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Title: Optima
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Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: November 1988
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PTIMA

MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER


N 25


NOVEMBER 1988


Tokyo Symposium Report



T HE first MPS meeting in Asia proved to
be a big success with 668 mathematical programmers
from 36 countries attending the 13th International Sym-
posium at Chuo University, Tokyo, August 28 through
September 2. The meeting was jointly sponsored by
the Operations Research Society of Japan, Hidinori Mo-
rimura, President, with cosponsorship by the Interna-
tional Federation of Operational Research Societies
(IFORS), the Asian-Pacific Operational Research
Societies, and several domestic societies. The organiz-
ing committee headed by Masao Iri, Hiroshi Konno
and Kaoru Tone drew high praise for the smoothly-run
technical and social programs.


MPS Chairman Michel Balinski addressed the member-
ship at the opening session on Monday which also
featured the awarding of prizes and a plenary lecture
by Dantzig Prize recipient Michael Todd (Comell).
Fulkerson Prizes went to Eva Tardos (MIT and E6tv6s
Univ.) for her development of strongly-polynomial
algorithms for the minimum cost flow problem and
Narendra Karmarkar (AT&T Bell Labs) for his break-
through work on interior methods for linear program-
ming. The second Orchard-Hays prize was awarded to
Tony J. Van Roy (Bank Brussels Lambert) and Laurence


continues, page two


OPTIMA
number 25


TOKYO SYMPOSIUM REPORT 1-3
PRIZES & AWARDS 4-7
CONFERENCE NOTES 8
TECHNICAL REPORTS &
WORKING PAPERS 9
88 ELECTION RESULTS 9
JOURNALS 10
BOOK REVIEWS 11-15
GALLIMAUFRY 16


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




PAGE 2


Tokyo Symposium Report

CONTINUED

Wolsey (Universite Catholique de
Louvain) for their work on automatic
reformulation of mixed integer pro-
grams. The first A. V. Tucker Student
Prize went to Andrew V. Goldberg
(Stanford) for his MIT doctoral thesis
entitled "Efficient Graph Algorithms for
Sequential and Parallel Computers."
(See accompanying articles.)
The scientific program consisted of 557
presentations in 185 sessions including
seven tutorial/survey sessions, one
special session for prize recipients, and
two memorial lectures honoring Martin
Beale and L. V. Kantorovich. The Beale
lecture was given by John Tomlin
(Ketron), and I. V. Romanovskii (Lenin-
grad Univ.) delivered the lecture
honoring Kantorovich.
Publications from the meeting will
include a conference proceedings of
principal lectures and a special issue of
Mathematical Programming, Series B,
containing high-quality application
papers selected from those presented at
the Symposium.
At the General Assembly on Thursday
it was announced that Robert E. Bixby
(Rice) will be the new editor of Mathe-
matical Programming Series A begin-
ning in mid-1989 and that W. R. Pul-
leyblank (Waterloo) is the first editor of
Series B, which replaces the Studies
series. Also, Karla Hoffman (George
Mason) announced plans for a member-
ship drive; the Committee on Algo-
rithms announced that Faiz Al-Khayyal
(Ga. Tech.) will be the new US co-editor
of the COAL newsletter; and Jan Karel
Lenstra announced that the 14th
Symposium will be held in Amsterdam
in 1991.
The social program included a welcom-
ing reception on Monday evening at the
Symposium site and a lavish Official
Banquet on Wednesday evening at the
famous Chinzan-so Restaurant and
gardens, a tranquil island in the center
of busy Tokyo.


Tutorials and Survey Lectures at MPS Symposium



Below is a list of the tutorial and survey lectures given at the Tokyo meeting:


Thomas L. Magnanti Mathematical Programming and Network Design
Michael Florian Mathematical Programming Applications in National, Regional
and Urban Planning
Andreas Griewank Automatic Differentiation
Bernhard Korte Applications of Combinatorial Optimization
Robert B. Schnabel Sequential and Parallel Methods for Local and Global Uncon-
strained Optimization
Philip E. Gill Nonlinear Programming
Freerk Auke Lootsma Multi-Objective Programming
Krzysztof C. Kiewiel Nondifferentiable Optimization
Roger J-B Wets Stochastic Programming: State-of-the-Art Survey
Yoshitsugu Yamamoto Fixed Point Algorithms for Stationary Point Problems
Laszlo Lovasz Recent Development of Number Theory and its Applications


r




\E


ABOVE LEFT: INCOMING MPS CHAIRMAN
GEORGE NEMHAUSER AND GEORGE
DANTZIG CELEBRATE AT BANQUET.
RIGHT: MPS CHAIRMAN MICHEL
BAINSKI. BELOW: SYMPOSIUM
CHAIRMAN MASAO IRI


