MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
AND~ D 1W~PEADLINES~
April 29, 1994
May 1, 1994
June 1, 1994
July 15, 1994
July 18, 1994
July 22, 1994
Last date to send
paper copy of abstract
Last date to send
abstract by e-mail
Last date for conference-
booked hotel or
D.- ,!i -.. to request
Fee for ... .. l I i
of Residence Hall
Copies of the Second Announcement have
been mailed to all MPS members. The
symposium coordinators' address is:
University of Michigan
Conferences and Seminars, Room 112
541 Thompson St.
Ann Arbor, MI 48109-1360 USA
Tel. (313) 764-5305
Fax: (313) 764-2990
Nominations for 1994 Elections
The Constitution of the Mathematical Pro-
gramming Society sets the term of office for
all officers of the Society at three years. Elec-
tions for all offices (Chairman, Treasurer and
four at-large members of the Council) are
held four months prior to each triennial
International Symposium. Therefore, the
next election will be held in April 1994. the
new members-at-large of the Council will
take office at the time of the symposium,
while the Chairman-elect andTreasurer-elect
will take office one year later.
Candidates must be members of the Society
and may be proposed either by Council or by
any six members of the Society. No proper
nomination may be refused, provided the
candidate agrees to stand. The procedure to
be followed is:
1 Nomination to any office is to be sub-
mitted to Jan Karel Lenstra, Chairman,
by April 1, 1994. Such nomination is
to be supported in writing by the
nominator and at least five other
members of the Society.
2 In keeping with tradition, the next
chairman preferably should be a North
American resident. The membership
is asked to consider no residents from
other continents to be Chairman.
3 When the ballots are counted, the four
at-large candidates for Council receiving
the highest number of votes will be elected,
except that not more than two members
having permanent residence in the same
country may be elected.
Jan Karel Lenstra, Chairman
Department of Mathematics and
Eindhoven University of Technology
P.O. Box 513, 5600 MB Eindhoven,
I'~~~`~~' ~--- ~~~--'~~~ ~~-- '----~
The 1thl Interna-
triennial scientific meet-
ing of the Mathematical
will be held Aug. I -19,
1994, on the central cam-
pus of the University of
Michigan at Ann Arbor,
Conference activities be-
gin on Sunday afternoon
Aug. 14 with a golf
game, planned for those
interested, at the univer-
sity golf course. Tee
times begin at 1 p.m.
The registration desk for
the symposium will be
on the second floor con-
course of the Michigan
League, and will be open
from noon to 8 p.m. on
Sunday. The desk will be
staffed Monday through
Friday from 7:30 a.m. to
5 p.m. The symposium's
inaugural ceremony will
begin Sunday at 8 p.m.,
with welcoming ad-
dresses and a musical
invocation by Pulitzer
William E. Bolcom.
The opening session on
Monday, Aug. 15 will be
held from 8:30-10:45 a.m.
in the Rackham Building.
William Cook of Bellcore
will address the state of
the art in large-scale
L.. -. i. rIl i!, -'C Z,-o-
t .i ., i J '.. I i .', r- . II
speal .,o, tl... ....:'nli.,-
tions i .,....,r i .n, .'lL
who will receive an hon-
orary award. The
Dantzig, Fulkerson and
prizes will be awarded
and the finalists for the
A.W. Tucker Prize will
Seminar sessions will be
held Monday through
Friday. The opening re-
ception for all registered
participants and their ac-
companying guests will
be held Monday in the
Michigan Union from
6:15-8 p.m. Refreshments
will be served.
One hour, state-of-the-
art tutorial and survey
lectures, dealing with
covering the broad spec-
trum of mathematical
programming and re-
lated areas, have been
organized. These will be
held in special sessions,
two in parallel, 1:30-2:30
p.m. and 4:30-5:30 p.m.
on Monday, 9:45-10:45
a.m. and 1:30-2:30 p.m.
