Title: Optima
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Permanent Link: http://ufdc.ufl.edu/UF00090046/00026
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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: April 1989
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Volume ID: VID00026
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1r I 1 VI N 26



Highlights of

Mike Todd's Research
SIAM and MPS have announced the award of
the 1988 George B. Dantzig Prize to Michael
J. Todd, Leon C. Welch Professor in the
School of OR and IE at Cornell University.
This award, as specified in its charter, is "...
for original work, which by its breadth and
scope, constitutes an outstanding contribution
to the field of mathematical programming."
The 1988 Dantzig Prize Committee consisted
of O. Mangasarian (Chair), K. Murty, G.
Nemhauser and M. Wright. This article
highlights certain of Todd's major research

number 26

continues, page two



i --~Bll~b~IIC~

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A 2u re A

TODD received the B.A. degree in
mathematics from Cambridge
University in 1968 and the Ph.D.
degree from Yale University (De-
partment of Administrative Sciences) in
1972. His early research interest was in
complementarity and fixed points, initially
from an abstract viewpoint. Todd's Ph.D.
dissertation [1] introduces an abstract com-
binatorial setting for complementary
pivoting algorithms. These combinatorial
structures are also the subject of [2], where a
generalized pivoting algorithm is presented
and shown to subsume contemporary
procedures for complementary pivoting. A
convergence proof for the generalized
algorithm is also given, along with perform-
ance bounds based on the "diameter" of an
abstract pivoting system, a notion similar to
the pivoting diameter of polytope. This line
of research culminates in [3], where dual
pairs of abstract pivoting systems are
demonstrated to be the circuits of a dual
pair of binary matroids, thus linking the
subject of complementarity to the rich area
of matroid theory. Todd's interest in the
combinatorial structure of polytopes is
evident once again in [4] with the presenta-
tion of a counterexample to the monotonic
bounded Hirsch conjecture.
Practical issues related to the actual compu-
tation of fixed points quickly began to
dominate Todd's work. His research
monograph [6] on this subject, The Computa-
tion of Fixed Points and Applications, is widely
known. The presentation of this monograph
stresses the basic triangulation as a funda-
mental c tomponent in algorithms for
computing fixed points. In order to describe
Todd's contributions in this area, the
following description of the problem setting,
taken from [9], is useful. "Suppose that we
seek a zero of a continuous function f from
R" to itself... We then choose a one-to-one
affine function f: R" -_ Rn defined by f(x) =
Gx -g and construct the homotopy h: R" x
[0,1] Rn by h(x,o) = of(x) + (1 0)f(x).
Next we make a piecewise-linear approxi-
mation Q to h; generally this is done by
choosing a triangulation T of R" x [0,1],
letting E agree with h on the vertices of T
and then extending linearly on the sim-
plices... Then a [piecewise-linear] path of
zeroes of Q is traced from the known zero

(x,0) where xo = G-'g. This path either
diverges to infinity or produces a zero (x',1)
of Q... x' is then an approximate zero of f.
The step from xo to x' is one major itera-
tion." Continuing from [10]: "In order to
obtain a sequence converging to a zero off,
a suitable sequence f f, ..., of functions is
chosen, e.g., [piecewise-linear] approxima-
tions with respect to finer and finer
Todd's initial contributions to fixed point
calculation address the dependence of
computational efficiency on the particular
triangulation used. In [5] Todd begins to
quantify this relationship with the introduc-
tion of theoretical measures of the efficiency
of a triangulation. These measures are
based on counting the number of simplices
of a triangulation met by certain straight line
segments. Such measures are shown to be
useful not only in comparing well-known
triangulations, but also in suggesting
guidelines for the construction of new

A natural question raised by the effi-
ciency measures introduced in [5] is
whether "optimal" triangulations can be
constructed. Though the sense of "optimal-
ity" changes, Todd returns to this theme in
[7], pointing out that the simple notion of
interval bisection provides a minimax
procedure for locating a fixed point of
continuous function f: [0,1] [0,1]. Thus
this procedure is "optimal" in the sense that
"... the algorithm is guaranteed to find, after
any given number of function evaluations,
an interval that contains a fixed point and
that has length at most e; for any other
algorithm, there is a function f for which the
interval found has length greater than e."
In [7] the bisection procedure is generalized
to obtain an "optimal" dissection of an n-
simplex into subsimplices through the
insertion of one new point; the procedure
then iterates on a subsimplex guaranteed to
contain a solution.
Improvement to the efficiency of fixed point
algorithms remains the subject of interest in
[8], though the emphasis now is on imple-
mentation. As described in [9]: "... it is
important to note that the pieces of linearity

of 2 are usually much larger than simplices,
even if f has no special structure; this
conglomeration of pieces can be enhanced if
f has special structure by an appropriate
choice of triangulation T." In [8] Todd
shows how this observation can be ex-
ploited globally, obtaining improved
algorithms that traverse several simplices in
a single step. Much of the improved
efficiency here is generally applicable, due
to large regions of linearity induced by the
affine function f. But Todd also shows that
additional computational advantage accrues
when f has special structure relating to
separability of variables. Effecting the
improvements described in [8] requires
operations similar to, though more compli-
cated than, linear programming pivots.
Thus in [9] techniques from numerical linear
algebra for maintaining numerical stability
and exploiting sparsity are adapted to the
present setting.
Note that the thrust of Todd's research on
fixed point methods has now clearly shifted
from the triangulation T to the function f.
He emphasizes this in the survey [10] on
piecewise-linear homotopy algorithms: "It
now seems more natural to state the
problem in terms of f, and because of the
mode of operation of recent algorithms we
shall call them piecewise-linear homotopy
methods.... The newer algorithms retain the
properties of global convergence under very
weak boundary conditions that hold
naturally in several applications.... In
addition, techniques have been devised to
take advantage of smoothness...." Smooth-
ness of f leads to quadratically convergent
algorithms via the realization that the kth
major iteration of the basic piecewise-linear
homotopy algorithm yields both an approxi-
mate zero xk for fand an approximation of
the derivative of fat xk (see 9 of [10]). A
further advantage of piecewise-linear
homotopy methods is that they are also
applicable when f is a point-to-set mapping.
In [11] such an application is addressed for a
convex union of smooth functions, a setting
arising in nonlinear programming and
economic equilibrium problems. Superlin-
ear convergence to a zero of fin this general
setting is established.

