 2 1 ) ( 3 2 2 ) ( 3 2 3 ) ( 3 2 4 ) ( 3 2 5 ) ( 3 2 6 ) (3 2 7 ) PAGE 56 56 where, , , and where 3 4 Facet Defining Inequalities and Convex Hull s As described in Section 3 2 the convex hull of int eger bilinear covering sets is polyhedral. To study further the polyhedral structure of these sets w e next describe some basic notions about polyhedra and polytope s that we will use in the remainder of this thesis. We focus on polyhedra of the form: (3 2 8 ) N ot all inequalities in the definition of a polyhedron are necessary in describing In order to differentiate the inequalities that are necessary from those that are not, we introduce a few definitions; we refer to [ 3] for more details. Affine i ndependence. P oints in are affinely independent if the system and has as unique solution Next we introduce the concept of dimension of a polyhedron The dimension of ( ) is one less than the maximum number of affinely independe nt points in A polyhedron is said to be full dimensional if An inequality is said to valid for if PAGE 57 57 or equivalently if Given a valid inequality for a polyhedron we define the face of induced by to be A valid inequality for is said to be facet defining for if In other words, is a facet defining inequality for if there exists affinely independent points in Given a full dimensional polyhedron and a valid inequalit y proving that is a facet defining inequality simply requires displaying affinely independent points in Several variants of the idea have been proposed that are sometimes eas ier to us e in practice. W e refer the i nterested reader to [ 2 ] for a description 3 4 1 Facet Defining Inequality for Case 0 In our previous section, we conjectured that in case 0 is given by: (3 2 9) (3 3 0) (3 3 1) for (3 3 2) for (3 3 3) where, PAGE 58 58 We will show next that these inequalities are facet defining for To this end, w e first show that inequalities (3 2 9), (3 3 0), (3 3 1), (3 3 2), and (3 3 3) are val id for and therefore for It is easily verified that is full dimensional when W hen the set is trivial, i.e., where has a convex hull that is given by and This shows that for is not full dime nsional. Proposition 3 4 1 1 : Inequalities (3 3 0), (3 3 1), (3 3 2), and (3 3 3) are valid inequalities for Proof: Inequalities (3 3 0), (3 3 1), (3 3 2), and (3 3 3) are valid for since they belong to the initial d escription of the problem. Proposition 3 4 1 2 : Inequality (3 2 9) is valid for Proof: Consider any feasible solution s.t. (3 3 4) where and Denote and Case 1 : For Inequality (3 2 9) reduce s to The inequality is valid for since PAGE 59 59 where the first inequality hold s since and the second inequality holds because of (3 3 4) Case 2: For Inequality (3 2 9) reduce s to or equivalently This inequality is clearly valid for as, when Case 3: For We consider two subcases. Case 3a: Assume that the solution of satisfies Then showing that satisfies (3 2 9) Case 3b: A ssume that the solution of satisfies It follows from (3 3 4) that if then Then showing that satisfies (3 2 9). PAGE 60 60 Next, we are going to show that (3 2 9), (3 3 0), (3 3 1), (3 3 2), and (3 3 3 ) are facet defini ng inequalities for In all subsequent proofs, we let be any inequality that induces a face in dimensional space that contain s the face of defined by the above mentioned inequalities Proposition 3 4 1 3 : I nequality (3 2 9) is facet defining for Proof: We consider three cases : Case 1: Assume , i.e., (3 2 9) is of the f orm For all ( ) is a feasible solution that satisfies at equality. Therefore, it must belong to which yields ( 1) F or all and the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (2 ) It follows from (1) and (2) that T herefore the face is defined by For all and the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (3) For all the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding PAGE 61 61 (4 ) or It follows from (3) and (4) that and so the face is defined by For all the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (5) and