0 P T IM A number twenty-five


NOVEMBER 1988









Opening Address of the Chairman, Mathematical Programming Society


13th Symposium on Mathematical Programming, Tokyo, Japan, August 29, 1988


T HE FIRST International Sympo-
sium of our community, now
organized and called the
Mathematical Programming
Society, bore the number 0 and
was held in Chicago in 1949.
Today, 39 years later, after eight symposia in
the United States and one each in the United
Kingdom, the Netherlands, Hungary, Canada
and the Federal Republic of Germany, we
begin the thirteenth here in Japan.
What are we? What is the nature of the in-
tellectual endeavor in which we are engaged?
In 1946, at the first post-war international
gathering of mathematicians held at Prince-
ton', Herman Weyl railed against a prescrip-
tion offered years before by Minkowski,
which was: "... to face problems with a mini-
mum of blind calculation, a maximum of see-
ing thought." He proffered a counter recipe in
the ti1.l. .. ing words, "I find the present state
of mathematics, that has arisen by going full
speed ahead under this slogan, so alarming
that I propose another principle: whenever
you can settle a question by explicit construc-
tion, be not satisfied with purely existential ar-
guments." Fifteen years before Weyl had ex-
pressed his fears that the mathematical sub-
stance in the formulization of which mathema-
ticians had exercised their powers showed
signs of exhaustion. By 1946 he saw some
hope but "... the deeper one drives the spade
the harder the digging gets; maybe it has
become too hard for us unless we are given
outside help be it even by such devilish de-
vices as high-speed computing machines."
We have those devilish devices. They have
helped considerably. But the problem re-
mains and will forever recur. The need to keep
our ship afloat maintaining theoretical depth
yet steering it with empirical demands and
practical uses, remains and must forever
remain a paramount concern. As Karl Marx so


aptly said (at the tender age of 17), "And so we
must be on guard against allowing ourselves
to fall victim to that most dangerous of all
temptations: the fascination of abstract
thought."
As evidence of the power avoiding that
temptation, it is fitting for me to recall two
outstanding members of our community who
have died since our last Symposium and
whose memories will be celebrated in special
sessions this week: Martin Beale and Leonid
Kantorovich. They offer striking contrasts,
yet remarkable similarities, and they were
both central to mathematical programming.
Martin Beale's distinctive flavor was his
untiring devotion to making mathematical
programming work in practice: he made con-
tributions to theory but more important he
brought an exceptional insight and flair and
insistence to modelling problems, imple-
menting algorithms, developing systems for
honest-to-God users. He came from a pecu-
liarly British some might say Cambridge -
mathematical tradition that emphasized
what I would call engineering analysis, not to-
pology- or algebra. He was, to borrow a
phrase, the outstanding mathematical pro-
gramming "numbers engineer".
Leonid Kantorovich, on the other hand, is-
sued from the continental school with its deep
roots in the traditional areas of mathematical
analysis. A prolific mathematician with far-
ranging interests, named a full professor in
his early twenties, his fundamental contribu-
tions to functional analysis are considered by
many his outstanding contribution. But moti-
vated by economic problems he was led to
linear programming and to interpretations of
duality that were not well received in Stalin's
Russia. And yet this may well have led to this
study of the Monge-Kantorovich problem, to
his formulations of duality in infinite pro-
gramming, and the important results in


probability that were obtained by him and his
collaborators and which are still a major area
of research. Indeed, as his wife stated at the
Beale Memorial Conference one year ago,
only now have many of his ideas in economics
been finally accepted in his country, too late
for him to know. So, he was an applied man in
another sense, not numbers, but practical eco-
nomic theory having important real implica-
tions.
Both men mixed theory with engineering,
or mathematics with computation, or mod-
elling with interpretation; their inspirations
and motivations came from spheres other
than mere mathematics; and this I believe is
central to the intellectual endeavor of our
subject. It seems that this mix of concerns is
very well represented at this Symposium,
perhaps better than ever before, although it is
a pity that so few economists are now in-
volved as versus their heavy participation in
the early years. They do, after all, need help!
It is my hope that a more thorough blend, a
more precarious balance on the razor's edge
between theory and use and computation,
will be realized by our journal in Series A and
B and in the activities of a growing member-
ship in our Society, and I cannot but hold up
for all to see the examples set by Kantorovich
and Beale.
We honor in them a Russian and an Eng-
lishman, and this first Symposium in the Far
East (as versus what may well be the Far West
in the eyes of our hosts) underlines the univer-
sal nature of our endeavors and the interna-
tional cast of characters that are involved. It is
my hope that an increasingly more thorough
blend of nationalities will be represented in
our Society, with the enhanced participation
of the relatively underrepresented Asian na-
tions, and in particular, those of the large
nations of China, Japan and the U.S.S.R.
-Michel L. Balinski


'The Princton University Bicentennial Conference on the Problems of Mathematics, Princeton, NJ, 1946.


PAGE 3


O P TI M A number twenty-five


NOVEMBER 1988






Tokyo Symposium Report




Prizes and Awards


Andrew Goldberg Awarded Tucker Prize


HAROLD KUHN PRESENTS TUCKER PRIZE TO ANDREW GOLD

The A. W. Tucker Prize, established in 1985,
and awarded for the first time at the Tokyo
Symposium, is for the outstanding paper by
a student. The paper must be solely au-
thored and completed since the beginning of
the calendar year of the preceding Sympo-
sium. The paper and the work on which it is
based should have been undertaken and
completed in conjunction with a degree
program.
Professor Tucker's contributions to our field
have been outstanding. These contributions
have come not only in the form of funda-
mental research but, in a very substantial
way, from his role as teacher, mentor and
counselor. The group of individuals who
have been Al Tucker's students, either
literally or figuratively, has had a remarka-
bly broad and deep impact on mathematical
programming. The entire field has benefited
from his stimulation and guidance. There-
fore, it is especially appropriate that this
prize, named in honor of Al Tucker should
be for student research.