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number twenty-six

APRIL 1989

PAGE 3 n u m b er ten-i APRIL19

In the early 1980s linear programming
emerges as a second major theme in Todd's
research. The extraordinary success of linear
programming, ranging from broad practical
applicability of the linear model to its
supporting combinatorial and computational
methodology, serves as a cornerstone for the
field of mathematical programming. Today,
some four decades after Dantzig's introduc-
tion of the simplex method, linear program-
ming remains a very active and fertile
research area, as evidenced by recent work
on polynomial-time ellipsoidal and interior-
point algorithms for linear programming, on
average-case analysis of the simplex algo-
rithm in order to explain its observed
computational efficiency and an oriented
matroid as an abstract combinatorial model
for linear programming and the simplex
method. Todd's more recent work contrib-
utes to each of these topics of current interest
in linear programming.
Oriented matroids arise as a natural abstrac-
tion of the combinatorial properties of signed
linear dependence among the columns of a
matrix. These structures thus provide a
setting for combinatorial consideration of the
theory and algorithms of linear program-
ming. In [12] the goal is ". .. to show that the
natural setting of [the linear complementar-
ity problem] is that of oriented matroids."
This is achieved by extending the concepts of
"P-matrices" and "completely Q-matrices,"
objects central to study of the linear comple-
mentarity problem, to oriented matroids.
Indeed, [12] extends well-known characteri-
zations of these concepts to the setting of
oriented matroids and the algorithmic tools
developed by Todd for achieving this
extension generalize well-known algorithms
of linear complementarity. As an added
benefit, the combinatorial setting provides a
natural duality, distinct from linear (or
quadratic) programming duality, yet
applicable to the linear complementarity
problem. The combinatorial setting is also
shown to be an appropriate framework for
studying relations among algorithms for
linear complementarity. This vein of
research is continued in [13], where it is
shown that ideas of [12] lead to constructive
proofs of linear and quadratic programming
duality results for oriented matroids, and in
[14], where combinatorial abstractions (i.e.,

oriented matroid analogues) of well-known
matrix properties such as symmetry and de-
finiteness are investigated.
Much of the recent development in the field
of analysis of algorithms has been in terms
of worst-case analysis and, indeed, it has
been well-known for over 15 years that in
the worst case many variants of the simplex
algorithm require an exponential number of
computational steps. Yet, equally well-
known is the excellent performance of
simplex algorithms in practice, an observa-
tion now supported by over three decades
of computational experience. Thus, in the
words of [15], "It has been a challenge for
mathematicians to theoretically confirm the
extremely good performance of simplex al-
gorithms for linear programming." The
manuscript [15] is thus of particular
significance, as it provides an average-case,
polynomial-time bound for a simplex
algorithm applicable for every linear
programming problem. This paper indi-
cates, under nonrestrictive assumptions on
the probabilistic model, that the "lexico-
graphic self-dual simplex method" solves a
linear programming problem of order m x n
with an average number of iterations
proportional to (min(m,n))2. Todd's
(independent) work [16] leading to this
result provides a probabilistic analysis of a
certain pivoting algorithm for the linear
complementarity problem. Of particular
significance is the specialization in [16] to
linear programming problems, yielding the
results cited in [15]. Todd's work also
addresses implementations of the simplex
method for structured, large-scale program-
ming. In [17] linear programming problems
with variable upper bound constraints, i.e.,
of the form x. < Xk, are considered and a
revised simplex implementation is pre-
sented that handles these constraints
implicitly. In contrast to earlier implemen-
tations for such problems, Todd's method is
based on a numerically stable, triangular
factorization of the basis matrix. In [18] this
work is continued through the adaptation of
Forrest-Tomlin and Saunders updating
schemes to this setting. Todd's basic work
on variable upper bounds also led to the
treatment in [19] ". . for linear program-
ming problems in which many of the
constraints are handled implicitly by

requiring that the vector of decision
variables lie in a polyhedron ..." In this
paper the insight gained from the variable
upper bounded algorithm of [17] is used to
obtain a unified geometric understanding of
several methods of large-scale program-
Khachiyan's indication, scarcely less than a
decade ago, that an ellipsoid method could
be implemented in polynomial-time to solve
linear programming problems settled a
theoretical question of long standing and
stimulated renewed algorithmic research in
linear programming. The feature article [20]
provides an important early survey on the
ellipsoid method and its theoretical and
practical significance, as well as its setting
within the context of earlier research. This
paper, aside from its technical and historical
contribution, is widely known for the im-
portant role it played in helping to disspell
the early confusion that ensued as word of
the ellipsoid method spread more rapidly
than technical understanding of its conse-
quences and limitations. Work on the
ellipsoid method continues in [21], where
the method is implemented in such a way
that (linear programming) dual variables are

M uch of the stimulus for current
research on polynomial-time algorithms for
linear programming is provided by Karmar-
kar's interior-point algorithm, introduced
only five years ago. It appears that, for the
first time, a potential competitor of the
simplex algorithm has emerged. Todd is
contributing to this development as well. In
[22], for instance, the question of generation
of dual variables is again addressed, and an
implementation of the basic interior-point
algorithm is given that generates dual
solutions. This approach also proves useful
in extending the basic algorithm's applica-
bility. Of particular significance to this line
of research is the manuscript [23], where a
unification of ellipsoidal and interior-point
projectivee) methods is achieved by estab-
lishing ". .. that in fact the heart of each
iteration of either algorithm is the solution
of a weighted least-squares subproblem,
and that these subproblems are very closely
related. This viewpoint allows further

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number twenty-six

APRIL 1989

11 ; 1 1 1 1 1 g ; l
PAGE~~~~5 4 nRb e twnt-iAPI 18

insights into the two methods, in particular
suggesting reasons for the very slow
convergence of the ellipsoid method
compared to the apparently very fast
convergence of the projective algorithm ...
Both the ellipsoid and the projective
algorithms appear at first sight not to
provide solutions to the dual linear pro-
gramming problem, but a closer examina-
tion shows that dual solutions are indeed
generated during the course of the methods,
essentially from the least-squares prob-
lems." Also significant is [24], where it is
shown that specially structured constraints
(including, e.g., the variable upper bound
constraints studied in [17, 18, 19] can be
exploited to computational advantage in
implementations of the projective algorithm.
The citation for the award of the Dantzig
Prize to Todd notes that, "The entire field of
mathematical programming has been
enriched by his valuable contributions in
algorithmic methodology and analysis for
topics ranging from linear programming to
fixed points." Indeed, Todd's research
publications over the past 15 years, now
numbering in excess of 60, constitute an
impressive eclectic array of individual
advancements. And this work in its entirety
makes a significant contribution as well, for
it reveals fundamental principles shared by
disparate areas of mathematical program-
ming. Abstract combinatorics [1, 2] is used
to link complementarity with matroid
theory [3], while insight gained from the
study of abstract pivoting systems later
finds application in invalidating the
monotonic bounded Hirsch conjecture [4].
A further connection between complemen-
tarity and matroid theory [121 results in
algorithms [13] that ultimately provide key
motivation for average-case analysis of
simplex algorithms for linear programming
[15, 16]. Work on improving efficiency of
fixed point algorithms by exploitation of
linearity [8, 9] motivates later material on
large-scale implementation [17, 18,19] that,
in turn, leads to advances in the most recent
linear programming methology [24]. And
so on. Thus the award of the 1988 Dantzig
Prize to Michael J. Todd provides appropri-
ate recognition to a body of work contribut-
ing to the very identity and unity of the field
of mathematical programming.