so the face is defined by w hich shows that the inequality defining is a scalar multiple of Case 2: Assume I nequality (3 2 9) r educes to For all ( ) is a feasible solution that satisfies This yields (1) F or al l , and the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (2 ) It follows from (1) and (2) that T herefore face is defined by For all the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding PAGE 62 62 (3) or and so the face is defined by w hich shows that the inequality defining is a scalar multiple of Therefore is a facet defining ine qu ality for Case 3: For I nequality (3 2 9) reduces to For all ( ) is a feasible solution that satisfies at equality and therefore must belong to the face yielding (1) F or all , and the feasibl e solution ( ) satisfies at equality and therefore must belong to the face yielding (2 ) It follows from (1) a nd (2) that T herefore the face is defined by For all the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (3) and so the face is defined PAGE 63 63 which shows that the inequality d efining is a scalar multiple of inequality Therefore is a facet defining inequality for Combining case 1 case 2 and case 3 we have shown that all inequalities described in (3 2 9 ) are facet defining for Proposition 3 4 1 4 : I nequality (3 3 0) is facet defining for Proof: Consider For all ( ) is a feasible solution that satisfies at equality and therefore must belong to the face yielding (1) For all the feasible solution ( ) satisfies at equ ality and therefore must belong to the face yielding (2 ) It follows from (1) and (2) that T herefore the face is defined by For all the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (3 ) It follows from (1) and (3) that .T herefore the face is defined by For any the feasible sol ution ( ) satisfies at equality and therefore must belong to the face yielding PAGE 64 64 ( 4 ) It follows from (1) and ( 4 ) that therefore the fa ce is defined by This shows that is facet defining fo r Proposition 3 4 1 5 : I nequality (3 3 1) is facet defining fo r Proof: Consider For ( ) is a feasible solution that satisfies at equality and therefore must belong to the face yielding (1) For all the feasible solution ( ) satisfies at equalit y and therefore must belong to the face yielding (2 ) It follows from (1) and (2) that T herefore the face is defined by For all the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (3 ) It follows from (1) and (3) that T herefore the face is defined by For all the feasible soluti on ( ) satisfies at equality and therefore must belong to the face yielding PAGE 65 65 (4 ) It follows from (1) and (4) that therefore the face is defined by or equivalently This shows that is facet defining for Proposition 3 4 1 6 : For I nequality (3 3 2) is facet defining fo r Proof: W e use the direct approach of fin ding affinely independent points in the face induced by We assume without loss of generality that we consider The following points for = = for = for = = are affinely independent and satisfy at equality This shows that is facet defining for Proposition 3 4 1 7 : For I nequality (3 3 3) is facet defining fo r Proof: We use t he direct approach of finding affinely independent points in the face induced by We assume without loss of generality that we consider The following points fo r PAGE 66 66 = = for = for = = are affinely independent and satisfy at equality. This shows that is facet defining for In th e next section we consider the linear inequalities we developed for case 1 and sketch that they are facet defining for 3 4 2 Facet Defining Inequality for Case 1 Previously, in this chapter we conjectured that of case 1 is given by: (3 3 5 ) (3 3 6 ) (3 3 7 ) (3 3 8 ) (3 3 9 ) (3 4 0 ) (3 4 1 ) where, , , PAGE 67 67 Similar to the previous section, we fir st prove that Inequalities (3 3 5), (3 3 6 ), (3 3 7 ), (3 3 8 ), (3 3 9 ), (3 4 0 ), and (3 4 1 ) are valid. Then we proceed to prove that they are facet defining inequalities for It is easily verified