As the first recipient
of the A. W. Tucker
Student Prize, the
Award Committee
has chosen Andrew
Vladislav Goldberg
for his doctoral thesis
at MIT, completed in
January 1987, and

-erPatr aI compulters.
entitled "Efficient
Graph Algorithms for
Sequential and
Parallel Computers."
The committee found
his work to be a deep
and impressive
contribution to the lit-
erature. His algo-
BERG rithms or extensions
of his algorithms for
the maximum flow problem and for the
minimum cost flow problem are the best in
terms of worst case complexity. Goldberg's
work on parallel algorithms is also very
important. Both his contributions to
treatments of network flows and to
"symmetry breaking techniques" are
improvements on previous results in terms
of worst case complexity for parallel
algorithms.
Andrew Goldberg was born in 1960 in
Moscow and emigrated to the United States
in 1979. He received a Bachelor of Science
degree in Mathematics at MIT in 1982, a
Master of Science degree in Computer
Science at the University of California,
Berkeley, in 1983 and a Doctor of Philoso-
phy degree in Computer Science at MIT in
1987. He is now on the faculty of Stanford
University in the Department of Computer
Science.
The members of the Award Committee are
Robert Bland, chair, Harold Kuhn, Alan
Tucker and Laurence Wolsey.
-Harold Kuhn


The 1988 Fulkerson Prizes


The Fulkerson Prize for outstanding
papers in the area of discrete
mathematics is sponsored jointly by
the Mathematical Programming
Society and the American Mathe-
matical Society. Beginning in 1979,
up to three awards are being pre-
sented at each (triennial) Interna-
tional Symposium of the Mathe-
matical Programming Society. The
prize was established to encourage
mathematical excellence in the
fields of research exemplified by the
works of Delbert Ray Fulkerson.
The specifications for the Fulkerson
Prize read:
Papers to be eligible for the Fulkerson
Prize should have been published in a
recognized journal during the six
calendar years preceding the year of the
Congress. This extended period is in
recognition of the fact that the value of
fundamental work cannot always be
immediately assessed. The prize will be
given for single papers, not series of
papers or books, and in the event of
joint authorship the prize will be
divided. The term "discrete mathemat-
ics" is intended to include graph theory,
networks, mathematical programming,
applied combinatorics and related
subjects. While research work in these
areas is usually not far removed from
practical applications, the judging of
papers will be based on their mathe-
matical quality and significance.
The Selection Committee for the Fulkerson
Prize of 1988 consisted of Manfred Padberg
(chairman), Martin Gr6tschel and Gian-
Carlo Rota and recommended two awards.
The recommendations (printed below) were
accepted by the Mathematical Programming
Society and the American Mathematical
Society.


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


PAGE 4


OPTIM A number twenty-five


NOVEMBER 1988






Tokyo Symposium Report


First Award
The Selection Committee recommended
that one award be given to tva Tardos for
her paper, "A strongly polynominal
minimum cost circulation algorithm,"
Combinatorica 5 (1985) 247-256.
tva Tardos is a member of the Department
of Computer Science, E6tv6s Lorand
University, Budapest, Hungary, currently
visiting the Department of Mathematics,
M.I.T., Cambridge, USA.
The minimum cost flow problem requires
finding, in a directed graph with arc
capacities and costs, a flow of given value
and minimum cost. It is one of the central
problems with respect to theory and
practice in combinatorial optimization.
A first successful phase of research on this
problem was in the late fifties and early
sixties when Ford, Fulkerson, Minty and
Yakovleva developed efficient algorithms
for its solution.
Edmonds and Karp (1972) found an
algorithm for the solution of the minimum
cost flow problem that provably runs in
polynomial time. A shortcoming of their
algorithm is that the number of arithmetic
operations performed is not bounded by
any function of the combinatorial size
(number of nodes and arcs) of the input and
that it depends on the number of digits of
the input numbers (costs). In their own
words:
"A challenging open problem is to give a
method for the minimum cost flow problem
having a bound of computation which is a
polynomial in the number of nodes, and is
independent of both costs and capacities."
[Edmonds & Karp, 1972]
Algorithms satisfying this requirement
have later been termed "strongly polyno-
mial". (For an exact definition of this term
see, e.g., Johnson (1987) or Gr6tschel,
Lovdsz, Schrijver (1988).)
The problem posed by Edmonds and Karp
remained open for 13 years despite the fact


that several eminent combinatorial optimiz-
ers had tried to solve it. Little progress was
made in the direction of strong polynomial-
ity until this problem was resolved by Eva
Tardos in 1985.
The method of Tardos is based on original
and beautifully simple ideas. Tardos'
solution is iterative, but in contrast to the
scaling techniques, her iterations reduce the
number of constraints in the problem rather
than increase the number of bits of accuracy
in the solution. So the number of iterations
is bounded by
the number of
constraints and
S'. independent of
the data
?'~ themselves.
tva Tardos'
work shows a
S deep under-
S standing of net-
\1 jE work flow
techniques and
throws new
EVA TARDOS RECEIVES light on the
AFULKERSONPRIZE
notion of
strong polynomiality. Apart from the
evident theoretical significance of the notion
of strong polynomiality, argued in detail,
e.g., by David Johnson (1987) in his column,
the impact of Tardos' breakthrough is
indicated by the rapid subsequent develop-
ments (with Tardos herself leading the
way). An improvement by Fujishige (1986)
followed almost instantly. Tardos (1986)
extended her method to solve any linear
program with polynomially bounded
integer coefficients in strongly polynomial
time (independent of the real numbers that
appear on the right hand side of the
constraints and as coefficients of the
objective function). Fujishige's work was
further improved by Galil and Tardos (1986)
and more recently by Orlin (1988) and the
bound obtained became competitive, even
in the classical sense, with the Edmonds-
Karp bound.