[1] Abstract Complementary Pivot Theory,
Ph.D. Dissertation, Yale University, Depart-
ment of Administrative Sciences, New
Haven, CT, 1972.
[2] "A Generalized Complementary Pivot
Algorithm," Mathematical Programming 6
(1974) 243-263.
[3] "A Combinatorial Generalization of
Polytopes," Journal of Combinatorial Theory
(B) 20 (1976) 229-242.
[4] "The Monotonic Bounded Hirsch
Conjecture is False for Dimension at Least
Four," Mathematics of Operations Research 5
(1980) 599-601.
[5] "On Triangulations for Computing
Fixed Points," Mathematical Programming 10
(1976) 322-346.
[6] The Computation of Fixed Points and
Applications, Lecture Notes in Economics
and Mathematical Systems, No. 124 (M.
Beckmann, ed.), Springer-Verlag, Berlin,
[7] "Optimal Dissection of Simplices,"
SIAM Journal of Applied Mathematics 34
(1978) 792-803.
[8] "Traversing Large Pieces of Linearity in
Algorithms that Solve Equations by Follow-
ing Piecewise-Linear Paths," Mathematics of
Operations Research 5 (1980) 242-257.
[9] "Numerical Stability and Sparsity in
Piecewise-Linear Algorithms," Analysis and
Computation of Fixed Points (S. M. Robinson,
ed.), Academic Press, New York, 1980,1-24.
[10] "An Introduction to Piecewise-Linear
Homotopy Algorithms for Solving Systems
of Equations," Topics in Numerical Analysis
(P. R. Turner, ed.), Lecture Notes in Mathe-
matics 965, Springer-Verlag, Berlin-Hei-
delberg-New York, 1982, 149-202.
[11] "An Efficient Simplicial Algorithm for
Computing a Zero of a Convex Union of
Smooth Functions" (with S. A. Awoniyi),
Mathematical Programming 25 (1983) 83-108.
[12] "Complementarity in Oriented
Matroids," SIAM Journal on Algebraic and
Discrete Methods 5 (1984) 467-485.

[13] "Linear and Quadratic Programming in
Oriented Matroids," Journal of Combinatorial
Theory (B) 39 (1985) 105-133.
[14] "Symmetry and Positive Definiteness in
Oriented Matroids" (with W. D. Morris, Jr.),
European Journal of Combinatorics 9 (1988)
[15] "New Results on the Average Behavior
of Simplex Algorithms" (with I. Adler and
N. Megiddo), Bulletin of the American
Mathematical Society 11 (1984) 378-382.
[16] "Polynomial Expected Behavior of a
Pivoting Algorithm for Linear Complemen-
tarity and Linear Programming Problems,"
Mathematical Programming 35 (1986) 173-192.
[17] "An Implementation of the Simplex
Method for Linear Programming Problems
with Variable Upper Bounds," Mathematical
Programming 23 (1982) 34-49.
[18] "Modifying the Forrest-Tomlin and
Saunders Updates for Linear Programming
Problems with Variable Upper Bounds,"
Annals of Operations Research 5 (1985/6) 501-
[19] "Large-Scale Linear Programming:
Geometry, Working Bases and
Factorizations," Mathematical Programming
26 (1983) 1-20.
[20] "The Ellipsoid Method: A Survey"
(with R. G. Bland and D. Goldfarb), Opera-
tions Research 29 (1981) 1039-1091.
[21] "The Ellipsoid Method Generates Dual
Variables" (with B. Burrell), Mathematics of
Operations Research 10 (1985) 688-700.
[221 "An Extension of Karmarkar's Algo-
rithm for Linear Programming Using Dual
Variables" (with B. Burrell), Algorithmica 1
(1986) 409-424.
[23] "Polynomial Algorithms for Linear
Programming," Advances in Optimization and
Control (H. A. Eiselt and G. Pederzoli, eds.),
Springer-Verlag, Berlin, 1988, 49-66.
[24] "Exploiting Special Structure in
Karmarkar's Linear Programming
Algorithm," Mathematical Programming 41
(1988) 97-113.
L. E. Trotter

__~__I ~ _~ 1___~________1______I___~


number twenty-six

APRIL 1989

PAGE 5numbertwenty--x-APIL-198

EAR President,
Ladies and Gentlemen:
While thinking of the Or-
chard-Hays Prize, I am remembering
those years Laurence and I tried, success-
fully to some extent, tackling difficult op-
timization problems, gearing our efforts
toward MPSARX.
The awarded publication
culminates a series of seven papers cover-
ing our joint research where I was influ-
enced in three ways:
First, I am grateful to
Geoffrion, Graves and Erlenkotter who
taught me decomposition theory and
helped my research in this field. From
this research, particularly from the prop-
erties exhibited by Cross decomposition, I
think I learned to understand why and
when problems get hard to solve, the key
difficulty being the duality gap to close in
an effective way.

Second, from my experience
in practise, again with the help of
Geoffrion and Graves, I learned that not
special but general purpose software
must be developed in order to be useful in
solving real-life problems effectively.
Finally, the success of
Manfred Padberg with Crowder and
with Crowder and Johnson in solving
pure zero one programming problems by
using facet defining cutting planes has
been the third stimulating factor. I am
very grateful to Manfred for helping us in
the earlier stages of our research while he
was at Core.
I am very honoured to re-
ceive the Orchard-Hays Prize and to
share it with Laurence whom I would like
to thank for those nice years of profes-
sional work at Core. I deeply regret,
however, that I cannot be present at your
meeting in Tokyo.
Tony Van Roy
Bank Brussels Lambert
Brussels, Belgium
August 1988

(The above letter was received too late for reading at the Tokyo Symposium Ed.)

New problem-
solving system
could save
customers millions

A superfast problem-solving
system derived from an AT&T
Bell Laboratories research break-
through is now available to
customers. The AT&T KORBX
System combines software and
hardware to solve business and
government resource allocation
problems involving several
hundred thousand variables.
The system is priced in the mil-
lions but can potentially save
customers tens of millions of
dollars. It is based on a mathe-
matical programming method,
the Karmarkar algorithm,
named for Bell Labs researcher
Narendra Karmarkar.

(Reprinted from the
AT&T Quarterly Report
dated October 30, 1988 Ed.)

_ I ~I_ __s____s~______________~ ~


number twenty-six

APRIL 1989

PAG 6numertwetysixAPIL 98


Programming: A Tool for
Faculty Polytechnique de Mons
Mons, Belgium
May 17-19, 1989

This meeting is the second in a cycle of
conferences devoted to Mathematics for
engineers and organized by the Facult6
Polytechnique of Mons in Belgium. The
theme of the conference will be mathemati-
cal programming and its applications in all
the fields of engineering. Indeed, this op-
erations research tool is used in chemical,
electrical, mechanical engineering, manage-
ment, etc., for the analysis of models as well
as for the optimization of systems.
All subjects related to mathematical pro-
gramming and numerical optimization will
be welcome. Emphasis will be put on real
applications in the field of engineering and
on advances in mathematical programming.
The aims of the conference are: (1) to show
the potential of the various methods of
mathematical programming in the field of
engineering, thus the use of these tech-
niques in real applications will be empha-
sized and (2) to describe the advances in
applied mathematical programming, linear
and nonlinear programming, unconstrained
optimization, integer programming, combi-
natorial optimization, multi-objective opti-
mization, software developments, etc.
To meet these two objectives a large part of
the programme will be devoted to tutorial
sessions given by some distinguished guest
speakers. Further information on these
speakers is given in the invitation pro-
gramme sent in March.

