Title: Optima
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00090046/00027
 Material Information
Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: August 1989
 Record Information
Bibliographic ID: UF00090046
Volume ID: VID00027
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.


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N 27


years have been the establishment of Mathematical
Programming Series B and the reduction from three
volumes (nine issues) to two for MPA. As a result of
the considerable efforts of Michel Balinski, Jan Karel
Lenstra, Bill Pulleyblank and Laurence Wolsey and
their negotiations with North-Holland, I believe we
have a much more rational structure for the Society's
publications, with more control over quality and
scheduling for MPB than with the Studies and a more
realistic frequency of publication for MPA, with no loss
to the members.
The editorial board consists of distinguished and
dedicated members of the mathematical programming
community and reflects the international nature of the
Society. We have two Co-Editors from Europe and two
from the U.S. Of the 26 associate editors, 12 reside in
the U.S., eight in Europe, three in Canada, and one
each in Japan, South America and the Soviet Union.
MPB has eight associate editors, including three who
also serve on the editorial board of MPA, as does its
Editor-in-Chief, Bill Pulleyblank. It is clear that there
should be a close relationship between the Editors-in-
Chief of the two publications to allow the flexibility of
transfers when appropriate and agreed to by the
Two special issues have appeared in MPA, 35(2) on
probabilistic analysis of the simplex algorithm and
41(2) on nonconvex optimization. Both were published
in order to alleviate scheduling difficulties (lack of
material for MPA and backlogs for the Studies) and


Report on



Series A





--l-W9110 RY -MP011

with the new structure I do not antici-
pate the need for future special issues in
The quality of the journal remains high,
and the mix of papers appears to be
roughly constant. For example, in the
nine issues in 1987, a crude characteri-
zation reveals 14 papers each in nonlin-
ear programming algorithms and com-
binatorial optimization, 13 in nonlinear
programming theory, 10 in complemen-
tarity and homotopy theory and
methods, and seven on linear program-
ming. The comparative figures for 1986
were 17, 15, 10 and 19 (including the
special issue on probabilistic analysis of
the simplex method). We are attracting
a reasonable number of good papers in
combinatorial and integer optimization
(with strong competition from Combi-
natorica, Discrete Mathematics,
Journal of Combinatorial Theory and
the computer science journals) and
computationally oriented nonlinear
programming (competing with SIAM's
journals on Numerical Analysis
(SINUM), and on Control and Optimi-
zation (SICOPT), and with the Journal
of Optimization and Applications
(JOTA)). As always we have stiff
competition from Mathematics of
Operations Research on more theoreti-
cal papers and Operations Research
and Management Science on more
applied papers. (While the division
above is on methodological lines, we
publish a small but reasonable number
of good applications.) Finally, we have
been able to attract some good papers
related to parallel computation, and we
are in excellent shape with regard to
articles on new linear programming
methods, with three of the five papers
on standard-form variants of the
projective method, five of the six on
path-following methods, etc.

It was pointed out at the Council
meetings in Tokyo that the SIAM
journals SINUM and SICOPT have
become if anything less serious com-
petitors to MPA; SINUM has moved
more towards differential equations,
SICOPT to control theory. Perhaps
closer are SISSC, the SIAM Journal on
Scientific and Statistical Computing,
and certainly the new Journals on
Discrete Mathematics and on Matrix
Analysis and Applications. Further-
more, SIAM is splitting SICOPT and
developing a new journal devoted to
I have heard from a number of mem-
bers that they are concerned that the
Society and its publications are chang-
ing their emphasis more and more
away from continuous and towards
discrete optimization. I do not believe
this is the case and so informed the
members; however, there is certainly a
perception among some that the
balance has changed considerably. I
believe it is important that this balance
be maintained and seen to be main-
We receive about 200 papers each year.
Of the 212 received in 1986, 68 (31%)
were accepted, 114 (54%) rejected or
withdrawn, and 32 (15%) are still active.
For 1987, the figures are 37 (19%)
accepted, 71 (35%) rejected/withdrawn,
and 94 (46%) in process, out of a total of
204. In 1988, we received 201 papers
and so far in 1989 we have 89 papers.
The current backlog is about nine
Overall, I judge that the quality of MPA
remains very high and that the new
structure has alleviated the pressing
problems of the past. I would like to
wish Bob Bixby all the best as the new

Vol.44, No.2

F. Barahona, M. Junger and G. Reinelt,
"Experiments in Quadratic 0-1
Y. Crama, "Recognition Problems for
Special Classes of Polynomials in 0-1
Y. Ye and E. Tse, "An Extension of
Karmarkar's Projective Algorithm for
Convex Quadratic Programming."
A. Sassano, "On the Facial Structure of
the Set Covering Polytope."
M.E. Dyer and A.M. Frieze, "A
Randomized Algorithm for Fixed-
Dimensional Linear Programming."
W. Kern, "A Probabilistic Analysis of the
Switching Algorithm for the Euclidean
S. Wright, "An Inexact Algorithm for
Composite Nondifferentiable
D. deWerra, "Generalized Edge Packings."