In another direction, Frank and Tardos
(1987) gave a different and more general
solution to the problem posed by Edmonds
and Karp, using Lovasz' algorithm for
simultaneous diophantine approximation
(cf. Lenstra, Lenstra, Lovasz (1982)). This
latter approach yields more general results
as it hardly takes notice of the combinatorial
structure of the constraints. Indeed, it works
in combination with the ellipsoid method
where polynomial-time optimization over
certain combinatorially defined polytopes
with exponentially many facets is achieved
(see Gr6tschel, Lovisz, Schrijver (1988)).
The insights gained from Tardos' fundamen-
tal results now highlight the problem of
whether or not the general linear program-
ming problem can be solved in strongly
polynomial time. Tardos' paper represents
one of the most significant advances in
recent years in combinatorial optimization
as well as in the theory of algorithms. It
demonstrates that the new algorithmic
paradigm used can successfully be em-
ployed to gain new insight into the structure
of central problems of combinatorial
optimization.


Second Award
The Selection Committee recommended that
one award be given to Narendra Karmarkar
for his paper, "A new polynomial-time
algorithm for linear programming,"
Combinatorica 4, (1984), 373-395.
Narendra Karmarkar is a Bell Labs Fellow at
AT&T Bell Laboratories, Murray Hill, New
Jersey, USA.
The linear programming problem is the
problem of optimizing a linear objective
function of several variables subject to linear
constraints in the form of equations and
inequalities. It is a problem of central
importance in modern applied mathematics
and provides a widely used model for the
planning of economic activities in the public
and private sectors. One of the reasons for


CONTINUES


11-~1------~


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


PAGE 5


0 P T IM A number twenty-five


NOVEMBER 1988






Tokyo Symposium Report









Prizes and Awards continued


the success of linear programming rests with
the fact that it has become possible to solve
the very large problem sizes that arise in
today' applications. The most frequently
used method to solve linear programming
problems to date is the simplex method.
The simplex method was invented around
1947 by George Dantzig. It starts by finding
an extreme point of the polyhedral set
defined by the linear constraints and
proceeds by moving on an edge of the
polyhedral set from one extreme point to a
"better" adjacent one until a sufficient
criterion for optimality is satisfied. It is thus
an iterative method that finds an optimum
by proceeding on the boundary of the
polyhedral set. In his own words:


X _


MARTIN GROTSHEL (RIGHT) AWARDS FULKERSON PRIZE
TO NARENDRA KAMAR

"While the simplex method appears to be a
natural one to try in the n-dimensional
space of the variables, it might be expected,
a priori, to be inefficient as there could be
considerable wandering on the outside
edges of the convex of solutions before an
optimal extreme point is reached." [Dantzig,
1963, p.1601
Nevertheless, early computational studies
beginning with Hoffman et al. (1953)
established empirically the efficiency of the
simplex method and its superiority to


several alternative methods for the resolu-
tion of linear programming problems. The
basic question of the worst-case behavior of
the simplex method was settled in the
negative by Klee & Minty (1972). It took ten
more years before the fundamental question
of polynomial solvability of linear program-
ming problems was answered in the
positive by the ellipsoid method of
Khachiyan (1979). (Fulkerson prizes for this
work were awarded in 1982.)
Narendra Karmarkar's work presents a new
polynomial-time algorithm for linear pro-
gramming that requires O(n35L) arithmetic
operations on O(L) bit words where n is the
number of variables and L is the number of
bits in the input. To achieve this result the
S general linear programming
problem is first brought into a
standard form that permits a
feasible starting point in the
center of a simplex. The iterative
step consists of optimizing a
linear function subject to a
homogeneous system of equa-
tions over a sphere inscribed into
the simplex and the subsequent
application of a projective trans-
formation that maps the new
iterate back into the center of the
simplex. The linear objective
function that is optimized at each
step changes from iteration to it-
eration. The sequence of points
generated this way is shown to
converge to an optimal solution
in polynomial time using a surrogate
objective function that ensures the necessary
monotonicity and decrease from step to
step.
Different from the simplex method and the
ellipsoid method Narendra Karmarkar's
algorithm is an interior point method that
realizes the intuitively most appealing idea
of "shooting" through the polyhedron to an
optimal point. It is based on a genuinely
new idea and has revived interest in the
search for faster and ever more efficient


ways of solving large-scale linear program-
ming problems. Within a short span of four
years, Karmarkar's new method has already
generated numerous subsequent develop-
ments, given a fresh impulse to the field of
mathematical programming and increased
the hope of extending linear and nonlinear
programming problem-solving capabilities
to problems of sizes bigger than those
solved today.
An entire issue edited by Nimrod Megiddo
(1986) of the journal Algorithmica is
devoted to related nonlinear programming
approaches to linear programming. Papers
by Iri and Imai (1986), de Ghellinck and Vial
(1986) and others explore Newton-type and
related descent methods for linear program-
ming. Karmarkar's work stimulated
renewed interest in the "barrier methods" of
nonlinear programming, see e.g. Gill,
Murray, Saunders, Tomlin and Wright
(1986). Connections to areas of classical
mathematics such as partial differential
equations, differential geometry and
convexity are investigated in several papers
by Bayer and Lagrias (1986). Affine variants
of Karmarkar's algorithm are discovered
and rediscovered, see Dikin (1967) and
Barnes (1986). Huard's method of analytic
centers (1967) is re-examined in a new light,
see Sonnevend (1985) and Mehrotra and Sun
(1988).
In summary, Narendra Karmarkar's work
represents one of the most significant ad-
vances in recent years in mathematical
programming as well as in the theory of
algorithms and has already left a major
imprint on the field of linear and nonlinear
computation.

References
[1] E. Barnes (1986), A Variation on Karmarkar's
Algorithm for Solving Linear Programming
Problems. Mathematical Programming, 36:174-182.
[2] D.A. Bayer and J.C. Lagarias (1986), The
Nonlinear Geometry of Linear Programming.
Preprints AT&T Laboratories, Murray Hill, NJ.