that is full dimensional when W hen which shows that is not full dimensional Proposition 3 4 2 1 : Inequalities (3 3 8), (3 3 9), (3 4 0 ), and (3 4 1 ) are valid for Proof: Inequalities (3 3 8), (3 3 9), (3 4 0), and (3 4 1 ) are valid for since they belong to the initial description of P roblem ( 3 1 5 ) Proposition 3 4 2 2 : Inequality (3 3 5 ) is valid for Proof: Consider any feasible solution We argue that satisfies Assume for a contradiction that T hen a contradiction to the feasibility of Therefore, is valid for Proposition 3 4 2 3 : Inequality (3 3 6 ) is valid for Proof: Consider any feasible solution of i.e where It can be easily verified that and for any We consider 3 subcases: Case 1: For Inequality (3 3 6 ) reduce s to PAGE 68 68 which is satisfied by any feasible solution. Case 2 : For Case 2 a: We assume first that and Then which shows that satisfies this inequality Case 2 b: A ssume next that and Then, i t is easy to verify that for any feasible solution if and if Then which shows that satisfies this inequality. Case 2c : A ssume that and Then This shows that satisfies this inequality. Case 2d : A ssume that and Then it is easy to verify that It follows that PAGE 69 69 showing that satisfies this inequality. By considering the cases described above, we have shown that all fe asible solutions to Inequality (3 3 6 ) i.e, Inequality (3 36) is valid for Because and any solution satisfying : also satisfies (S P) Since we know that for all it follows from our study of case 0 that any solution to ( SP) also satisfies Proposition 3 4 2 4 : Inequality (3 3 7) is valid for Next, we show that (3 3 5), (3 3 6), (3 3 7), (3 3 8), (3 3 9), (3 4 0 ), and (3 4 1 ) are facet defining inequalities for In the proofs below we let be any inequality that induces a fa ce in dimensional space that is contains the face of under consideration. Proposition 3 4 2 5 : I nequality (3 3 5 ) is facet defining for Proof: Let be For all ( ) is a feasible solution that satisfies at equality and therefore must belong to the face yielding (1) F or all , and the feasible solution ( ) satisfies PAGE 70 70 at equality and therefore must belong to t he face yielding (2 ) It follows from (1) and (2) that and therefore the face is defined by For the feasible solution ( ) satisfies at equality and therefore must belong to the face yielding (3) and so the face is defined by For any the feasible solution s ( ) satisfy at eq uality and therefore must belong to the face yielding (4) It follows from (3) and (4) that T he face is therefore defined by proving that ( 3 3 5 ) is facet defining inequality for Propo sition 3 4 2 6 : I nequality (3 3 6 ) is facet defining for Proof [sketch] : In Proposition 3 4 1 3 we proved that inequalities are facet defining. We follow the same steps to prove that inequalities described in (3 3 6 ) are facet defining with only one difference, which is the coefficient of Below, we discuss how to obtain as the coefficient of instead of PAGE 71 71 Similar to Proposition 3 4 1 3 we reduce to the form Now, assume that and consider the feasible solution ( ) that s atisfies at equality. This point implies that The proofs that Inequalities (3 3 8), (3 3 9), (3 4 0), and (3 4 1 ) are facet defining for are similar to those we presented in Proposition s 3 4 1 4 3 4 1 5 3 4 1 6 and 3 4 1 7 respectively. 3 4 3 Convex Hull Proof for Case 0 We define (3 4 2) We denote the convex hull of by We also define (3 4 3) We next prove that provides a polyhedral description of Proposition 3 4 3 1 : Proof: We have shown earlier that Inequalities (1), (2), and (3) are valid for It follows that Since is convex, we conclude that PAGE 72 72 We now show that to complete the proof. To this end, we define the following two optimization prob lems (3 4 4) and (3 4 5) Observe that is a mixed integer nonlinear program and is a linear program. To show that we show that for all choices of objective coefficients in and We differentiate two cases. Case 1: Assume that there exist s such that and In