Reporting Guidelines

The draft report of an ad hoc committee on
the guidelines for reporting computational
experiments has been published in the
COAL newsletter dated March, 1989, and
sent to all Society members. The report
reviews existing guidelines and discusses
the issues of performance claims, measure-
ments of performance and computational
testing on the new computer architectures.
Researchers in computational mathemati-
cal programming are urged to study the
report and communicate their comments to
the committee: Richard H. F. Jackson
(NIST), Chair; Paul T. Boggs (NITS),
Stephen G. Nash (George Mason) and
Susan Powell (London School of Econom-
ics). Full addresses are given in the report.

---- --~------


number twenty-seven


Edilor-in-Chiet: Peter L. Hammer, RuIcor. Hill Center for the Mathcmatical Sciernes.
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\ ol S-9: Albin. S L and C \1 Harmr, '[rilr-ria[l 3rd
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Price per %olumi incl. poaIe: 5 153.61. or S 114.411 fiir menhbers, (IRS%. Plra.e reque't e\teni'le prriipeclu for while
series: mil. 1-21. 1984-1989
Hluo to order:
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10 H408Bses1teln


number twenty-seven

Conference Notes

Mathematical Sciences Institute
Workshop Announcement
Cornell University
Ithaca, New York
October 19-20, 1989

The Mathematical Sciences Institute (MSI) at
Cornell University is sponsoring a work-
shop on Large-Scale Numerical Optimiza-
tion. This workshop will discuss recent
algorithmic and software developments in
numerical optimization with a special focus
on large-scale problems. Particular empha-
sis will be on practical methods, specific
applications, and parallel computation. In
addition, recent advances in parallel
methods for sparse linear systems will be
considered and discussed with respect to
their relevance for large-scale optimization.
Approximately 18 half-hour, invited talks
will be delivered by leading researchers in
the field. There will be no contributed talks.
The workshop will conclude with a discus-
sion session on the topic, "What should
optimizers do with parallelism?"

Published proceedings will be .w\.ill., bl.
soon after the conclusion of the workshop
which will be held at Cornell immediately
following the ORSA-TIMS meeting in New
York City.
For more information on the scientific
content contact:
Tom Coleman
Department of Computer Science
311a Upson Hall
Cornell University
Ithaca, NY 14853
(607) 255-9203
Yuying Li
Department of Computer Science
311b Upson Hall
Cornell University
Ithaca, NY 14853
(607) 255-9203
To attend the workshop, contact MSI at 201
Caldwell Hall, Cornell University, Ithaca,
NY 14853-2602, (607)255-7740, 8005, or

Computational Aspects of Combinatorial Optimization
January 9-13, 1989

The conference was organized by R. E.
Burkard (Technical University of Graz)
and M. Gr6tschel (University of
Augsburg). The participants came from
13 countries and presented (in 52 talks)
new results on the following topics:
* Generalized traveling salesman and
routing problems;
" Design of survivable networks;
* New algorithms for network flows;
* Combinatorial problems in VLSI-
* Solving NP-hard problems on
supercomputers or on distributed

* Scheduling problems;
* Probabilistic analysis of simple
Several participants provided a
demonstration of their software
packages, featuring:
a Linear programming codes;
* Algorithms on graphs;
* Codes for scheduling problems;
* A CAM-system for manufacturing.
The unique setting of the research
institute in Oberwolfach was, as usual,
very inspiring for all the participants
and contributed considerably to the
success of the conference.


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s--~- ~-~------

PAGE,5 number twent-seve AUGUST,1989

Technical Reports &

Working Papers

Northwestern University
Department of Industrial
Engineering and Management
Evanston, IL 60208

S. Mehrotra and J. Sun, "A Method of
Analytic Centers for Quadratically
Constrained Convex Quadratic Programs,"
TR 88-01.
S. Mehrotra, "A Method for Solving Piece-
Wise Linear Programs by Shrinking
Polytopes," TR 88-04.
J.T. Simon and W.J. Hopp, "Availability
and Average Inventory of Balanced
Assembly-Line Flow Systems," TR 88-05.
E.S. Theise and P.C. Jones, "Alternative
Implementations of a Diagonalization
Algorithm for Multiple Commodity Spatial
Price Equilibria," TR 88-06.
N. Pati and W.J. Hopp, "Optimal
Inventory Control in a Production Flow
System with Failures," TR 88-07.
S. Mehrotra and J. Sun, "An Interior
Point Algorithm for Solving Smooth
Convex Programs Based on Newton's
Method," TR 88-08.
W-L. Hsu and W-K. Shih, "An
Approximation Algorithm for Coloring
Circular-Arc Graphs," TR 88-09.
W-L. Hsu and W-K. Shih, "An
0(minlm*n, n2loglog n]) Maximum Weight
Clique Algorithm for Cicular-Arc Graphs,"
TR 88-10.
S. Mehrotra and J. Sun, "On Computing
the Center of a Quadratically Constrained
Set," TR 88-11.
W-L. Hsu and W-K. Shih, "An 0(N15)
Algorithm to Color Proper Circular-Arc
Graphs," TR 88-12.
E.S. Theise and P.C. Jones, Thi ty