--~-s


PAGE 6


0 P T I A numbEer twenty-f~ive


NOVEMEVBER 1988




PAGE 7


[3] G.B. Dantzig (1963), Linear Programming and
Extensions. Princeton University Press, Princeton,
NJ.
[4] G. de Ghellinck and J.-P. Vial (1986), A
Polynomial Newton Method for Linear
Programming. Algorithmica, 1:425-453.
[5] I.I. Dikin (1967), Iterative Solution of Problems
of Linear and Quadratic Progamming. Soviet
Mathematics Doklady, 8:674-675.
[6] J. Edmonds and R.M. Karp (1972), Theoretical
Improvements in Algorithmic Efficiency for
Network Flow Problems. Journal of Association for
Computing Machinery, 19:248-264.
[7] A. Frank and t. Tardos (1987), An Application
of Simultaneous Approximation in Combinato-
rial Optimization. Combinatorica, 7:49-65.
[8] S. Fujishige (1986), A Capacity-Rounding
Algorithm for the Minimum-Cost Circulation
Problem: A Dual Framework of the Tardos
Algorithm. Mathematical Programming, 35:298-308.
[9] Z. Galil and t. Tardos (1986), An O(n2(m + n
log n) log n) Min-cost Flow Algorithm. Proc. 27th
IEEE FOCS Symp., Toronto, 1-9.
[10] P.E. Gill, W. Murray, M.A. Saunders, J.A.
Tomlin and M.H. Wright (1986), On Projected
Newton Barrier Methods for Linear Program-
ming and An Equivalence to Karmarkar's
Projective Method. Mathematical Programming,
36:183-209.
[11] M. Gr6tschel, L. Lovdsz and A. Schrijver
(1988), Geometric Algorithms and Combinatorial
Optimization. Springer Verlag, Heidelberg.
[12] A.J. Hoffman, M. Mannos, D. Sokolowsky
and N. Wiegman (1953), Computational
Experience in Solving Linear Programs. J. Soc.
Indust. Appl. Math., 1:17-33.
[13] P. Huard (1967), Resolution of Mathematical
Programming with Nonlinear Constraints by the
Method of Centers. In J. Abadie, Editor, Nonlinear
Programming, North-Holland, Amsterdam.
[14] M. Iri and H. Imai (1986), A Multiplicative
Barrier Function Method for Linear
Programming. Algorithmica, 1:455-482.
[15] D.S. Johnson (1987), The NP-Completeness
Column. Journal of Algorithms, 8:285-303.
[16] L.G. Khachiyan (1979), A Polynomial
Algorithm in Linear Programming. Soviet
Mathematics Doklady, 20:191-194.
[171 V. Klee and G. Minty (1972), How Good is
the Simplex Method? In O. Shisha, Editor,
Inequalities-III, Academic Press, New York and
London, 159-175.


RIGHT: DANTZIG PRIZE
WINNER MICHAEL TODD
DELIVERS PLENARY
LECTURE.
BELOW: LAURENCE
WOLSEY ACCEPTS THE
ORCHARD-HAYES PRIZE
FROM JOHN TOMLIN ON
BEHALF OF HIMSELF AND
TONY VAN ROY FOR THIER
JOINT WORK IN INTEGER
PROGRAMMING.


[18] A.K. Lenstra, H.W. Lenstra, Jr. and L. Lovasz
(1982), Factoring Polynomials with Rational
Coefficients. Mathematische Annalen, 261:515-534.
[19] N. Megiddo (1986), Introduction: New
Approaches to Linear Programming. Algorith-
mica, 1:387-394.
[20] S. Mehrotra and Jie Sun (1988), A Method of
Analytic Centers for Quadratically Constrained
Convex Quadratic Programs. Preprint, North-
western University, Evanston, IL.
[21] J.B. Orlin (1988), A Faster Strongly Polyno-
mial Minimum Cost Flow Algorithm. In: Proc,
20th ACM Symp. on Theory of Computing, 377-387.
[22] G. Sonnevend (1985), An "Analytical Center"
for Polyhedrons and new Classes of Global
Algorithms for Linear (Smooth, Convex) Pro-
gramming. In A. Prekopa, J. Szelezsan and B.
Strazicky, editors, Systems Modelling and
Optimization. Lecture Notes in Control and
Information Sciences 84, Springer, Berlin, 866-
875.
[23] t. Tardos (1986), A Strongly Polynomial
Algorithm to Solve Combinatorial Linear
Programs. Operations Research, 34:250-256.


Mike Todd Receives 1988
Dantzig Prize



The 1988 George B. Dantzig Prize has been
awarded to Michael J. Todd, Leon C. Welch
Professor, in the School of Operations Re-
search and Industrial Engineering at Cornell
University. The character for this award
species that, "The prize is awarded for origi-
nalwork, which by its breadth and scope, con-
stitutes an outstanding contribution to the
field of mathematical programming." It is
precisely the breadth and scope of Professor
Todd's many important contributions in
mathematical programming which form the
basis for this award.
The extraordinary success of linear program-
ming, ranging from broad practical applica-
bility of the linear model to its supporting
combinatorial and computational methodol-
ogy, is the cornerstone of the field of mathe-
matical programming. Indeed, it is primarily
Dantzig's seminal accomplishments in this
area which this prize seeks to commemorate.
Today, some four decades after the introduc-
tion of the simplex method, linear program-
ming remains a very active and fertile area for
research, as evidenced by recent work on (i)
oriented matroids as an abstract combinato-
rial model for linear programming and the
simplex method, (ii) average-case analysis of
the simplex algorithm as an approach to ex-
plain its observed computational efficiency
and (iii) polynomial-time ellipsoidal and inte-
rior-point algorithms for linear program-
ming.
Professor Todd is perhaps the only researcher
to have made fundamental advances on each
of these three topics of present-day research.
His work, however, is not limited to the topic
of linear programming. He has also made
important contributions to the study of large
structured mathematical programming prob-
lems and to the development and analysis of
algorithms for solving systems of nonlinear
equations, particularly with application of
economic equilibrium problems.
The award, which is joint with SIAM, was
presented at the SIAM annual meeting in July
and announced at the opening session of the
MPS Tokyo meeting.
-Les Trotter