Furthermore, some parallel sessions of
contributed papers will give the participants
the opportunity to present and discuss more
recent and specialized research in the
various fields of engineering. A software
fair in computing environment and a book
exhibition will also be organized.
Papers and software presentations are
welcome. For additional information, please
J. Teghem / M. P. for Eng.
Faculty Polytechnique de Mons
9, rue de Houdain
7000 Mons BELGIUM
Telephone: (065)37-40.48 37.40.05
Telefax: (065)37.42.00
Telex: 57764 uemons b

Fifth International
Conference on Stochastic
August 13-18, 1989
Ann Arbor, Michigan, U.S.A.

This conference is being organized by the
Committee on Stochastic Programming of
the Mathematical Programming Society and
is co-sponsored by ORSA, TIMS, Technical
Committee 7 of IFIP, and the Department of
Industrial and Operations Engineering and
the College of Engineering, The University
of Michigan. The focus will be on stochastic
programming theory and applications with
particular emphasis on computation.
Specific topics will include numerical
integration, Monte Carlo approaches,
stability and sensitivity analysis, approxi-

mation and model simplification, uses of
parallel processors, and applications in
production, manufacturing, transportation,
finance, natural resources, power systems,
and long-range planning. A tutorial session
will introduce new investigators and users
to the field.
Anyone interested in attending the confer-
ence or submitting a paper should contact:
Professor J. R. Birge
Department of Industrial and Operations
1205 Beal
The University of Michigan
Ann Arbor, Michigan, USA, 48109-2117
Telephone: (313) 764-9422
E-mail: John_R. Birge@um.cc.umich.edu

CORE Announces Lecture

A new series of lectures and reports entitled
the CORE Lecture Series is announced. The
first speaker will be Professor Martin
Gr6tschel of the University of Augsburg. He
will lecture at the Center for Operations
Research and Econometrics (CORE),
University Catholique de Louvain in
October 1989 on the subject of Modelling,
Algorithms and Practical Problem Solving
in Optimization. The series is entitled
"Postmen, Ground States of Spin Glasses,
Via Optimization and Cycles in Binary
Anyone wishing to attend the Series is
invited to write to the above address for
further details. Some limited financing is
available to help students attend the lectures
(applications should include a reference
letter from one professor).
-Laurence Wolsey

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number twenty-six

APRIL 1989

Pn e yA L


Integer Programming and
University of Waterloo
Waterloo, Ontario
May 28-30, 1990

This meeting will highlight recent develop-
ments in the theory of integer programming
and combinatorial optimization. Topics will
include polyhedral combinatorics, integer
programming, geometry of numbers,
computational complexity, graph theoretic
algorithms, network flows, matroids and
submodular functions, approximation
algorithms, scheduling theory and algo-
rithms, and algorithms for solving counting
problems. In all these areas we welcome
structural and algorithmic results. The
latter may be sequential or parallel, deter-
ministic or probabilistic.
The meeting will be patterned after the
highly successful Foundations of Comput-
ing Science (FOCS) meetings and Symposia
on the Theory of Computing (STOCS)
organized by the ACM and IEEE. During
the three days, approximately thirty papers
will be presented in a series of sequential
nonparallell) sessions. The program
committee will accept the papers to be
presented on the basis of extended abstracts
to be submitted as indicated below. The
proceedings of the conference will contain
full texts of all presented papers and will be
published by the UW Press. Copies will be
provided to all participants at registration

time. Papers appearing in the proceedings
will not be refereed, and it is expected that
revised versions of most papers would be
submitted for publication in appropriate
This meeting is being organized under the
auspices of the Mathematical Programming
The Program Committee members are V.
Chvatal, Rutgers University; W. Cunning-
ham, Carleton University; R. Kannan,
Carnegie-Mellon University; R. Karp,
University of California, Berkeley; G.
Nemhauser, George Institute of Technology;
W. Pulleyblank, University of Waterloo; and
P. Seymour, Bell Communications Research.
Accommodations for participants will be
available in the student residences of the
University of Waterloo or in local hotels, if
For more information, please contact the
organizers as follows:
R. Kannan
Department of Computer Science
Carnegie Mellon University
W. R. Pulleyblank
Department of Combinatorics and Optimi-
University of Waterloo
Deadline for submission of extended
abstracts of papers is October 31, 1989.

Ninth International
Conference on Analysis and
Optimization of Systems
June 12-15,1990
Antibes, France

The purpose of this Conference is to present
the advanced research in the field of
Systems Analysis and Control where the
most promising applications may be
This meeting organized every other year by
INRIA will take place near the INRIA
Sophia Antipolis Center on the French
The organizers strongly encourage the
authors to forward proposals of communi-
cations describing: the most recent results
of research in the field, new applications.
Software demonstrations and industrial
products will also be presented in an
The deadline for submission of papers is
October 1, 1989. The deadline for descrip-
tive papers for the exhibition is May 1, 1990.
For information concerning the conference
please contact the conference secretariat:
T. Bricheteau / S. Gosset
Public Relations Department
Domaine de Voluceau
B.P. 105
78153 Le Chesnay Cedex
Phone: 33(1) 39 63 56 00
Telex 697 033 F

- - ----------------~-~ ssmss~-----------


number twenty-six

APRIL 1989

PAGE numbllljllei$kr twent-i API 198

Technical Reports &

Working Papers
L-: 2 --2-- 2 i 'Z-' .. :-' .rT ._ --a 2--- & 9 T_,- ,-

Cornell University
School of Operations Research
and Industrial Engineering
Upson Hall
Ithaca, NY 14853

W. Morris, A. Morton, and M.J. Todd, "An
Implementation of Todd's Algorithm for
Variable Upper Bounds," TR 799.
R.O. Roundy, "Efficient, Effective Lot-Sizing
for Multi-Product, Multi-Stage Production/
Distribution Systems with Correlated De-
mands," TR 802.
R.O. Roundy and J.A. Muckstadt, "Analysis
of Multistage Production Systems," TR 806.
J. Renegar and M. Shub, "Simplified Complex-
ity Analysis for Newton LP Methods," TR 807.
E.M. Arkin and R.O. Roundy, "A Pseudo-
Polynomial Time Algorithm for Weighted-
Tardiness Scheduling with Proportional
Weights," TR 812.
M. Hartmann, "Cutting Planes and the
Complexity of the Integer Hull," TR 819.
E. Arkin, "Complexity of the Multiple Product
Single Facility Stockout Avoidance Problem,"
TR 822.

Operations Research Group
The Johns Hopkins University
Baltimore, MD

M. Schneider and S. Zenios, "A Comparative
Study of Algorithms for Matrix Balancing," 88-
A.T. Benjamin, "Graphs, Maneuvers and
Turnpikes," 88-03.
C. ReVelle and K. Hogan, "The Maximum
Reliability Location Problem and e-Reliable P-
Center Problem: Derivatives of the Probabilistic
Location Set Covering Problem," 88-04.
Z.-P. Zhu, C. ReVelle and K. Rosing,
"Adaption of the Plant Location Model for
Regional Environmental Facilities and Cost
Allocation Strategy," 88-05.
M. Heller, J.L. Cohon and C. ReVelle, "The
Use of Simulation in Validating a Multiobjective
EMS Location Model," 88-06.
C. ReVelle, "Review, Extension and Prediction
in Emergency Service Siting Models," 88-07.
C. ReVelle and K. Hogan, "The Maximum
Availability Location Problem," 89-01.