on Wednesday, and 9:45-
'" -1 a.m. and 3:15-4:15
p.m. on Tuesday, Thurs-
day and Friday. Follow-
ing is a list of speakers
who have accepted the
invitation to lecture in
RlLLi t r ii lt-11 linear programming computation
Laig,-scal, combinatorial optimization
Strongly polynomial algorithms
Interior and pathfollowing methods for LP
Interior methods in combinatorial optimization
Mathematical programming in manufacturing
Approximate solutions to NP-optimization problems
Number theory, the algebra of polynomials, M and MP
Large-scale nonlinear optimization
Integer programming solution strategies
Interior methods for convex programming
R.T. Rockafellar Nonsmooth optimization
Newton methods and coniiil,/' iii
Routing and transit
These speakers have been requested to prepare state-of-
the-art survey articles or extended abstracts, based on
their talks. A paperback booklet of these articles will be
handed out to each registered participant on arrival at
John Birge, General Chair
Katta G. Murty, Program Chair
N *2 MARCH 1994
PAGE 3 N
Berlin, Aug. 30-Sept. 2, 1994
Operations Research 1994 is the sec-
ond conference in a series of quadren-
nial international conferences that
take place under the auspices of the
OR societies, DGOR, GM OOR,
OGOR, SVOR, of the German-
speaking countries and (for the first
time) the Special Interest Group on
Optimization, SIGOPT, of the
The conference serves as a forum for
scientists and practitioners in all areas
of Operations Research. Conference
languages are English and German.
The scientific program includes in-
vited plenary and semiplenary lec-
tures, as well as contributed papers.
I *IlI .i i ... ....i. .. .
P1, ,,,,i- ,,, ,,,,f ;,,,,,, ,,Il r,,
1. I , 1,1, 1
I i , r ,1r I I r, I.l
, ,, I r ,,, 1, , ,
r .. .... i l.. ,, .I.
W, qMW "O
Fiftdi Stockhohn Optimiz.iation Days
i .--I I. I I ' I I. I... FI,. .,
tion Days, a two-day conference
on optimization, to be held at
KTH (Royal Institute of Tech-
nology) in Stockholm, Sweden,
June 27-28, 1994. Plans call for
sessions on crew and vehicle
scheduling, dual optimization
methods and nonlinear pro-
gramming, among other areas.
Abstracts (a maximum of 200
words) should be sent by May
1 (preferably by e-mail) to:
email@example.com by mail to
Optimization Days, Division of
Optimization and Systems Theory,
KTH, S-100 44 Stockholm,
Sweden. The FAX number is
+46 8-22 53 20.
Further information can be ob-
tained from the same addresses.
The conference is financially sup-
ported by the Goran Gustafsson
Foundation and the Swedish Na-
tional Board for Industrial and Tech-
nical Development ,t.;i 'I I.I i Orga-
nizers are Ulf Brannlund, Anders
Forsgren (head), Per Olov Lindberg
and Krister Svanberg from the Divi-
sion of Optimization and Systems
Theory, Department of Mathematics,
Royal Institute of Technology
In ( 1c .I r &. II)_I I r 1,,,. I I
I IDL, AvduLi.ii, U.,lrd,
AT&T, Murray Hill, CA, USA
University of California, San Diego, CA, USA
McGill University, Montreal, Canada
University of Florida, Gainesville, FL, USA
Systems Research Institute, Warsaw, Poland
Washington State University, Pullman, WA, USA
Stanford University, Palo Alto, CA, USA
Courant Institute, NY, USA
Cambridge University, UK
IIASA, Laxenburg, Austria
FUNDP, Namur, Belgium
University of Colorado, Boulder, CO, USA
Virginia Tech, Blacksburg, VA, USA
FUNDP, Namur, Belgium
This symposium is designed to
bring together practitioners who
use mathematical programming
optimization models and deal
with questions of sensitivity
analysis, with researchers who are
developing techniques applicable
to these problems.
Contributed papers in math-
ematical programming are solic-
ited in the following areas:
~Sensitivity and stability analysis re-
sults and their applications;
ISolution methods for problems in-
volving implicitly defined problem
ISolution methods for problems in-
volving deterministic or stochastic
parameter changes; and
ISolution approximation techniques
and error analysis.
Clinical presentations that describe
problems in sensitivity or stability
analysis encountered in applications
also are invited.
should be sent in triplicate to Profes-
sorAnthony V Fiacco. Abstracts
should provide a good technical sum-
mary of key results, avoid the use of
mathematical symbols and references,
not exceed 500 words and include a
title and the name and full mailing
address of each author. The deadline
for submitting abstracts is March 15,
Approximately 30 minutes will be
allocated for presenting each paper.
Anthony V. Fiacco, organizer
Sponsored by the Department of Op-
erations Research and the Institute
for Management Science and Engi-
neering, School of Engineering and
Applied Science, The George Wash-
ington University, Washington, D.C.