I-- Is~E~I~ -----~-(------sl --------------------~~


SP T I M A number twenty-five


INk V L .IVIL mUI /o'U




PAGE 8


NOVEMBER 19885


6ote c Note


14th IFIP Conference on System SIAM Conference on Optimization

Modeling and Optimization April 3-5, 1989; Boston, MA
July 3-7, 1989: LeivziP. GDR


The conference will be held in the Congress Centre of the
Karl-Marx-University in the city centre of Leipzig. The
aim of the conference is to discuss recent advances in the
mathematical representation of engineering, socio-
technical and socio-economical systems as well as the op-
timization of their performances. The language of the
conference will be English.
Abstracts and software descriptions should be submitted
by December 15, 1989. Contact: Dr. K. Tammer, Leipzig
University of Technology, Department of Mathematics
and Informatics, PF 66, Leipzig, 7070, GDR.


The third SIAM Conference on Optimization will be held April 3-5, 1989, at 57 Park Plaza
Hotel, Boston, MA. Themes of the conference include Interior Point Methods for Linear
Programming, Network Optimization, Constrained Optimization and Large-Scale Optimiza-
tion. The Organizing Committee is Donald Goldfarb (Columbia) and Michael Todd (Cornell),
Co-chairs, with David Gay (AT&T Bell Labs) and Jorge More (Argonne).

Preceding the conference there will be a short course on Recent Developments in Linear and
Nonlinear Programming organized by Richard Stone and Margaret Wright of AT&T Bell Labo-
ratories.

Deadline for contributed abstracts and poster sessions is November 4, 1988.
Contact: SIAM Conference Coordinator, 117 South 17th Street, 14th Floor
Philadelphia, PA 19103-5052
Telephone (215) 564-2929
Email: siam@wharton.upenn.edu; Fax: 215-564-4174.


FS

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OP TI M A number twenty-five


~ ~~


NOVEMBER 1988


J





PAGE 9i 0 P T MB

KIM


Faculty of Mathematical Studies
University of Southampton
Southampton S09 5NH
United Kingdom

H. Belouadah, M.E. Posner and C.N. Potts,
"A Branch and Bound Algorithm for Schedul-
ing Jobs with Release Dates on a Single
Machine to Minimize Total Weighted Complete
Time,"OR14.
H.P. Williams, "Orthogonality in Linear
Congruence Duality," OR9.
V.J.D. Baston, M.K. Rahmouni and H.P.
Williams, "The Practical Conversion of Linear
Programmes to Network Flow Models," OR10.
K.I.M. McKinnon and H.P. Williams, "Con-
structing Integer Programming Models by the
Predicate Calculus," OR11. Y.
Ouinten and A.K. Shahani, "Modelling for
an Early Detection of Breast Cancer," OR13.
Y. Maghsoodi, "Design and Computation of
Near-Optimal Stable Observers for Bilinear
Systems," OR15.
V.J.D. Baston and F.A. Bostock: "A Continu-
ous Game of Ambush," "Discrete Hamstrung


Squad Car Games," "Deception Games," "An
Evasion Game with Barriers," "A Simple Cover-
up Game," "A One Dimensional Helicopter-
Submarine Game," "The Number Hides Game,"
(with T. Ferguson).




Mathematisches Institut
Universitit zu K6ln
Weyertal 86-90
5000 Klin 41
West Germany

M. Leclerc, "Constrained Spanning Trees and
the Travelling Salesman Problem," WP 88.55.
M. Leclerc, "A Class of Latices," WP 88.54.
M. Leclerc, "Fast Polynominal Arithmetic and
Exact Matching," WP 88.53.
M. Leclerc, "A Linear Algorithm for Breaking
Periodic Vernam Ciphers," WP 88.52.
A. Bachem, B. Korte and R. Schrader,
"Mathematische Modelle far Bausparkollektive,"
WP 88.51.
J. Rieder, "The Lattices of Matroid Bases and


Exact Matroid Bases," WP 88.50.
M. Alfter, W. Kern and A. Wanka, "On Ad-
joints and Dual Matroids," WP 88.49.
U. Riltsch, "The Arborescence Lattice," WP
87.48.
M. Leclerc, "Algorithmen far Kombinatorische
Optimierungsprobleme mit Partitionsbes-
chrankungenm," WP 87.47.
B. Fassbender, "Kriterien vom ORE Typ far
langste Kreise in 2-zusammenhangendern
Graphen," WP 87.46.
W. Hochsttitter and W. Kern, "Matroid
Matching in Pseundomodular Lattices," WP
87.45.
M. Leclerc, "Eine Min-Max Beziehung far das
Exakte Matroid Problem," WP 87.44.
B. Fassbender, "A Generalization of a Theorem
of Nash-Willimas," WP 87.43.
M. Hofmeister, "Counting Double Covers of
Graphs," WP 87.42.
M. Leclerc, C.H.C. Little and F. Rendl,
"Constrained Matching Problems and Pfaffian
Graphs," WP 87.41.
M. Leclerc and F. Rendl, "A Multiply Con-
strained Matroid Optimization Problem," WP
87.40.