this case, it is easy to verify that the solution is an optimal solution for It is feasible since and PAGE 73 73 It is optimal since is uses only variables with nonpositive objective c oefficients. Therefore, the optimal value of is equal to We next prove that First observe that every feasible solution satisfies (3 4 6) (3 4 7) (3 4 8) (3 4 9) Then define If we multiply (3 4 6) by for all (3 4 7) by for all (3 4 8) by for all and (3 4 8 ) by for all and sum the resulting inequalities, we obtain the valid inequality for : (3 5 0) PAGE 74 74 Since every feasible solution of satisfies (3 5 0), we obtain that Case 2: Assume that for all either or In this case, the natural solution is not directly usable since it is not feasible as It is easy to see that it is sufficient to modify one of the above term in order to obtain an optimal so lution to The term to modify is clearly the one for which the change in objective value is smallest. We define From our assumption in Case 2 For the value of Making this term feasible would require changing it to resulting in an increase of objective value of PAGE 75 75 For the value of Making this term feasible would require changing it to resulting in an increase of objective value of For the value of Making this term feasible would require changing it to resulting in an increase of objective value of Clearly, we want the smallest increase in objective value. This is obtained by finding any index as follows: (3 5 1) An optimal solution to the problem is therefore given by for and The optimal value of is therefore equal to We next prove that by consider ing 3 subcases. PAGE 76 76 Case 2a: Assume that Since variables can be reordered, we may assume that , and We define such that Because of the definition of in (3 5 1) we have that (3 5 2) ( 3 5 3) (3 5 4) C onsider now the following valid inequalities for for (3 5 5) (3 5 6 ) (3 5 7 ) (3 5 8 ) (3 5 9 ) F or c onstraint (3 5 5) define the following weights : for for for which are positive because of the definition of and because are sorted. Now, u sing these positive weights on (3 5 5). We obtain (3 6 0) PAGE 77 77 Now multipl y constraint s (3 5 8) for with positive weights and add to (3 6 0). Also multiply constraint s (3 5 6) for with positive weights and add to (3 6 0) to obtain (3 6 1) Now multiply constraint s (3 5 6) for with positive weights and add to (3 6 1) to obtain (3 6 2) Next multiply constraint s (3 5 6) for with positive weights and add to (3 6 2) to obtain (3 6 3) Then multiply constraint s (3 5 9) for with positive weights and add to (3 6 3) to obtain (3 6 4 ) Next multiply constraint s (3 5 8) for with positive weights and add to (3 6 4 ) to obtain (3 6 5 ) Finally multiply constraint s (3 5 7) for with positive weights and add to (3 6 5) to obtain PAGE 78 78 (3 6 6) Inequality (3 6 6) shows that every feasible solution to satisfies i.e., Case 2b: Assume that Since variables can be reordered, we may assume that , and We define such that Because of the definition of in (3 5 1) we have that (3 6 7) (3 6 8) (3 6 9) C onsider the following valid inequalities for for (3 7 0) (3 7 1) (3 7 2) (3 7 3) (3 7 4) F or constraint (3 7 0) define the following weights : PAGE 79 79 for for for which are positive because of the definition of and because are sorted. Now, using these positive weights on (3 7 0). We obtain (3 7 5) Now multiply constraint s (3 7 1 ) for with positive weights and add to (3 7 5 ). Also multiply constraint s (3 7 3 ) for with positive weights and add to (3 7 5 ) to obtain (3 7 6 ) or equivalently, (3 7 7) since is positive. Now multiply constraint s (3 7 3 ) for with positive weights and add to (3 7 7 ) to obtain (3 7 8 ) Next multiply constraint s (3 7 4 ) for with positive weights and add to (3 7 8 ) to obtain PAGE 80 80 (3 7 9) Multiply constraint s (3 7 3 ) for with positive weig hts and add to (3 7 9 ) to obtain (3 8 0) Finally multiply constraint s (3 7 2 ) for with positive weights and add to (3 8 0 ) to obtain (3 8 1) Inequality (3 8 1) shows