20052. Tel. (202) 994-7511
16th Symposium on Mathematical
Programming with Data Perturbations
May 26-27, 1994
XGI 5 N0 42 MARl~h 1994
EVERY year since 1954 the
Council of the Operations
Research Society of
America has offered the
Lanchester Prize for the best pub-
lished contributions in operations
research in the English language. For
1993, the prize is $5,000 and a com-
Books and papers for the 1993 prize
will be screened by a committee ap-
pointed by the Council of the Soci-
ety. To be eligible for consideration,
the book or paper must be nomi-
nated to the Committee. Anyone
can make nominations.
To be eligible for the Lanchester
Prize, a book, paper, or group of
books or papers must meet the fol-
TIt must have been on an opera-
tions research subject;
?It must have been published in
1993; or two years prior to 1993,
or, in the case of a group, at least
one member of a group must have
been published during that time
I'It must have been written in the
English language; and
'It must have appeared in the open
literature. Books or papers may be
case histories, reports of research
representing new results, or prima-
rily exposition. For a nominated
set (group of either articles or
books) published over more than
one year, it is expected that each
element in the set is part of one
continuous effort, such as a
multiyear project or a continu-
ously written, multivolume book.
The Committee will use the ..II..
ing criteria in making judgments:
'The extent to which the contribu-
tion advances the state of the art of
IThe originality of the ideas or
IThe new areas of application it
'The degree to which existing
theory or method is unified or sim-
IThe clarity and II ...... of the
Nominations may be in any form,
but must include the titles of
papers) or bookss, authorss, place
and date of publication, and six cop-
ies of the material. Supporting state-
ments bearing on the worth of the
publication, in terms of the five crite-
ria, will be helpful, but are not re-
quired. Each nomination will be
carefully reviewed by the Committee.
Nominations must be received by
March 30, 1994, to allow adequate
time for review.
The decision by the Committee and
the ORSA Council will be an-
nounced, and any prizes approved
will be awarded, at the National
Meeting of the Society, Oct. 23-26,
1994, in Detroit, MI.
Nominations should be sent to:
John J. Bartholdi III, Chairman
Lanchester Prize Committee
School of Industrial and Systems
Georgia Institute of Technology
Atlanta, GA 30332-0205
1 9 9 3Lanchester
Vol. 62 No. 3
David B Shmoys and Eva Tardos,
"An approximation ii,.. ithm for
the generalized assignment
Arie Tamir, "The least element
property of center location on tree
networks with applications to
distance and precedence con-
Sanjay Mehrotra and Yinyu Ye,
"Finding an interior point in the
optimal face of linear programs."
Kurt M. Anstreicher, "Strict
monotonicity and improved
complexity in the standard form
projective ,il,, i tll for linear
Yinyu Ye and Kurt M.
Anstreicher, "On quadratic and
O(AiL) convergence of a predictor-
corrector algorithm for LCP."
B. Curtis Eaves, "Pivoting to
normalize a basic matrix."
C. Delorme, "Laplacian eigen-
values and the maximum-cut
Gabriele Danninger and
Immanuel M. Bomze, "Using
(.*.' -it. i i, for global optimality
criteria in concave quadratic
Vol. 63 No. 1
D. Burton and Ph.L. Toint, "On
the use of inverse shortest paths
algorithm for recovering linearly
Dimitris Bertsimis and James B.
Orlin, "A technique for speeding
up the solution of the Lagrangean
J. Scott Provan, "Efficient enu-
meration of the vertices of polyhe-
dra associated with network LPs."
Kelly T. Au, Julia L. Higle and
Suvrajeet Sen, "Inexact
subgradient methods with applica-
tions in stochastic programming."
J.B.G. Frenk and J. Gromicho,
"A deep cut ellipsoid algorithm for
convex programming: Theory and
B.M. Glover, V. Jeyakumar and
W. Oettli, "A Farkas lemma for
difference sublinear systems and
R E V I E W S
New Trends in Discrete and Computational
J. Pach, ed.
Algorithms and Combinatorics 10
Springer-Verlag, Berlin, 1993
Computational geometry has come a long way. When it emerged as a discipline in its own right in
the early '80s, it looked like a branch of computer science. It dealt with problems that, for example,
I; .. ; 1. Ti.. I-.1 ;, 1 i ...l .. I. .... I ,.. . .. ; I com putergraphics, butalso in dealing
with multidimensional data structures. To start with, there was not that much geometry in it: the main
did not use too much knowledge about the geometry of the objects under consideration.