Vol.43, No.3

R. Fletcher and E. Sainz de la Maza, "Nonlin-
ear Programming and Nonsmooth Optimization
by Successive Linear Programming."
M.C. Ferris and A.B. Philpott, "An Interior
Point Algorithm for Semi-Infinite Linear
J.V. Burke and S-P. Han, "A Robust Sequen-
tial Quadratic Programming Method."
N.R. Patel, R.L. Smith and Z.B. Zabinsky,
"Pure Adaptive Search in Monte Carlo
M. Sniedovich, "Analysis of a Class of
Fractional Programming Problems."
J.C. Bernard and J.A. Ferland, "Convergence
of Interval-Type Algorithms for Generalized
Fractional Programming."
S.M. Ryan and J.C. Bean, "Degeneracy in
Infinite Horizon Optimization."

Vol.44, No.1

M. Kojima, S. Mizuno and A. Yoshise,
"Polynomial-Time Algorithm for a Class of
Linear Complementarity Problems."
R.D.C. Monteiro and I. Adler, "Interior Path
Following Primal-Dual Algorithms. Part I:
Linear Programming."
R.D.C. Monteiro and I. Adler, "Interior Path
Following Primal-Dual Algorithms. Part II:
Convex Quadratic Programming."
A.C. Williams, "Marginal Values in Mixed
Integer Linear Programming."
N. Eagambaram and S.R. Mohan, "On
Strongly Degenerate Complementary Cones and
Solution Rays."
S.C. Fang and J.R. Rajaskera, "Quadratically
Constrained Minimum Cross-Entropy Analy-
Z. Win, "On the Windy Postman Problem on
Eulerian Graphs."
R.B. Bapat, "A Constructive Proof of a
Permutation-Based Generalization of Sperner's



number twenty-six

APRIL 1989


PAGE 9 number twenty-six APRIL 1989

Editor-in-Chief: Peter L. Hammer, Rutcor, Hill Center for the Mathematical Sciences,
Rutgers University, Busch Campus, New Brunsvick, NJ 08903.
New in 1988:

Editor: H.G. Jeroslow, Ass. Editors: E. Barrett,
H. Greenberg, F. Leimkubler, W. Marek,
Th. Morton, F. Tonge and A. Whinston
1988. 366 pages. ISSN 0254 5330
Annals of Operations Research, vol. 12
B. Jaumard, P.S.- Ow and B. Simeone, A Selected Artificial
Intelligence Bibliography for Operations Researchers
H. Wolfson, E. Schonberg, A. Kalvin and Y. Lamdan,
Solving Jigsaw Puzzles by Computer
K. Sycara, Utility Theory in Conflict Resolution
P.S. Ow and S.F. Smith, Viewing Scheduling as an
Opportunistic Problem-Solving Process
S. De, A Knowledge-Based Approach to Scheduling
in an F.M.S.
C.E. Bell, A least Commitment Approach to Avoiding
Protection Violations in Nonlinear Planning
T.L. Dean, Reasoning about the Effects of Actions in
Automated Planning Systems
D.P. .Miller, A Task and Resource Scheduling System for
Automated Planning
F. Glover and H.J. Greenberg, Logical Testing for
Rule-Base Management
J.N. Hooker, Generalized Resolution and Cutting Planes
R.G. Jeroslow, Alternative Formulations of Mixed Integer
D. Klingman el al., Intelligent Decision Support Systems:
A Unique Application in the Petroleum Industry
K. Funk, A Knowledge-Based System for Tactical Situation
R.R. Yager, A Note on the Representation of Quantified
Statements in Terms of the Implication Operation
L.D. Xu, A Fuzzy Multiobjective Programming Algorithm
in Decision Support Systems
D.P. Paradice, J.F. Courtney, Dynamic Construction of
Statistical Models in Managerial DSS
S.D. Burd, S.K. Kassicieh, A Prolog-Based Decision
Support System for Computer Capacity Planning

Editors: R.R. Meyer and S.A. Zenios
1988. 340 pages. ISSN 0254 5330
Annals of Operations Research, vol. 14
G.B. Dantzig, Planning Under Uncertainty Using Parallel
R. V. Helgason el al., A Parallelization of the Simplex
O.L. Mangasarian, R. De Leone, Parallel Gradient
Projection Successive Overrelaxation for Symmetric Linear
Complementary Problems and Linear Programs
J.-S. Pang, J.-M. Yang, Two-stage Parallel Iterative Methods
for the Symmetric Linear Complementarity Problem
A.T. Phillips, J.B. Rosen, A Parallel Algorithm for Solving
the Linear Complementarity Problem
D.P. Berlsekas, The Auction Algorithm: A Distributed
Relaxation Method for the Assignment Problem
M.D. Chang et al., A Parallel Algorithm for Generalized
S.A. Zenios, R.A. Lasken, Nonlinear Network Optimization
on a Massively Parallel Connection Machine
R.H. Byrd et al., Using Parallel Function Evaluations to
Improve Hessian Approximation for Unconstrained
Af.-Q. Chen, S.-P. Han, A Parallel Quasi-Newton Method
for Partially Separable Large Scale Minimization
M. Lescrenier, Partially Separable Optimization and Parallel
S. Wright, A Fast Algorithm for Equality-Constrained
Quadratic Programming on the Alliant FX/8
G.A.P. Kindervater, J.K. Lenstra, Parallel Computing in
Combinatorial Optimization
J. Plummer et al., Solving a Large Nonlinear Programming
Problem on a Vector Processing Computer
R.E. Haymond el al., A Shortest Path Algorithm in
Robotics and Its Implementation on the FPS T-20

Available 1984 1987:
Vol. 10-11: Ibaraki, T., Enumerative Approaches to
Combinatorial Optimization. 2 volumes, 1987. 602 pages.
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1987. 644 pages.
Vol. 7: Blaiewicz et al., Scheduling under Resource
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Vol. 6: Osleeb, J.P. and S.J. Rarick, Locational Decisions:
Methodology and Applications. 1986. 328 pages.