that every feasible solution to satisfies i.e., Case 2c: Assume that Since variables can reordered, we may assume that , and We define such that Because of the definition of in (3 5 1) we have that (3 8 2) (3 8 3) (3 8 4) PAGE 81 81 C onsider the following valid inequalities for for (3 8 5) (3 8 6) (3 8 7) (3 8 8) (3 8 9) F or constraint (3 5 5) define the following weights: for for for which are positive because of the definition of and because are sorted. Now, usin g these positive weights on (3 8 5). We obtain (3 9 0) Now multiply constraint s (3 8 8) for with positive weights and add to (3 9 0). Also multiply constraint s (3 8 6) for with positive weights and add to (3 9 0) to obtain (3 9 1) or equivalently, PAGE 82 82 (3 9 2) since is po sitive. Now multiply constraint s (3 8 6) for with positive weights and add to (3 9 2 ) to obtain (3 9 3) Next multiply constraint s (3 8 6) for with positive weig hts and add to (3 9 3 ) to obtain (3 9 4 ) Multiply constraint s (3 8 9) for with positive weights and add to (3 9 4 ) to obtain (3 9 5) Multiply constraint s (3 8 8) for with positive weights and add to (3 9 5 ) to obtain (3 9 6) Finally multiply constraint s (3 8 7) for with positive weights and add to (3 9 6 ) to obtain (3 9 7 ) PAGE 83 83 Inequality (3 9 7) shows that every feasible solution to satisfies i.e., This concludes the pro of. Proposition 3 4 3 1 provides a proof of convex hull for integer bilinear covering sets in case 0 We believe the polyhedral characterization for case 1 given in (3 3 5) (3 4 1) defines the convex hull for this case also. We do not however have a proof of this result at this time PAGE 84 84 CHAPTER 4 SUMMARY In this thesis, we studied the polyhedral structure of an integer bilinear covering set which appears in the formulation of several practical problems in optimization including staff scheduling. D uring the study, we established that the polyhedral structure of this covering set depends strongly on some of its parameters We conjectured polyhedral description of the convex hull for two different cases We then investigated these conjectures In part icular, for these two cases, we proved that several families of inequalities are facet defining For one of the cases, we also showed that these inequalities are sufficient to describe the convex hull. Proving that the inequalities we derived for case 1 a re sufficient to describe the convex hull in this case is a direct avenue of research in studying these covering sets. Developing polyhedral descriptions of the convex hull for the remaining cases is an other interesting direction of research. For these other cases we believe it will typically be hard to uncover a generic form of the convex hull but some families of valid inequalities might be possible to describe PAGE 85 85 APPENDIX MATLAB CODES 1 For Case 0 d =3, n = 2 x = [0 0 0 0] y = [0 0 0 0] for i = 1 : 63 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) x(j) = 0 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) m = 3; else m = 1; end end y=[y;x]; end x = [1 3 1 3] PAGE 86 86 for i = 1 : 63 if (y(i,1)*y(i,2) + y(i,3)*y(i,4) >=3) x = [x;y(i,:)] end end 2 For Case 0 d =4, n = 2 x = [0 0 0 0] y = [0 0 0 0] for i = 1 : 99 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) x(j) = 0 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) m = 4; else m = 1; end PAGE 87 87 end y=[y;x]; end x = [1 4 1 4] for i = 1 : 9 9 if (y(i,1)*y(i,2) + y(i,3)*y(i,4) >=4) x = [x;y(i,:)] end end 3 For Case 0 d =4, n = 3 x = [0 0 0 0 0 0] y = [0 0 0 0 0 0] for i = 1 : 999 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) x(j) = 0 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) PAGE 88 88 m = 4 ; else m = 1; end end y=[y;x]; end x = [1 4 1 4 1 4 ] for i = 1 : 999 if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=4 ) x = [x;y(i,:)] end end 4 For Case 1 d =4, n = 3, u 1 = 3 x = [0 0 0 0 0 0] y = [0 0 0 0 0 0] for i = 1 : 799 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) x(j) = 0 PAGE 89 89 