This has changed. Already, the first and basic books that have appeared about the subject observed
that a lot of "classical" discrete geometry is useful for the analysis, but even more so for constructing
geometric algorithms. At the time Goodman and Pollack started their journal, "Discrete & Compu-
tational Geometry," this may have looked like bringing two foreign (or at least alien) fields together.
The fields of discrete geometry and computational geometry have merged, for the betterment of both.
The volume under review, which borrows its title from the journal, is proof of this.
In 12 loosely linked chapters, the volume gives a "tour d'horizon" of what the field looks like today
and it makes for a lot of interesting reading. The topics range from geometrical problems and topo-
logical tools for analyzing them, to methods for the construction and analysis of deterministic and
randomized algorithms. Below the surface, however, the strong link between the combinatorial and
.IL.., 1n i ...;,. ..F .. ,,l .. r, I [r l,r..l ,,[..,,, ,T 1 ..f.i. I ;.. ,..l ... ,- m anyofthechapters
certainly could be worth an extended, separate review here is a list of the authors and chapter headings.
I. L. Guibas & M. Sharir: Combinatorics and Algorithms of Arrangements
II. R. Seidel: Backwards Analysis of Randomized Geometric Algorithms
III. J. Matousek: Epsilon Nets and Computational Geometry
IV. L. Khachiyan: Complexity of Polytope Volume Computation
V. J.E. Goodman & R. Pollack: il ,I I Sequences and Order Types in Discrete
and Computational Geometry
VI. N.M. Korneenko & H. Martini: Hyperplane Approximation and
VII. J.E. Goodman, R. Pollack & R. Wenger: Geometric Transversal Theory
VIII. K. Bezdek: Hadwiger-Levi's Covering Problem Revisited
IX. I. Bairny: Geometric and Combinatorial Applications of Borsuk's Theorem
X. G. Fejes T6th & W. i.,, ... I...,: Recent Results in the Theory of '.1 ... and
XI. W. Moser & J. Pach: Recent Developments in Combinatorial Geometry
XII. P. Komjith: Set Theoretic Constructions in Euclidean Spaces
h1[I.,.. I,, .I ., 11 .. I .;.... .1;,, ;, ..... 1. 1 I mostly agree with the publisher's notes on the back
of the book:
"The mostimportantnew developments inml; I .. *I 1' i .; I ... r i. ........ -i .1
inthisbook. ....Thechaptersareself-.. .I... i i. .I ..l... i' ." 1. d .. 1.The
book will be used by researchers, graduate students, and engineers interested in applications."
Of course, most of the material in this book is not directly useful for the "engineer interested in appli-
cations." However, it does make a lot of inspiring reading, and it is related more closely than one might
think at first glance to the problems and algorithms that actually "work" in some of the software for
optimization, graphics and robotics.
 H. Edelsbrunner: A' *. '. in Computational Geometry, Springer-Verlag, Berlin
 J.E. Goodman & R. Pollack, eds.: Discrete & Computational Geometry, Springer-
Verlag, New York, Vol. 1 (1986).
 K. Mehlhorn: Data structures and algorithms 3: Multi-dimensional searching and com-
putationalgeometry, Springer-Verlag, Berlin Heidelberg, 1984.
 F.P. Preparata & I.M. Shamos: Computational Geometry: An Introduction, Springer-
Verlag, New York, 1985.
-GONTER M. ZIEGLER
for Solving Linear Algebraic Equations:
The State of the Art
NATO ASI Series
Series F: Computer and Systems Sciences, Vol. 77
Emilio Spedicato, ed.
Springer-Verlag, Berlin, 1991
In 12 loosely linked
chapters, the volume gives
a "tour d'horizon" of what
the field looks like today
and it makes for a lot of
This book contains a selection of the contributions presented by some of the world' 1 ,li; ...ir....
ties at the NATO Advanced Study Institute meeting on computer algorithms for solving linear al-
gebraic equations. The meeting was held Sept. 9-21, 1990, at II Ciocco, Barga, Italy. Of the 14 con-
tributions in the book, some contain sequential algorithms, some contain both sequential and multipro-
cessor algorithms, and some are devoted solely to the implementation ofalgorithms on multiprocessor ar-
In the first paper, by C.G. Broyden, direct methods for the solution of thesystem of linear algebraic equations
Ax=b, where A E R"" is non-singular and b E R" is given, are described in a rather novel way. After con-
sideringC .. !;i... ., ,;. I ...,.. .... i.Idecomposition, themethodsofCholeski forsymmetricpositive
definite A, of Aasen for symmetric indefinite A, of Levinson for Toeplitz A, and of Bj6rck and Pereyra for
Vandermonde A are described. The last part of Broyden's paper is devoted to conjugate direction methods,
especially appropriate for large sparse positive definite A.