Vol. 4-5: Monma, C.L., Algorithms and Software for
Optimization. 2 volumes. 1986. 632 pages.
Vol. 3: Stecke, K.E. and R. Suri, Flexible Manufacturing
Systems: Research Models and Applications. 1985. 488 pages.
Vbl. 2: Thompson, R.G. and R.M. Thrall, Normative
Analysis for Policy Decisions, Public and Private. 1985.
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etp Iz Ca: l ze

PAGE10numbertwe sx

An Algorithmic Theory of Numbers,
Graphs and Convexity
by LAszl6 Lovisz
SIAM, Philadelphia, 1986
ISBN 0-89871-203-3

In combinatorial optimization one of the most outstanding re-
sults in the last decade is that under some appropriate assumptions
separation problems and the associated optimization problems are
equivalent from the complexity theoretical point of view. This theo-
rem was first published in "The ellipsoid method and its conse-
quences in combinatorial optimization," Combinatorica 1, 1981, 169-
197, by M. Gr6tschel, L. Lovasz and A. Schrijver, for which they
received the Fulkerson Prize in 1982 from the Mathematical Program-
ming Society. The technical details of this paper are rather sophisti-
cated, and it contains quite a lot of tricky but crucial steps.
If you want to learn more of these exciting
things, a good guide through the
subject-as this beautiful
book is going to be-
will help you to save
time and reveals the
decisive points
you have to
pay atten-

On the basis of that paper L. Lovhsz gave a series of

application of sophisticated funding procedures that are also dis-
lectures on that topic at the AMS-CBMS regional confer-
ence in Eugene, Oregon, August 1984, pointing out more
underlying material as well as the consequences and implications of
the results to combinatorial optimization. So the reader of these
lecture notes is at first introduced to the main ingredients, as there are
Sdiophantine approximation, bases reduction in lattices and the Ellip-
soid Method. The use of square roots and so forth compel the
application of sophisticated rounding procedures that are also dis-
cussed in detail. Here you will also learn how to deal with real
numbers and that in reality they are nothing more than a black box
with two slots. With these tools the notes show the complexity
theoretical equivalence of quite different descriptions of convex
bodies. Then it sketches also proofs of results like: linear program-
ming is ("partly strongly") polynomial, or for every fixed dimension
integer programming can be solved in polynomial time, and it evalu-
ates bounds of various measures of convex bodies like volume,

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I -

PAGE 11~ia number twenty- APRIL 1989


widths, diameter, etc. The above mentioned main theorem is proved,
of course, as well.
The last chapter surveys applications of these results to combina-
torial optimization with particular emphasis on cut problems, optimi-
zation in perfect graphs and minimization of submodular functions.
Here we find problems whose polynomial time solvability is known
only via the equivalence of separation and optimization.
Summarizing the style of representation of the book I completely
agree with the author when he says, "Throughout these notes, I have
put emphasis on ideas and illustrating examples rather than on
technical details or comprehensive surveys. A forthcoming book of
M. Gr6tschel, L. Lovisz and A. Schrijver (1985) will contain an in-
depth study of the Ellipsoid Method and the simultaneous dio-
phantine approximation problem, their versions and modifications,
as well as a comprehensive tour d'horizon over the applications in
combinatorial optimization." But these notes won't be obsolete when
the announced book is issued. They are definitely an excellent guide
for an advanced course in combinatorial optimization, while the joint
bookof Gr6tschel, Lovasz and Schrijver will rather serve as a reference
book containing elaborate proofs and further readings.
A. Wanka

Theory of Linear Integer Programming
by Alexander Schrijver
John Wiley and Sons, 1986
ISBN 0-471-90854-1

An impressive, indeed astounding, achievement of complete-
ness, conciseness, clarity, and depth, this book is a must-buy for any
researcher interested in linear and integer programming. The main
emphasis of the book is on theory, and the real strength derives from
this restriction since it permits an in-depth presentation of all relevant
theoretical results, both old and new.
The book consists of three parts. In the first one, relevant back-
ground material from linear algebra and the theory of lattices and
linear diophantine equations are introduced. The treatment includes
the important basis reduction method for lattices due to Lovasz. The
second part deals with polyhedra, linear inequalities, and linear
programming. Here too, all recent results are covered besides the
well-known ones, for example, Khachiyan's ellipsoid method, Kar-
markar's method for linear programming, Borgwardt's analysis of the
average speed of the simplex method, and Tardos' and Megiddo's
algorithms for linear programming. Finally, the third part covers

integer linear programming. Of the recent results included here, one
should mention Lenstra's algorithm for integer linear programming,
Seymour's decomposition theorem for totally unimodular matrices,
and the theory of total duality.
The number of references provided in the book is simply aston-
ishing. Every relevant reference is simply there, no matter which
portion of the book is examined. Clearly the author has made an
enormous effort to achieve such completeness.
In the preface, the author promises a companion volume on poly-
hedral combinatorics. Given such a wonderful book on linear and
integer programming, this reviewer looks forward with much antici-
pation to the publication of that second volume.
K. Truemper

Linear Programming in Infinite-
Dimensional Spaces
by Edward I. Anderson and Peter Nash
Wiley, Chichester, 1987
ISBN 0-471-91250-6

This monograph gives a systematic account of certain types of
infinite-dimensional linear programs and certain approaches to their
duality theory and their algorithmic solutions.
Chapter 1 is introductory and begins with a display of some
specific linear programs in which infinite-dimensional cones occur
most naturally, either the variable being taken from an infinite-di-
mensional space or the number of constraints being infinite, or both:
Bellman's bottleneck problem (1957), continuous-time network
flows, cutting and filling (Monge 1786 ("deblais et remblais"), Kan-
torovich 1942), the dual game. The classical theory of finite-dimen-
sional linear programs involving only a finite number of constraints
is sketched: duality, simplex method. Three possibilities of carrying
over the duality theory to infinite-dimensional cases are envisaged:
purely algebraic passage to an adjoint problem, finding a Lagrangian
which is suitable for infinite-dimensional generalization, or approxi-
mation by finite linear programs whose duals are to approximate
some kind of infinite-dimensional dual of the original problem.
Moreover, two essential features of the simplex method are pointed
out: restriction to basic solutions (i.e. extremal points of the solution
set, degeneracy and non-degeneracy of basic solutions, and passage
from onebasic solution to a better one (pivoting). To a large extent, the


II--~-~ ----- --- -- ---- ------------------~------~


number ftventy-six

APRIL 1989


^==P^ gst

book is devoted to generalizations of just thesetwo features to infinite-
dimensional cases.
Chapter 2 investigates the possibilities of a purely algebraic
approach to infinite-dimensional linear programs in general. Passage
to the dual program and some kind of weak duality are obvious here.
Basic solutions are defined in such a fashion that equivalence to
extremality and minimal support property are easily established.
Under some finite-dimensionality hypothesis, the existence of basic
solutions is proven. Non-degeneracy of basic solutions is defined by
a certain splitting property with respect to the underlying linear
space. Under the hypothesis that non-degenerate basic solutions
exist, strong duality is established.
With Chapter 3, topology comes in: dual pairs of topological
vector spaces, etc. The linear maps occurring in linear programs are
supposed to be continuous. Some weak duality results are easily ob-
tained. The possibility of a duality gap is displayed by an example of
Gale. In view of this possibility, a number of properties of linear
programs in the topological vector space setting are defined: consis-
tency (CONS), inconsistency (INC), boundedness (BD), unbounded-
ness (UBD), sub(in)consistency (SUBC/SUBINC),
sub(un)boundedness (SUBD/SUBUBD). A table of 6x6 entries ac-
cording to combinations of such properties of the primal and dual
program is set up and partly filled by examples. Upon introducing an
associated "modified homogeneous program," the classification
scheme is refined still further. The rest of the chapter is devoted to
conditions implying the absence of a duality gap or the existence of
optimal solutions. In addition to the continuity of the defining linear
map, closedness, existence of interior points, boundedness and com-
pactness conditions play a crucial role here.
Chapter 4 treats semi-infinite linear programs: either the number
of variables or the number of constraints remains finite. Special em-
phasis is given to such programs where the "infinite part" is either
countable or a continuum r R". An example of Karney (1981) displays
the possibility of a duality gap. Variants of the simplex algorithm are
discussed at length. One of the important applications is to uniform
The authors deal with the mass transfer problem in Chapter 5. Its
finite-dimensional version is a well-known linear program. For the
infinite-dimensional version a measure-theoretical setup is chosen
here, with a compact basic space F and a continuous transport cost
function on FxF. The dual problem is set up with pairs of continuous
functions on UxV, where U, V E F are the "deblai" or "remblai"
regions. This is natural in one way but leads to compactness problems
which are tackled by introducing L" and exploiting the continuity of