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) if(j==2) m = 3; else m = 4; end else m = 1; end end y=[y;x]; end x = [1 3 1 4 1 4] for i = 1 : 7 99 if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i ,5)*y(i,6) >=4) x = [x;y(i,:)] end end PAGE 90 90 5 For Case 1 d =4, n = 3, u 1 = 2 x = [0 0 0 0 0 0] y = [0 0 0 0 0 0] for i = 1 : 599 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) x(j) = 0 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) if(j==2) m = 2; else m = 4; end else m = 1; end end y=[y;x]; end PAGE 91 91 x = [1 2 1 4 1 4] for i = 1 : 599 if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=4) x = [x;y(i,:)] end end 6 For Case 1 d =5, n = 3, u 1 = 4 x = [0 0 0 0 0 0] y = [0 0 0 0 0 0] for i = 1 : 1439 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) x(j) = 0 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) if(j==2) m = 4; else PAGE 92 92 m = 5; end else m = 1; end end y=[y;x]; end x = [1 4 1 5 1 5] for i = 1 : 1439 if (y(i,1)*y(i, 2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=5) x = [x;y(i,:)] end end 7 For Case 1 d =5, n = 3, u 1 = 3 x = [0 0 0 0 0 0] y = [0 0 0 0 0 0] for i = 1 : 1151 x(1) = x(1) + 1 j = 1 m = rem(j,2) while (x(j) > m) PAGE 93 93 x(j) = 0 x(j+1) = x(j+1) +1 j = j + 1 k = rem(j,2) if (k == 0) if(j==2) m = 3; else m = 5; end else m = 1; end end y=[y;x]; end x = [1 3 1 5 1 5] for i = 1 : 1151 if (y(i,1)*y(i,2) + y(i,3)*y(i,4) + y(i,5)*y(i,6) >=5) x = [x;y(i,:)] end end PAGE 94 94 LIST OF REFERENCES [1 ] Al Khayyal, Faiz A. and Falk James E. (1983). Jointly Constrained Biconvex Programming. Mathematics of Operations Research 8 273 286. [2 ] Balas, E. (1997). Recognizing Facet Defining Inequalities. Acta Mathematica Vietnamica 22 7 28. [3 ] Boyd, Stephen and Vandenberghe Lieven (2 004). Convex Optimization Cambridge University Press, New York [4 ] Gadsby G. Neville (1965) The Army Operational Research Establishment Oper. Res 16 5 18. [5 ] Hillier, Frederick S. and Lieberman, Gerald J. (2004). Introduction to Operations Re search eighth edition. McGraw Hill, Inc., New York. [6 ] Horst, Reiner and Tuy, Hoang. (1990). Global Optimization Springer Verlag, Berlin. [7 ] Laderman, Julian D. (1976). A Noncommutative Algorithm for Multiplying 3 x 3 Matrices Using 23 Multiplication s. American Mathematical Society 82 126 128 [8 ] Luenberger David G. and Ye Yinyu. (2007). Linear and Nonlinear Programming third edition. Springer Science+ Business Media, LLC New York [9] McCormick G. P (1976). Computability of global solutions to factorable nonconvex programs: Part I. Convex underestimating problems. Mathematical Programming 10 147 175. [10 ] Strassen, Volker. (1969). Gaussian Elimination is not Optimal Numer. Math 13 354 356. [11] Tawarmalani, M. and Sahinidis N. V. (20 02). Convexification and Global Optimization in Continuous and Mixed Integer Nonlinear Programming Kluwer Academic Publishers [12] Winston, Wayne L. (1995). Introduction to Mathematical Programming second edition. Wadsworth Publishing Company, Californ ia. [13] Ziegler Gnter M. (1995). Lectures on Polytopes Springer Verlag, New York. PAGE 95 95 BIOGRAPHICAL SKETCH Nitish Garg was born in Hisar India in 1987 Nitish received a Bachelor o f Technology in Production and Industrial Engineering from Indian Insti tute of Technology, Delhi, India in 2009 Nitish joined the Department of Industrial and Systems Engineering at University of F lorida in August 2009. In summer 2010 Nitish participated in a research project with Operations Research Group at London School of Economics and Political Science (LSE) where he worked under guidance of Prof. Gautam Appa. At LSE, his work involved devising an algorithm to compute Mutually Orthogonal Latin Squares and implementing it with CPLEX CP Optimizer. Since August 2010, Nit ish has been working with Dr. Jean Philippe P. Richard. Upon completion of his degree, Nitish will start working for Bloomberg L.P. as a Financial Software Engineer. His current interests include m athematics o ptimization and a lgorithms 