In the second paper, by V. Pan, the complexity of algorithms for the solution of systems of linear algebraic
equations Ax=bis considered in the cases for which Ais dense and unstructured, sparse, and dense but struc-
tured. Attentionalso is given to iterative algorithmssuch asJacobi, Gauss-Seidel, S( .1.' I 1 I1.I1, I
methods, which are important in the solution of partial differential equations, also are considered.
The third paper, by A. Bj6rck, presents a survey of the singular value decomposition and its use for ana-
lyzing and solving linear least squares problems, and two r 1 .1I: ., ,, 1 1 I ..... ... .1 ... I .1. 1.. l
problems, based on QRfactorization, are presented. New backward stable methods, based on the modified
Gram-Schmidt factorization, are given for linear least squares problems and underdetermined linear sys-
tems. Iterative methods for the factored normal equations, A' (b-Ax) = 0, and preconditioners are surveyed,
and parallel block preconditioners for least squares problems in which the matrices have block structure
In the fourth paper, by E. Spedicato, the main properties of the ABS class of algorithms for the solution
of a general system of linear algebraic equations, Ax=b, with A E R""", b E R'" and rm
explained in a recently published monograph titled, "ABS Projection Algorithms," by J. Abaffy and E.
Spedicato (AS89), the majority of existing algorithms for the solution of systems of linear and nonlinear
algebraic equations are embedded in the ABS class where they correspond to particular choices of the free
parameters. In his paper, Spedicato states that if m>n, then there are several ABS approaches for solving
Ax=b in the least-squares sense without explicitly forming the normal equations. The paper also states that
extensive numerical experiments have shown that several of the methods that have been investigated are
compatible with LINPACK and with NAG codes in terms of accuracy on I I..... I....... I rank deficient
In the paper byJ. Abaffy, the application of.'.." i 1. . il,.... .Ax=b, where AE R"" is sparse, is considered.
The implicit LL'algorithm is applied to matrices havin ... ...... .... i ... i The implicit LUand QR
algorithms and the Huang method are applied to various types of band matrices. Numerical evidence is
S, ..iI ., 1 ,,, h. .,r .., ,,i l ,,, ,, i cases, tl '1 .' 1,_. ,; ..... .1 .... I;., i-m sofstorage,
compared to methods using standard factorizations.
The sixth paper, byW. Hackbusch, contains a comprehensive survey of multigrid methods, with examples
and pseudocode algorithms.
In the seventh paper, by H. Yserentant, the hierarchical basis method [Y86] of Yserentant and the
preconditioner of Xu [X89] and Bramble, Pasciak and Xu [BPX90] are described.
T1,... .;1,i, 1. I.., O. Axelsson, presents a relaxed incomplete L Ufactorization method without pivoting
for the solution of a linear system Ax=b, in which A is a block H-matrix. A pseudocode version of the fac-
torization is given.
The ninth paper, by G. Meurant, contains a review of some domain decomposition techniques for solving
symmetric sparse linear systems. These techniques are used to construct .I f 11 ,ll I preconditioners
for the conjugate gradient method.
In the 10th paper, I. C lii, ,. 1* -i.. ,.i. .,., .1. of the implementation, on a computer consisting of two
vector processors, of the preconditioned simultaneous displacements method for solving large sparse systems
of linear equations. The convergence properties of the method are treated under the assumption that the matrix
of the system is symmetric positive definite or is an irreducible L-matrix with strong diagonal dominance. Nu-
Of the 14 contributions
in the book, some contain
some contain both
sequential and multi-
processor algorithms, and
some are devoted solely to
the implementation of
I ~ _
medical results for a test matrix arising from the discretization of the diffusion problem on a rectangular
domain are given.
In the 11th paper, by M. Cosnard, a review of some of the results obtained in the last decade on the
d 1. ...1 .... I i. i ... il. I factorization ,I ...,'I.I .. for solving dense linear systems is presented.