the cost function. This leads to satisfactory duality results. The
variant of the simplex algorithm discussed here at length goes back to
Andy Philpott's Cambridge Thesis (1982), the key notion being
"piecewise continuous assignment;" convergence to an optimal solu-
tion is proven. The notes to this chapter trace the history from Monge,
Dupin, Appell and others to Kantorovich (1942), Levin-Milyutin
(1978) and others.
In Chapter 6 the continuous-time dynamic version of the net-
work-flow problem is treated: the network remains finite, but con-
straints and flow strength vary in time. Ford-Fulkerson's reduction of
the discrete-time variant of this problem to a ("time-augmented")
static problem is displayed and taken as a motive for the definition of
"dynamic cuts" in the original problem. A key lemma shows that a
(dynamic) flow may be augmented if the sink is reachable at some
time, and a dynamic max-flow theorem involving piecewise continu-
ous flows is proven.
Chapter 7 deals with continuous linear programs of the type of
the bottleneck problem: maximize some weighted integrals under a
(time) continuum of constraints, i.e. solve a linear optimal control
problem. Solvability by piecewise linear functions is proved for so-
called separated continuous linear programs with linear or constant
weight and constraint functions. The notes to this chapter trace a vast
and not very homogeneous literature, including Russian papers not
utilizing simplex methods.
A variety of further infinite linear programs is treated in Chapter
8: the "capacity problem" of maximizing total electric charge subject
to an upper bound for the potential, given a conducting body; a con-
tinuous-time minimum-cost problem in finite networks; a continu-
ous-space max-flow problem in a planar region with piecewise
smooth boundary. Much of the material covered in this chapter is
from Philpott's thesis (1982). The notes also mention related investi-
gations, e.g. by Hu, Gomory, Jacobs, Seiffert, Strang, Iri.
One of the basic features of this most valuable, stimulating and,
within the deliberate choice of its topics, comprehensive monograph
is on duality and on continuous simulation of the simplex algorithm.
The presentation is of excellent clarity, the notations are very well
chosen. Who among the workers in this field would deny himself the
pleasure of getting so much well-presented exciting information in
such a slim volume? The non-specialist might enjoy it likewise.
K. Jacobs

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number twenty-six

APRIL 1989

number twenty-six

Combinatorial Theory and Statistical
by G. M. Constantine
Wiley, New York (1987)
ISBN 0-471-84097-1

As the author states in his preface, his book "is addressed to those
who are interested in understanding the fundamental techniques of
discrete mathematics as applied to statistical design," with four major
topics predominating: enumeration, graphs and networks, statistical
and combinatorial designs, and M6bius inversion. The chapter on
statistical designs nicely shows how much of the material treated in
the other eight chapters ties in with statistical design. The remainder
of the book (by far the largest portion) can be read as an advanced
exposition of some of the central parts of combinatorial theory. In my
opinion, the present book is eminently suitable for courses at the
graduate level; I feel that it might be a bit too demanding at the
undergraduate level. A unifying theme is the strong emphasis on
algebraic methods, in my opinion one of the big advantages of this
text. Surely any more penetrating study of combinatorics soon
becomes algebraic in flavour and methods.
Regarding the presentation, the author usually starts out with
simple motivational examples but rapidly builds towards non-trivial
results and interesting theorems which go much beyond what is
usually offered in a general textbook on combinatorics. Being a
specialist in design theory myself, I was, for instance, quite impressed
with the selection of topics in the chapter on combinatorial designs.
The style employed is engaging, to a large part avoiding the strictly
static and formal presentation usual in mathematics. This is one of the
few books I know which reads more like an oral presentation than a
formal text. The author does not hide behind his topic, and we get an
impression of his personality. I am quite convinced that he must be a
stimulating and popular teacher. In short, this text is not only
mathematically interesting but also fun to read.
Let me indicate the topics covered in some detail now. We have
nine chapters: (1) Ways to choose (covering the essentials of counting
and introducing, e.g. binomial numbers, Stirling numbers, Bell
numbers, Lah numbers); (2) Generating functions (introducing for-
mal power series, generating functions for some of the topics of
Chapter 1, recurrence relations as well as for labelled spanning trees
and partitions); (3) Classical inversion (considering various inverse
relations, Taylor expansion, formal Laurent series, Lagrange inver-
sion, generating functions in more detail, Gaussian polynomials); (4)

Graphs (including theorems of Euler, Ramsye and Turan and empha-
sizing graph spectra, e.g. treating strongly regular graphs, the adja-
cency and Kirchhoff matrices and graphs with extreme spectra); (5)
Flows in networks (giving the theorems of Birkhoff-von Neumann,
K6nig, P. Hall, Dilworth, Ford and Fulkerson, including some algo-
rithms and an introduction to matroid theory); (6) Counting in the
presence of a group (giving the counting theory of Polya and de
Bruijn, with many interesting examples like enumerating the number
of isomorphism classes of graphs); (7) Block designs (including t-
designs, Fisher's inequality for t-designs, classes of 2-designs,
Cameron's theorem on extending symmetric designs, the Bruck-
Ryser-Chowla theorem, automorphism groups of designs, associa-
tion schemes and Bose-Mesner algebras); (8) Statistical designs
(studying random variables, factorial experiments, blocking, 2-de-
signs as optimal designs, E-optimality and line graphs, mixed facto-
rial designs and orthogonal arrays); (9) M6bius inversion (presenting
the general inversion theory on locally finite partially ordered sets,
with interesting applications, e.g. to determining the number of
automorphisms of an abelian group).
I have two more comments: It would have been advisable to give
a few comments about issues of complexity in Chapter5; indeed, I find
the statement on p. 137 that the Hamiltonian circuit problem is "much
of the same spirit" as the one for Eulerian circuits rather misleading.
Regarding Chapter 7, non-trivial (simple) t-designs are now known to
exist for all values of t by a result of L. Teirlinck ("Nontrivial t-designs
without repeated blocks exist for all t," Discr. Math. 65 (1987), 301-
311). Finally, there are some typographical errors, however, which do
not really affect the readability of the text.
Summing up, then, this is one of the best texts on combinatorics
for graduate students now available, and I am happy to highly recom-
mend it to everybody interested in this area.
D. Jungnickel