Linear, Single Commodity Spatial Price
Equilibrium Problems and Their
Solutions," TR 88-13.
E.S. Theise and P.C. Jones, "A
Computational Comparison Between an
Import Equilibration Algorithm and the
Expanding Equilibrium Algorithm for the
Linear, Single Commodity Spatial Price
Equilibrium Problem," TR 88-14.
E.S. Theise and P.C. Jones, "Nonlinear,
Single Commodity Spatial Price Equilibria
and the Expanding Equilibrium Algorithm:
Two Strategies for Implementation,"
TR 88-16.
R.R. Inman and P.C. Jones, "Economic
Lot Scheduling of Bottlenecks with External
Setups," TR 88-17.
M.L. Spearman, "An Analytic Congestion
Model for Closed Production Systems,"
TR 88-23.

Rutgers Center for Operations
Hill Center
New Brunswick, New Jersey 08903

L.J.Billera and L.L.Rose, "Gribner Basis
Methods for Multivariate Splines,"
RRR 1-89.
M.H. Rothkopf, T.J. Teisberg and E.P.
Kahn, "Why are Vickrey Auctions Rare?"
RRR 2-89.
A.S. Manne and M.H. Rothkopf,
"Analyzing U.S. Policies for Alternative
Automotive Fuels," RRR 3-89.
P. Hansen, B. Jaumard, S-H. Lu, "An
Analytical Approach to Global
Optimization," RRR 4-89.
J. Kahn and R. Meshulam, "On mod p

Transversals," RRR 5-89.
F. Harary, S. Kim and F.S. Roberts,
"Extremal Competition Numbers as a
Generalization of Turan's Theorem,"
RRR 6-89.
S.D. FlAm, "On Finite Convergence and
Constraint Identification of Subgradient
Projection Methods," RRR 7-89.
M. Zheng and X. Lu, "On the Maximum
Induced Forest of a Connected Cubic Graph
without Triangles," RRR 8-89.
B. Avi-Itzhak and S. Halfin, "Response
Times in Gated M/G/1 Queues: The
Processor-Sha-Ring Case," RRR 9-89.
P. Hansen, B. Jaumard and 0. Frank,
"An 0(N2) Algorithm for Maximum Sum-
of-Splits Clustering," RRR 10-89.
R.P. McLean, "Random Order Coalition
Structure Values," RRR 11-89.
P.L. Hammer, U.N. Peled and X. Sun,
"Difference Graphs," RRR 12-89.
P.L. Hammer, N.V.R. Mahadev and
U.N. Peled, "Bipartite Bithreshold
Graphs," RRR 13-89.
E. Boros, Y. Crama and P.L. Hammer,
"Upper Bounds for Quadratic 0 1
Maximization," RRR 14-89.
E. Boros and P.L. Hammer, "A Max-
Flow Approach to Improved Roof Duality
in Quadratic 0 1 Minimization,"
RRR 15-89.
P.L. Hammer, F. Maffray and M.
Preissmann, "A Characterization of
Chordal Bipartite Graphs," RRR 16-89.
P. Hansen, B. Jaumard and G. Savard,
"A Variable Elimination Algorithm for
Bilevel Linear Programming," RRR 17-89.

University of Southern California
Department of Industrial and
Systems Engineering
Los Angeles, CA 90089-0193

B.C. Tansel and E. Erkut, "On
Parametric Medians of Trees," 88-01.
B.C. Tansel and G.F. Scheuenstuhl,
"Facility Location on Tree Networks with
Imprecise Data," 88-03.



number twenty-seven

PAGE 6 number twenty-seven AUGUST 198

A.S. Kiran and P. Kouvelis, "The Plant
Layout Problem in Automated
Manufacturing Systems," 88-04.
A.S. Kiran, "A Tardiness Heuristic for
Scheduling Flexible Manufacturing
Systems," 88-06.
G. Nadler, J.M. Smith and C.E. Frey,
"Problem Formulation Methods in
Engineering Design," 88-08.
A.S. Kiran, "A Combined Heuristic
Approach to Dynamic Lot Sizing
Problems," 88-09.
M.H. Chignell and R.G. Narayan, "An
Empirical Evaluation of Efficient Ranking
Methods," 88-10.
G.F. Scheuenstuhl and B. Tansel, "Tree
Network Facility Location with Normal
Random Demands," 88-20.
E. Balas and S.M. Ng, "On the Set
Covering Polytope: II. Lifting the Facets
with Coefficients in (0,1,2,3)," 88-21.
A.S. Kiran and S. Karabati, "The Station
Location Problem on Unicyclic Material
Handling Networks," 88-25.