In the 12th paper, byY. Robert, the implementation, on distributed memory architectures, such as systolic
arrays and general-purpose hypercubes,l i 1 i i ,1I. ,;,I,,., is discussed.
In the 13th paper, by I.S. C it I i I I, l,,, for the solution of sparse linearsystems on parallel
architectures is discussed.
In the final paper, by L.C.W. Dixon, the task of forming and solving the sets of linear equations that
arise from nonlinear problems in optimization, in ordinary differential equations and in partial differ-
ential equations, is considered. It is shown that in each case the formation of the linear system can be
handled efficient tly by .. ....... .. .1 ..1 I .. .. i ,, .. .. . ... I... . I .. .... ....I.
performed on a parallel computer. 7 . ii., ,i,. .1,,;..,. of the set of equations, it is shown that, in
.. .,- ..,1 ,I I.. II ,i.. INewtonmethodout-performsmoretraditionaloptimizationcodesandthatwhen
the optimization problem arises in a two-dimensional finite element context, the SIMD/DAP processor
..' I ...' I I I.... ,. ,r difficultto obtaineffectivespee I1-" l. I .. I .. . .. 1...." II
MIMD machines of the Sequent or transputer network type. It is shown for large sparse optimization
that the use ofaccurated... ,, . 1. ,, i ,, 1... .. .. 1, usingaconjugate
[AS] J. Abaffy & E. Spedicato: ABS Projection Algorithms: Mathematical Techniques for
Linear and Nonlinear Equations, Ellis Horwood Ltd., ( I1 I,. T 1989.
[BPX90] J.H. Bramble, J.E. Pasciak & J. Xu: Parallel Multilevel Preconditioners, Math.
Comp. 55 (1990), 1-22.
[X89] J. Xu: Theory ofMultilevel Methods, Report No. AM48, Department of Mathemat-
ics, Pennsylvania State University, 1989.
[Y86] H. Yserentant: On the Multi-level "..' of Finite Element Spaces, Numer. Math.
49 (1986), 379-412.
-BY MA. WOLFE
Dynamic Economic Models and
G. Feichtinger, ed.
North Holland, Amsterdam, 1992
This book is the fourth in a series reporting selected presentations at the Viennese Workshop on Dy-
namic Economic Models and Optimal Control. 1 I..... .I I I. .... ..-.. ,. of the workshop
(held in 1981, 1984 and 1987), are remarkable achievements, both by the quality of the papers presented
and by the concentration of the conference on the modeling and analysis of dynamic systems in eco-
nomics and management science.
Thecurrents..l C .1 1i k..i i ..i.. i, .iz.ofnonlineardynamicsystemshasseveral
new and special features. For example, papers in chaos and chaotic economic systems provide an im-
portant linkage between the classical approach of control and the new ideas arising from bifurcation
theory and chaotic oscillations (11 contributions). There also are important collections of papers on
the application of differential game and optimal control theory to economics (11 and 20 contribu-
tions, respectively). The book is focused on deterministic systems and, conspicuously, no contribu-
tions in stochastic control theory and applications are included. This is an advantage, however, as it
__ _ _ __ _ _ I -
_ _ _ _ _I _
focuses on other aspects of uncertainty, arising from gaming situations and nonlinear dynamics.
The book has numerous contributions. Some examples include the paper by Lionard, "On the ubiq-
uity of trade in capital goods: Jumps in the sate variables," which ,.. i .... 1 ,... results by
Arrow and Kurz. Kemp et al. present a paper on a dynamic formulation of the foreign aid process,
a problem which has been dealt with previously in a static form. Feichtinger et al. discuss the limit
cycles of resource-employment in a regulated fishery (based on Hopf bifurcation theory).
T am ar B asar' i. 1 .. .!, ., ,! T'l " ... i .. ,,,, ,. ,i .. ,,. .i.. ... .. .. i .1 .
economic systems, an important contribution to H" control. Gradus and Kort discuss "Optimal taxa-
tion on profit and pollution within a macroeconomic framework," providing an insightful analysis to
a timely problem- taxation and pollution. J .. ... .. I .' I' ; .. II'. "Dynamics ofextramarital affairs,"
providing a stimulating presentation of a problem that is a permanent fixture of our social makeup.