APRIL 1989



_ I ------------~-------~~



Nonconvex Programming
by Forenc Forg6
Akademiai Kiad6, Budapest, 1988
ISBN 963-05-4459-9

The book deals with those types of nonlinear programs which
find less attention in standard nonlinear programming texts, namely
problems where a local maximum may not be a global maximum.
Emphasis is placed on solution methods. Theoretical results on the
models involved are only discussed to the extent they are needed to
describe the algorithms. Small numerical results illustrate the tech-
niques. Computational experience with and relative efficiency of
algorithms find little attention. Two topics are excluded from the
discussion: integer programming and global optimization (uncon-
strained programming).
The first two chapters provide the theoretical basis for the re-
maining eight chapters on algorithms. Optimality conditions and
duality as well as convex and concave envelopes of functions are
discussed. In Chapter 3 three major solution strategies in nonconvex
programming are introduced: complete and implicit enumeration,
branch and bound, and cuts. Chapters 4 and 5 deal with two classical
problems: maximization of a (quasi)convex function over a convex
polytope and maximization of a linear function subject to convex
greater-or-equal constraints. Various cutting methods are discussed.
In chapter 6 general nonconvex programs are addressed where con-
vexity of the objective function and/or the constraints is replaced by
continuity or differentiability. In addition to cutting-plane methods,
branch and bound methods are presented. Chapter 7 then focuses on
nonconvex quadratic programs. Here cutting-plane generating
methods are discussed which exploit the quadratic nature of the
objective function. Also special cases such as bilinear programs and
generalized linear programs are considered. Chapter 8 deals with
algorithms for fixed charge problems. In Chapter 9 SUMT-like proce-
dures are described, and Chapter 10 discusses decomposition proce-
dures for nonconvex programs.
The material is well organized. The reader is assumed to have a
knowledge of elementary calculus and linear algebra and familiarity
with basic linear and nonlinear programming. The book seems to be
particularly useful as an introduction to the field. The experienced
researcher in nonlinear programming may miss more up-to-date
results published in the 80s.

Graphen, Netzwerke und Algorithmen
by Dieter Jungnickel
Bibliographisches Institut, Mannheim 1987
ISBN 3-411-03126-3,68,-DM

Combinatorial optimization, along with graph algorithms and com-
plexity theory, is booming. This results in a great number of textbooks
devoted to the subject. However, there is a conspicuous lack of a good
German monograph on the topic. Dieter Jungnickel tries to fill this
gap for that part of combinatorial optimization that can be treated
within the language of graph theory.
The book treats the most prominent problems which are poly-
nomially solvable. The Traveling Salesman Problem is discussed as a
paradigm of an NP-complete problem.
Chapters 1 and 2 are of an introductory nature. The basic notions
of graph theory and some algorithmic topics are discussed. Hier-
holzer's algorithm for Euler circuits serves for introducing data
structures and polynomiality. Pidgin Pascal is used as the "language"
for presenting the algorithms. NP-completeness is mentioned; a
somewhat deeper discussion is postponed until the last chapter.
Chapter 3 deals with shortest paths. The traditional algorithms are
discussed. Two more advanced topics are mentioned: the use of more
complicated data structures in the design of fast algorithms and path
algebras. Trees and matroids are treated in the subsequent chapter.
Besides primal and dual greedy algorithms, more theoretical aspects
like the matrix tree theorem are considered. Chapters 5 to 9 are
concerned with various aspects and applications of network flows.
The most efficient max flow algorithms (without taking advanced
data structures into account) are presented in a very clear way. The
chapter on combinatorial applications of network flows contains, in
addition to more standard topics, the proof of Baranyai's theorem.
This result is even though quite pure in nature a true gem for
somebody in search of applications of graph algorithms in other areas
of mathematics. Circulations, network synthesis and algorithmic
connectivity is treated in detail. Chapters 10 and 11 are devoted to
matching. After a nice exposition of unweighted matching in
chapter 10, the author meets the limits he has imposed upon himself
in chapter 11. It is the declared aim of the book to explain everything
without using linear programming techniques. Here the author has
to concede that a treatment of Jack Edmonds' matching algorithm
makes little sense without LP duality. However, the development of
the Hungarian algorithm seems quite natural (and historically justi-
fied). The last chapter is definitely a highlight of the book. Using the
traveling salesman problem, it introduces the reader to many aspects

_ I-- --- -------------~-------~

number twenty-six

APRIL 1989


of and algorithmic approaches to NP-complete problems. It can serve
as a sound basis for a deeper study of hard decision problems.
The text is well written; most exercises (at the end of each chapter)
are quite enlightening and the hints are clear. Algorithms are de-
scribed very thoroughly. The list of references is impressive and gives
good guidance for further reading. At several points the reader finds
himself right at the frontier of current research (e.g. the section on
exact matchings. The book serves as a sound foundation for more
advanced topics, e.g. polyhedral combinatorics. It presents the matter
as a coherent and unified whole. I like the book a lot. It seems to fill
the gap I was talking about in the introduction. It might even be
desirable to compile an English translation of it.
The book can be recommended to beginners as an introductory
text as well as for researchers in academics and industry as a reference.
M. Leclerc

Application for Membership

Mail to:

c/o International Statistical Institute
428 Prinses Beatrixlaan
2270 AZ Voorburg
The Netherlands

Cheques or money orders should be made payable to
in one of the currencies listed below.
Dues for 1989, covering subscription to volumes 43-45 of
Mathematical Programming, are Dfl.115.00 (or $55.00 or
DM100.00 or 32.50 or FF345.00 or Sw.Fr.86.00).
Student applications: Dues are one-half the above rates.
Have a faculty member verify your student status and send
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I wish to enroll as a member of the Society. My subscrip-
tion is for my personal use and not for the benefit of any
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number twenty-six

APRIL 1989


PE6m w y A


George L. Nemhauser, MPS Chairman-Elect, received
the Kimball Medal for distinguished service to ORSA
and to the Operations Research profession at the
Denver meeting of ORSA....Richard H. F. Jackson has
been appointed the Deputy Director of the Center for
Manufacturing Engineering at the National Institute
of Standards and Technology (formerly National
Bureau of Standards)....Ralph E. Gomory, pioneer in
mathematical programming and more recently a
senior vice president at IBM, will become president of
the Sloan Foundation in June. Dr. Gomory received
the National Medal of Science in July 1988....Tom
Magnanti (MIT) will spend the Fall, 1989 semester at
CORE....Anna B. Nagurney (University of
Massachusetts) is spending 1988-89 at MIT.
I Deadline for the next OPTIMA is June 15, 1989.

Books for review should be
sent to the Book Review Editor,
Prof. Dr. Achim Bachem,
Mathematiches Institute der
Universitiit zu Kiln,
Weyertal 86-90, D-5000 K6ln,
West Germany.

Journal contents are subject
to change by the publisher.
Donald W. Hearn, Editor
Achim Bachem, Associate Editor
Published by the Mathematical
Programming Society and
Publication Services of the
College of Engineering,
University of Florida.

k P T I M A

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


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APRIL 1989

number twenty-six

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