Operations Research Group
The Johns Hopkins University
Baltimore, MD

M.H. Schneider, "Matrix Scaling,
Entropy Minimization, and Conjugate
Duality (I): Existence Conditions," 89-02.
M.H. Schneider, "Matrix Scaling,
Entropy Minimization, and Conjugate
Duality (II): The Dual Problem," 89-03.
H. Schneider and M.H. Schneider,
"Max-Balancing Weighted Directed
Graphs," 89-04.
R.D. Parker, "Calculating the Weights of a
Mask for Character Recognition," 89-05.
H. Schneider and M.H. Schneider,
"Towers and Cycle Covers for Max-
Balanced Graphs," 89-06.
J.R. Current, C.S. ReVelle and J.L.
Cohon, "An Interactive Approach to
Identify the Best Compromise Solution for
Two Objective Shortest Path Problems,"

W. Cook, M. Hartmann, R. Kannan and
C. McDiarmid, "On Integer Points in
Polyhedra," 89-08.
V.A. Hutson and C.S. ReVelle,
"Maximal Direct Covering Tree Problems,"
J-S. Shih and C.S. ReVelle, "Hedging
Rules for the Single Water Supply
Reservoir," 89-10.

University di Pisa
Dipartimento di Matematica
Sezione di Matematica Applicata
Gruppo di Ottimizzazione e
Ricerca Operativa
Pisa, Italy

J. Naumann, "Existence of Lagrange
Multipliers in Classical Calculus of
Variations," 150.
L. Pellegrini, "An Extension of Hestenes
Necessary Condition for Nondifferentiable
Constrained Extremum Problems," 151.
L. Favati, "Generalizzazione del Modello di
Mossin-Kupperman-Lisei per il Controllo
Ottimale," 152.
M. Pappalardo, "A Priori Bounds for
Strongly Convex Nondifferentiable
Extremum Problems," 153.
L. Martein, "An Approach to Lagrangian
Duality in Vector Optimization," 154.
K.-H. Elster, "Generalized Notions of
Directional Derivatives," 155.
D.T. Luc, "A Theorem of the Alternative
and Axiomatic Duality in Mathematical
Programming," 156.
O. Ferrero, "On a Property of the
Generalized Subdifferential," 157.
P. Favati, F. Tardella, "A Notion of
Convexity for Functions Defined Over the
Integers," 158.
G. Finke, E. Medova-Dempster,
"Combinatorial Optimization Problems in
Trace Form," 159.
E. Medova-Dempster, "The Circulant
Traveling Salesman Problem," 160.
L.F. Escudero, "On Solving a
Nondifferentiable Transshipment Problem,"






number twoenty-seven

PAE7nme wny-ee UUT18

Combinatorics of Experimental Design
A. P. Street and D. J. Street
Oxford University Press, Oxford, 1987
ISBN 0-19-853255-5
The book under review is intended as an introductory text (aimed at
3rd and 4th year undergraduates in both mathematics and statistics) on
the combinatorial and statistical aspects of Design Theory. To quote
from the introduction: 'There is an obvious dichotomy in the literature
of designs; they are considered as incidence structures by combinatori-
alists and as experimental plans or layoutsbystatisticians, and members
of each of these groups are sometimes unaware of related developments
and problems arising in the other area. We aim to bridge this gap by
providing the background necessary to make the combinatorial aspects
of statistical literature more easily accessible to combinatorialists, and
vice versa." Consequently, the book contains parts where the emphasis
is on combinatorics as well as parts where the statistical aspect domi-
nates. Unfortunately, there is quite often the feeling of a rather abrupt
transition between both points of view; thus Iam not quite sure whether
the authors have fully achieved their (difficult) goal. However, the book
certainly is quite interesting and worth studying.
Let me list the main topics covered as indicated by the titles of chap-
ters: 1. Introduction; 2. Balanced incompleteblock designs; 3. Difference
set constructions; 4. Isomorphism and irreducibility; 5. Latin squares
and triple systems; 6. Mutually orthogonal Latin squares; 7. Further
results on Latin Squares; 8. Resolvable designs and finite geometries; 9.
Symmetrical factorial designs; 10. Single replicate factorial designs; 11.
Designs with partialbalance; 12. Existence results: Symmetric balanced
designs; 13. Existence results: designs with index 1 and given block size;
14. Designs balanced for neighboring varieties; 15. Competition de-
signs. As this list indicates, the main concern is on the existence and con-
struction of various types of (pairwise) balanced designs; thus some
rather important basic topics like t-designs, automorphism groups,
characterizations, connections to coding theory have been (almost)
totally excluded. In viewof thevast amount of literature on designs, this
is justified for an introductory text, though a few more references to the
missing topics would have been welcome. Also, I would have liked to
have some other applications of Design Theory (except for statistical
ones) at least mentioned, in particular, those to computer science and
Still, the reader gets a
good introduction
at least to