Papers by Nishimura on "Factor intensity and Hopf bifurcations," Hommes on "Periodic, Quasi-
periodic and Chaotic dynamics in a simple macro model with Hicksian nonlinearities," Lorenz on
"Multiple attractors, complex basin boundaries, and transient motion in deterministic systems,
Fienkenstadt and Kuhbier on "Principle component analysis and I .'1. city: An application
of chaotic time series" provide further motivation to the stability analysis and a growing interest in
chaotic economic dynamics.
C ,11, this is a useful reference text for researchers and graduate students alike in the field of modeling,
and the analysis of dynamic systems and their applications in economics.
Optimality and Equilibria in
by E Thuijsman
Centre for Mathematics and Computer Science, Amsterdam, 1992
This monograph, which differs from the Ph.D. thesis of the author only in some minor points, deals
with two-person, .. .. .i ',,;,, i,;, . I , ; .. .. :, l finite state and action spaces. The introduc-
tory Chapter 1 discusses formal definitions of the stochastic game model (zero-sum versus general-sum)
with its solution concepts (value and optimal strategies versus equilibria), II as the major historic
results in the field. ( i -.1, I.. .I -n both the zero-sum and the general-sum case can be considered
for three evaluation criteria of incomes: the /-discontinued reward criterion, the limiting average re-
ward criterion and the total reward criterion. Fine solutions, in terms of stationary strategies (i.e. strat-
egies for which the action choices of the players only depend on the state and not the history), are known
to exist with respect to the 0-discontinued reward criterion, whereas, with respect to the limiting average
reward criterion, similar problems as for the total reward criterion occur.
In chapters 2,3 and 4, the emphasis is on stochastic games,' ;,1,, I ... -, .. .; .1
criterion (mainly because the existence of limiting average e-equilibria can be seen as the major open
problem in stochastic game theory nowadays). Involving the zero-sum case, Chapter 2 provides an
,r ,; .. .. 1 r _,f .. .. .....FF .rI I .. result that for each player there exists easy initial
states, i.e. starting states for which this player has a stationary limiting average optimal strategy. In
the same setting, with respect to some other set of initial states (with maximal or minimal limiting
average value),, ,, I .I.... .... 11,.... ; presented for each player to have stationary limiting average
E-optimal strategies. For the general-sum case, Chapter 2 also provides a similar result in that there
always is a non-empty set of initial states for which an "almost-satisfactory" limiting average e-equi-
librium exists, i.e. stationary strategies which are amplified with some threat to prevent profitable
Overall, this is a useful
reference text for researchers
and graduate students alike
in the field '* ,
and the analysis of
dynamic systems and their
applications in economics.
deviations of the opponent. So, if both players stick to their e-. i,;1;1 ,; ,,, , .. then
with probabilityclose to 1, theywill .. .. r ;. ,1 ... I ..i. l 1, I-. .
In Chapter 3, the general-sum result of Chapter 2 is extended by .... 1. ... .
conditions for the existence of an "almost-stationary" limiting average e-equilibrium (for
,*i ;. states). The relevant conditions are formulated in terms of asymptotic prop-
erties of sequences of stationary /-discounted equilibria. It is not clear whether these
ii. ....; conditions hold for any general-sum stochastic game.
In Chapter 4, it is established that tl... i. i.... .. conditions are automatically fulfilled
forthreespecialclasses o r. I .. I .. ..... i l- ,, 1. .. ,.. 1 I i, .. ; .
stochastic games (satisfying the property that, for any pair of stationary strategies, there
is just one irreducible set of states); stochastic games with state-independent transitions
(SIT); and repeated games with absorbing states.
Chapter 5 focuses on the total reward criterion for zero-sum stochastic games. Because
the total value may fail to exist in general, the emphasis is on games satisfying the con-
dition of limiting average value 0 for all initial states and both players possessing station-
ary limiting average optimal strategies. For such a game, the total reward criterion can
be seen as a refinement of the limiting average reward criterion, since a total (e-)optimal
strategy is necessarily limiting average optimal. By means of an example, it is illustrated
that, even with the above property, history-dependent behavior strategies are indispens-
able for the player to achieve total E-optimality. The existence of stationary total optimal
strategies for stochastic games with the above property is characterized.
Chapter 6 is devoted to mathematical programs connected to stochastic games. In the
framework of all three evaluation criteria, nonlinear programs are presented that com-
pletely characterize the existence of stationary equilibria or (E-)optimal strategies.
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