Election Results of

the 1988

Mathematical

Programming Society


Chairman: George L. Nemhauser (USA);
Treasurer: Leslie E. Trotter, Jr. (USA);
Council Members-at-Large: Egon Balas (USA), William Cunningham (Canada),
Claude Lemarechal (France), Alexander Schrijver (Netherlands).
By the terms of the constitution of the Society this determines the following future
periods of service of elected officers:
Chairman: Michel L. Balinski (through 8/1989),
George L. Nemhauser (9/1989- 8/1992);
Treasurer: Albert C. Williams (through 8/1989), Leslie L. Trotter, Jr.
(9/1989-8/1992);
Council: Egon Balas (9/1988-8/1991), Michel L. Balinski (9/1988-8/1989
Chairman and 9/1989-8/1991 Vice-Chairman), William H. Cunningham
(9/1988-8/1991), Claude Lemarechal (9/1988-8/1991),
George L. Nemhauser (9/1988- 8/1989 Vice-Chairman and 9/1989-8/1992 Chair-
man, 9/1992-8/1994 Vice-Chairman),
Leslie L. Trotter, Jr. (9/1989-8/1992), Alexander Schrijver (9/1988-8/1991) and
Albert C. Williams (9/1988-8/1989).


PAGE 9


0 PT IM A number twenty-five


NOVEMBER 1988




P0 P T I M A number tent-fiveNV


J 0 U RNIA L


Series A
Vol.41, No. 3

R. Fourer, "A Simplex Algorithm for Piecewise-
Linear Programming II: Finiteness, Feasibility
and Degeneracy."
T.L. Magnanti and J.B. Orlin, "Parametric
Linear Programming and Anti-Cycling Pivoting
Rules."
H. Kawasaki, "The Upper and Lower-Second
Order Directional Derivatives of a Sup-Type
Function."
R.E. Moore and H. Ratchek, "Inclusion
Functions and Global Optimization II."
P.C. Jones, J.L. Zydiak, and W.J. Hopp,
"Stationary Dual Prices and Depreciation."
K.M. Anstreicher, "Linear Programming and
the Newton Barrier Flow."
J-M. Bourjolly, "An Extension of the Konig-
Egervary Property to Node-Weighted Bidirected
Graphs."




Series A
Vol.43, No. 1

S. Chopra, D.L. Jensen and E.L. Johnson,
"Polyhedra of Regular p-Nary Group Prob-
lems."
R.J. Vanderbei, "Affine-Scaling for Linear
Programs with Free Variables."
G. Cornuejols and A. Sassano, "On the 0,1
Facets of the Set Covering Polytope."
D.M. Ryan and M.R. Osborne, "On the
Solution of Highly Degenerate Linear Pro-
grams."


E. Balas and S.M. Ng, "On the Set Covering
Polytope: I. All the facets with Coefficients in
(0,1,2)."
B.F. Lamond, "A Generalized Inverse Method
for Asymptotic Linear Programming."
A. Ech-Cherif, J.G. Ecker and M. Kupfer-
schmid, "A Numerical Investigation of Rank-
Two Ellipsoid Algorithms for Nonlinear
Programming."
R.J. Caron and J.F. McDonald, "A New
Approach to the Analysis of Random Methods
for Detecting Necessary Linear Inequality
Constraints."
B. Ram and M.H. Karwan, "A Result in
Surrogate Duality for Certain Integer Program-
ming Problems."




Series A
Vol.43, No. 2

R. Mifflin and J.-J. Strodiot, "A Bracketing
Technique to Ensure Desirable Convergence in
Univariate Minimization."
J.R. Birge and R.J-B. Wets, "Sublinear Upper
Bounds for Stochastic Programs with Recourse."
C.C. Gonzaga, "Conical Projection Algorithms
for Linear Programming."
F. Guder, "Pairwise Reactive SOR Algorithm
for Quadratic Programming of Net Import
Spatial Equilibrium Models."
G. Li, "Successive Column Correction Algo-
rithms for Solving Sparse Nonlinear Systems of
Equations."
K.M. Anstreicher, "A Combined Phase 1-Phase
II Projective Algorithm for Linear Program-
ming."
J.L. Noakes, "Fitting Maps of the Plane to
Experimental Data."


Series B
Vol.42, No. 1

G.B. Dantzig, P.H. McAllister and J.C. Stone,
"Formulating an Objective for an Economy."
K.C. Bowen, "A Mathematician's Journey
Through Operational Research."
P.J. Green, "Regression, Curvature and
Weighted Least Squares."
B.D. Ripley, "Uses and Abuses of Statistical
Simulation."
J.A. Tomlin, "Special Ordered Sets and an
Application to Gas Supply Operations Plan-
ning."
A. Orden, "Model Assessment Objectives in
OR."
B.R.R. Butler, "Applications of OR in the Oil
Industry."
L.F. Escudero, "S3 Sets. An Extension of the
Beale-Tomlin Special Ordered Sets."
T. Gal, H.-J. Kruse and P. Z6rnig, "Survey of
Solved and Open Problems in the Degeneracy
Phenomenon."
R.S. Hattersley and J. Wilson, "A Dual
Approach to Primal Degeneracy."
D. Kennedy, "Some Branch and Bound
Techniques for Nonlinear Optimization."
B. Nygreen, "European Assembly Constituen-
cies for Wales Comparing of Methods for
Solving a Political Districting Problem."
M.J.D. Powell, "An Algorithm for Maximizing
Entropy Subject to Simple Bounds."
B.M. Smith, "IMPACS-A Bus Crew Schedul-
ing System Using Integer Programming."
J. Sullivan, "The Application of Mathematical
Programming Methods to Oil and Gas Field
Development Planning."