aspects regarding block designs and Latin squares. The presentation is
generally clear and well-written. As always, one finds minor faults; e.g.,
the terminological confusion between difference families and sets is
annoying. The proof of the first multiplier theorem given in Ch. 4 is the
original involved one, even though a much more transparent approach
(due to Lander) is known now. In Ch. 7, the three mutually orthogonal
Latin squares of order 14 should not have been displayed explicitly, as
they are in fact constructed by a difference method which would have
simplified the presentation considerably. These imperfections are
balanced by some highlights not yet found in any other text book: e.g., a
proof of the sufficiency of the necessary conditions for triple systems
based on Latin squares or Stinson's proof forTarry's theorem (i.e., for the
non-existence of a pair of orthogonal Latin squares of order 6).
There are two other recent books on Design Theory: Another
introductory text by D. R. Hughes and F. C. Piper ("Design Theory,"
Cambridge UniversityPress, 1985) and one co-authored by the reviewer
(T. Beth, D. Jungnickel and H. Lenz: "Design Theory," Bibliographis-
ches Institut Mannheim, 1985, and Cambridge University Press, 1986)
which aims at graduates and experts in the area. All three books stress
quite different aspects: While Hughes and Piper also is of an introduc-
tory nature, this text emphasizes the algebraic aspects of designs (treat-
ing e.g. Witt designs and Mathieu groups) and quite neglects the
existence question. Thus this book and the one under review rather
nicely complement each other and together provide an introduction to
all the most important parts of Design Theory. As already mentioned,
myown (and my co-authors') efforts were more concerned with provid-
ing a somewhat deeper treatment and a reference for the expert working
in the area. Thus all three books serve their different purposes: If one
wants to specialize in Design Theory, all three books are needed; if one
only wants to get acquainted with designs, either the present text or that
by Hughes and Piper or both (depending on your personal preference
for a more constructive or more algebraic treatment) are well worth

Surveys in Game Theory and Related Topics
Edited by H. J. M. Peters and 0. J. Vrieze
CWI Amsterdam, 1987
ISBN 90-6196-322-2
This book is a collection of 13 survey papers on game
theory and related topics and was dedicated to Profes-
sor Stef Tijs of the Catholic University of
Nijmegen in The Netherlands on the occa-
sion of his 50th birthday. The forward tells
us the wonderful history of how he has
developed the Dutch school of game
theory, now one of the leading



ACIarall llrlIllrB




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P. PAGE 8 number twenty-seven AUGUST 198~

research groups in this field, since he finished his Ph.D. thesis, "Semi
infinite and infinite matrix games and bimatrix games," in 1975. All the
authors of the papers were introduced to game theoryby Professor Tijs,
and 10 of them have finished or are preparing their Ph.D. theses under
his supervision.
The papers cover a wide range of game theory and related topics:
equilibrium points in noncooperative games (by Eric van Damme and
by MathijsJansen), games with incomplete information (by Peter Borm),
stochastic games (by Koos Vrieze and by Frank Thuijsman), a relation-
ship between game theory and decision theory (by Peter Wakker),
cooperative games in characteristic function form (by Theo Driessen
and by Jean Derks), cooperative games arising from combinatorial and/
or linear optimization problems (by Imma Curiel and byJos Potters), the
bargaining theory (by Hans Peters) and the theory of social choice (by
Ton Storcken). All papers are clearly written and provide concise
surveys of recent developments in their respective topics. They also
include both new results by the authors themselves and useful refer-
ences. The readers can obtain a review of the state of the art in the areas
of game theory mentioned above and also can learn that many tools in
mathematical programming play an important role in those areas. Since
most papers emphasize mathematical aspects of the results such as the
proof methods for the existence of various solution concepts and their
computation, I think that the book should be accessible to many re-
searchers in the field of mathematical programming who have little
knowledge of game theory.
This book is recommended to researchers and graduate students
who are interested in recent developments in various fields of mathe-
matical game theory.

Recent Advances and Historical Development of
Vector Optimization

by J. Jahn and W. Krabs
Springer, Berlin, 1987
ISBN 3-540-18215-2
In August 1986, J. Jahn and W. Krabs organized an international
conference on vector optimization in Darmstadt, West Germany. Be-
sides four state-of-the-art tutorials, numerous talks cover various as-
pects and purposes of vector optimization, such as: abstract theory,
duality, sensitivity, numerical methods, parametric optimization, multi-
criteria-decision-making, application of MCDM, etc. Several talks are
collected in this proceedings. In the following, the state-of-the-art
tutorials of the Professors W. Stadler, J. M. Borwein, P. L. Yu, and H.
Eschenauer will be discussed in detail.
In his article "Initiators of Multicriteria Optimization," Stadler de-
scribes the historical development of vector optimization. A vector
optimization problem is a problem with several objective functions. In





general, there is conflict between them. Usually, one solves such
problems by introducing a so-called utility function such that a solution
of the problems with this utility function is a solution of the initial
problem. Such questions were first handled in economic theory. The
economists A. Smith (The Wealth of Nations, 1776), F. Edgeworth (The
Edgeworth Box in Mathematical Physics, 1881), and V. Pareto (Pareto-
optimality, 1906) can be said to be the founders of multicriteria optimi-
zation as an inherent part of economic equilibrium.
Stadler presents the fundamental statements in the work of Edge-
worth and Pareto for welfare theory. In their original papers, one can
already see the well-known scalarization of the weighted objective
functions. Instead of n objectives and m variables, they speak about n
consumers and m goods. The notion of "efficiency" occurs for the first
time in Koopmans' work on production theory in 1951.
Mathematically, as Stadler writes, the first formulation of a vector
optimization problem is due to Kuhn and Tucker. In their famous paper
from 1951, they give a necessary condition for "proper" solutions. The
first basic treatment of vector optimization can be found in Hurwicz's
paper in 1958 where he considered optimization problems in linear
In a conclusion section, some areas of future research are pointed
out: (1) vector optimization theory with respect to partial orders and
preorders; (2) development of computational algorithms to generate the
efficient point set; (3) multicritera aspects of natural phenomena; (4)
further applications.
This paperisvery interesting to read becauseof itsdetailed historical
information and the biographies and pictures of the founders of mul-
ticriteria optimization which are added to the article. In my opinion, the
only missing thing is a discussion about the general development and
movement in vector optimization after Hurwicz's paper to the present.
In the article "Convex cones, minimality notions, and consequences,"
Borwein presents vector optimization problems in arbitrary vector
spaces. In every section and subsection, he uses the following scheme:
(a) definitions, (b) properties and relations between the introduced
concepts, (c) theorems, and (d) examples and applications in some
special spaces as L and Qp. Most proofs are omitted (but references are
cited); some proofs are sketched.
The section on cone structures deals, among other things, with order
intervals, monotone sequences and nets, normalityof acone, the Daniell
property of a cone, the base of a cone, and Banach lattices. In every
subsection, Borwein shows the relationships between the introduced
concepts in complete detail. Thus, this section is a reference-book on
where to find conditions on whether a cone with property (A) has
property (B), or property (C) is equivalent to property (D).
Minimality notions are introduced in the next section: Pareto-
optimal (or efficient, non-dominated, minimal) points, least elements
(or strong minimum point, dominating point), weak-efficient point,
proper efficient point are optimal points of a set with respect to a given
cone. Figures illustrate the differences between these concepts. Some
existence theorems of efficient points and a characterization of proper

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efficiency are given.
Finally, Borwein outlines the theory of lattice complementary prob-
lems, i.e. to solve minK (x,F(x)) = 0 for F:X-X, X a Banach lattice, K a cone.
For instance, a standard linear programming pair can be rewritten as
such a problem.
The paper of Borwein treats the introduced concepts very compre-
hensively, but in my opinion it is a little bit too compact. More
discussion would be better for a reader who is not familiar with vector
optimization problems in abstract spaces. Nevertheless, the complete
discussion of the broached questions is impressive as was Borwein's
excellent talk at the conference.
In their article, "Foundations of Effective Goal Setting", Yu and
Chien give a readable, detailed introduction to the field of effective goal
setting. Their purpose is to formulate a complex multicriteria optimal
control system in which problems of effective goal setting can be
Usually in multicriteria decision making there is a fixed set of
objectives and alternatives, and the aim is to find "optimal" solutions.
But many decision problems have alternative sets and criteria functions
which are not fixed, but change, for instance, with time. For example, in
reaching a great goal one gives oneself a series of "increasing" goals for
For their purpose, the authors use the concepts of human behavior
mechanism and habitual domain: each individual is endowed with an
internal information processing and problem solving capacity and has a
set of goals to reach and maintain. In comparison between real and ideal
values, one tries to find goals and alternatives which produce great
charges and reduce the level of charges by selecting other alternatives or
by active problem solving or avoidance justification. This dynamic
behavior mechanism, although changing with time, can stabilize and
can have stable habitual patterns for processing information. This lead s
to habitual domains, divided into potential domains, actual domains,
and reachable domains. An essential role is played by the cores of the
habitual domains, i.e. the set of central ideas or concepts.
Utilizing these concepts, the authors can formulate problems of
effective goal setting into a complex multiple criteria optimal control
system by (a) selecting measurable goal functions, (b) setting goal
achievement levels, and (c) determining effective supportive systems as
control variables. The stated variables are working conditions, charge
structures and confidence; the objectives are to maximize the attention
allocation of time to job-related works, to maximize the efficiency and
effectiveness of work performance, and to maximize the favorability of
the working environment. Finally, some empirically known results are
In his paper, "Multicriteria Optimization Procedures in Application
on Structural Mechanics Systems," Eschenauer presents a computer
program package called SAPOP (Structural Analysis Program and
Optimization Procedure) to support a decision maker by solving vector
optimization problems for structural analysis. What objectives are to be
considered for these problems? Of course, a decision maker wants to


minimize the costs of developing and manufacturing machines. But
other criteria may shape accuracy and reliability of the systems, among
The program SAPOP coordinates the three main parts of the optimi-
zation process and the data exchange between them: (1) optimization
algorithms, (2) optimal modelling, and (3) structural analysis.
In part (1), SAPOP makes available a lot of optimization algorithms
because there is no procedure which is at the same time efficient and
applicable for all problems.
Part (3) is the starting point of every structural optimization prob-
lem. As Eschenauer indicates, this first step must be done very carefully
because the computation depends essentially on the quality of the
mathematical-mechanical model. Among these models one distin-
guishes between (a) ordinary differential equation models, (b) differ-
ence equation models, (c) partial differential equation models, and (d)
algebraic (non-difference) equation models.
Part (2) is the link between the other two parts. Here, strategies to
find efficient solutions were created. Some scalarizations (weighted
objective functions, distance functions, trade-off-method, min-max-
formulation) are discussed.
Finally, some applications and numerical results are quoted. Esch-
enauer describes how to find an optimal layout of a shell structure.
These problems arise, for instance, in the field of antenna and telescope
This paper gives a good discussion of problems of structural analysis
with several objectives and a brief introduction to the program SAPOP.
For deeper insight, references are cited.

Algorithmic Information Theory
by G. J. Chaitin
Cambridge Tracts in Theoretical Computer Science 1
Cambridge University Press, Cambridge, 1987
ISBN 0-521-34306-2
One way of stating certain famous theorems of G6del, Church, and
Turing is that there is a function f such that no computer program can
decide, for all natural numbers a, whether there is a natural number x
f(x) = a.
The f in these results typically examined an x intended to encode a
proof (or a computation) with f(x) being an encoding of the final result.
The functions f constructed in these results involved many definitions
by cases using one formula if x was even, another if
x = 4k + 1, and so forth. The search for results involving f with
"neater" definitions culminated in Matijasevic's 1970 solution of Hil-
bert's tenth problem: a polynomial in several variables f (x,a,) was





number twentty-sevuen

PAGE 0 numer tenty-evenAUGUS 198

constructed such that no program can tell, for all natural number vectors
a, whether there is a natural number vector x with f (x,a) = 0. The
construction is quite complex, in spite of simplifications by Davis,
Robinson, and others. In 1984, Jones and Matijasevic gave a much
simpler construction for f which included expressions involving expo-
The author presents this most recent construction and explores its
implications. No advanced results from number theory or theory of
equations are required. In order to establish results about computer
programs, one must have a precise definition. The author has chosen for
this purpose a version of LISP which he develops from scratch in twelve
pages. Perhaps fortunately, the reader unfamiliar with LISP can accept
on faith the construction of f and proceed to the half of the book dealing
with implications.
The proof of non-computability is different from the usual one. The
author obtains contradictionsby focussing on the smallest program that
will print a specified string. The author has done much work in which
finite strings which require long programs are considered to be pseudo-
random, and this aspect receives considerable attention here.
In both halves of the book, the author has taken the trouble to supply
motivational remarks and exhortations ("Initially, the material will
seem completely incomprehensible, but all of a sudden the pieces will
snap together into a coherent whole"). It would help the reader if he has
seen previous work by the author on these issues (for example, "Ran-
domness and Mathematical Proof," Scientific American, 1975).
Matijasevic's work was used in a paper by Jeroslow, "There Cannot
be any Algorithm for Integer Programs with Quadratic Constraints"
(Operations Research, 1973). I suspect that most Mathematical Pro-
gramming Society members are more interested in establishment of
lower bounds on the difficulty of problems for which computer pro-
grams exist. This book does not directly address such issues, but my
impression is that the type of reasoning developed here might help on
some problems of this kind.

Mathematical Programming: An Introduction to

Pure and Applied Mathematics Series
by Melvyn W. Jeter
Marcel Dekker, Basel, 1986
ISBN 0-8247-7478-7
Whenever a new textbook with a title like "Mathematical Program-
ming" or "Introduction to Optimization" is published, my first reaction
is usually a rather cynical comment on this (n+1)si[n --*] book of its
With Jeter's book my reaction was different. Students will enjoy
using this bookbecause of two main reasons: (1) Most o fthe mathemati-




number twenty-seven


cal facts which are stated are proved in a clear and understandable way.
There are hardly any of these "obviously" or "as can easily be seen"
sequences which scare so many of our students away; (2) Every detail of
the presented material is accompanied by worked examples and further
supported by exercises. But Jeter avoids the flaw of many mathematical
programming textbooks of replacing theory completely by examples.
Because of the thoroughness of the presentation of the chosen
material, the author evidently had to make some sacrifices in the
material selection. In Chapter 1 different types of mathematical pro-
grams are introduced. Chapter 2 reviews elementary linear algebra and
affine and convex sets. Furthermore, LPs and their basic properties are
introduced. Chapters 3, 4 and 5 cover various versions of the simplex
method including a chapter on duality and linear complementarity. The
cycling phenomenon is discussed, but I was surprised not to see Bland's
simple cycle avoiding rule.
The sixth chapter on network programming is somewhat disap-
pointing. The classic Ford/Fulkerson algorithm for finding maximal
flows is discussed without any reference to more efficient procedures.
Sections on network programming problems different from flow prob-
lems are missing.
Chapter 7 provides the mathematical tools needed in dealing with
convex functions of one or more variables. The last three chapters give
an overview of nonlinear, continuous programs. In Chapter 8 optimal-
ity conditions are discussed. Chapter 9 deals with search techniques for
unconstrained problems, and Chapter 10 introduces penalty methods.
As the preceding summary shows, most instructors will add supple-
mental material to various parts ofJeter's book. Since thebook is written
so nicely, one may in an advanced course actually concentrate on
supplemental material and assign large parts of Jeter's text as reading

Fractional Programming
by B. D. Craven
Heldermann Verlag, Berlin, 1988
ISBN 3-88538-404-3
The book deals with nonlinear programming problems where the
objective function is a ratio of two functions or involves even several
ratios. These so-called fractional programs often have properties which
they do not share with general nonlinear programs. A linear fractional
program is one where both numerator and denominator are affine-
linear and the constraints are linear. The book covers applications,
theory and algorithms for linear and nonlinear fractional programs.
In Chapter 1 several (potential) applications of fractional program-
ming are surveyed. These include planning problems in production,
scheduling, finance as well as stochastic programming and stochastic
processes. Chapter 2 is devoted to linear fractional programs where

number twenty-seven


equivalent programs and duality are discussed. Chapter 3 focuses on
the more general problem of maximizing the ratio of a concave and a
convex function. Equivalent problems and the relationship to general-
ized convexity are dealt with. Duality and sensitivity of nonlinear
fractional programs are presented in Chapter 4. In Chapter 5 the author
discusses algorithms in linear and nonlinear fractional programming.
The final chapter addresses three problems in multi-ratio fractional
programming: maximizing the sum of ratios, maximizing the smallest
of several ratios and multiobjective fractional programming.
Each chapter ends with exercises and a selective bibliography. The
book can serve as a textbook for students who are famr ni.,' with the
basics of linear and nonlinear programming and who are acquainted
with the fundamentals of linear algebra and calculus. The book is an
introduction to fractional programming rather than a detailed survey of
the extensive literature. But it reaches a depth that makes it attractive
also to the researcher in the field. It is the first book on fractional pro-
gramming that appeared after the initial monograph of the reviewer in
1978. I warmly recommend it to anyone interested in fractional pro-
gramming or general nonlinear programming.

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PAGE 2 numbr~ twnysvn UUT18


Kurt Anstreicher (Yale) will spend the 1989-90
academic year at CORE...Carl Harris (George
Mason University) has issued a call for
nominations for the 1988 Lanchester
SPrize...The IFORS '90 conference will be
held June 25-29, 1990 in Athens, Greece.
Jens Clausen has taken over production
and distribution of the COAL newsletter
and Faiz A. Al-Khayyal (Georgia Tech) is
the U. S. co-editor.
I OPTIMA Wine is available from Op-
tima Vineyards, Sonoma County, CA.
II Deadlinefor the next OPTIMA is Octo-
ber 1, 1989.


303 Weil Hall
College of Engineering
University of Florida
Gainesville, Florida 32611 USA



number t~wenty-seven


Books for review should be
sent to the Book Review Editor,
Prof. Dr. Achim Bachem,
Mathematiches Institute der
Universitfit zu Kiln,
Weyertal 86-90, D-5000 Kiln,
West Germany.

Journal contents are subject
to change by the publisher.

Donald W. Hearn, EDITOR

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