. ....... d .....


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O P T I A/I A numbeP twenhr-flve


NOVEM~lkBER 1988





PAGE 11 0_ Ps T Anmertetyfv NVMER18


BOOK

REVIEWS


Parallel Computers and Computations
Edited by J. van Leeuven and J. K. Lenstra
CWI Syllabus 9
Amsterdam, Netherlands, 1986
ISBN 90-6196-297-8

While hardware is becoming cheaper, one
approach to obtain faster computing machines is to provide
several instances of functional units in a system that can
operate concurrently. But as there are a number of possible bot-
tlenecks in a system (memory, communication paths, arithme-
tic units, etc.) and to judge which components are strongly
depends on the type of task to be performed a corresponding
large number of architectural designs with varying degrees of
parallelism and applicability has emerged on the market.
This volume contains eight papers covering
various aspects of parallel computers and computation which
were presented during the fall of 1983 at the University of
Utrecht.
The first paper introduces several models of
parallel machines, techniques of a parallelization and hints at
the benefits that can or cannot be expected from parallel
algorithms even if designed close to the hardware and commu-
nication schemes involved. A comparison of two supercom-
puters, namely the Cray-1 and the Cyber 205, which can,
roughly spoken, both be categorized as vector processing
machines, with respect to the impact of the "minor" architec-
tural differences on the optimal design of programs for some
basic problems in linear algebra, is discussed in the second
paper. The third paper presents some techniques to introduce
parallelism into algorithms for linear algebra and for the solu-
tion of partial differential equations.
In order to be able to exploit the parallelism of
a hardware scenario, a suitable operating system has to be
provided. The fourth paper presents some internal details of
the CDC 205 operating system nucleus, while the fifth gives an
overview of the Amoeba distributed and capability based
operating system.
Trace theory as a formal specification scheme
for the design and synthesis of parallel programs is presented


in the sixth paper, and the seventh discusses various ap-
proaches to introduce parallelism into complexity theory and
the consequences on the results obtainable by using different
machine models. Finally, a tutorial introduction to the litera-
ture on parallel computers and algorithms relevant to combi-
natorial optimization is given.
As an introduction to the various aspects of
parallelization of computing machinery and algorithms, this
book can be recommended to everyone who is interested in a
closer look at this important topic.
- O. Holland


Theory of Matroids
Edited by Neil White
Cambridge, 1986
ISBN 0-521-30937-9

This book is the first of a three-volume series
which is to present essential results about matroids. Since
matroid theory accelerated rapidly during the last 50 years, it
is of course impossible to treat all aspects in only one book. The
main points of the first volume are matroid cryptomorphisms
and constructions of matroids. Most of the theorems are
proved in detail; however, the more difficult results and most
of the new results derived during the last 10 years are stated
without proof but with references given to the original works.
The book consists of 10 chapters contributed by
several authors. The first chapter by Henry Crapo gives some
motivations for matroid theory. In particular, it is explained
that matroids comprise many configurations met in pure and
applied mathematics. Here and throughout the whole book,
the examples of vectorial, projective and affine geometries,
graphical matroids and transversal matroids are examined in
great detail.
In the second chapter by Giorgio Nicoletti and
Neil White some of the major axiom systems for matroids (in
terms of bases, independent sets, circuits etc.) are proved to be
equivalent. In an appendix to the book some further axiom


CONTINUES


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NOVEMBER 1988


PAGE 11



0 P I MA number twenty-five





PAGE 12 0 P T I M A number twenty-five NOVEMBER 1 qR~


i BOOK REVIEWS


systems for matroids are given and the relations between all of
them are described in detail.
In the third chapter, Ulrich Faigle presents the
lattice-theoretic approach to matroids. One of the main results
is the well-known theorem that the lattices of flats of matroids
are exactly the geometric lattices up to isomorphism.
The fourth chapter by Joseph P. S. Kung deals
with exchange properties of bases of matroids. In particular,
many well-known theorems of linear algebra concerning bases
of vector spaces are generalized. Furthermore, it is remarked
that several exchange theorems, which can be proved for vec-
torial matroids by using identities concerning determinants,
up to now have only been generalized in special cases.
In the fifth chapter Henry Crapo explains the
duality concept in matroid theory. Several descriptions of the
dual of a matroid are given based on the different axiom
systems for matroids. One of the main theorems states that a
matroid is representable over some field F if and only if its dual
is representable over F too.
James Oxley examines graphical matroids in
the sixth chapter. Every graph is related to some polygon
matroid in a canonical way. A very important result states that
a graph is planar if and only if the dual of its polygon matroid
is itself a polygon matroid. This theorem is reduced to Kura-
towski's well-known theorem which states that a graph is
nonplanar if and only if it contains a subgraph homeomorphic
from the complete graph K5 or the complete bipartite graph
K3,3.
The seventh chapter by Thomas Brylawski is
the longest in the book and deals with constructions of ma-
troids from given matroids. The main themes are restrictions,
contractions, truncations and extensions of matroids as well as
direct sums and several generalizations. For all these construc-
tions many equivalent definitions are given in terms of the
different descriptions of matroids. Furthermore, this chapter
includes the concept of matroid bracing, unpublished before,
which yields certain canonical extensions of matroids.
In the eighth and ninth chapters Joseph P. S.
Kung and Joseph P. S. Kung together with Hien Q. Nguyen, re-
spectively, discuss two suggestions of morphisms between
matroids, namely strong maps in the eighth and weak maps in


the ninth chapter. Both types may be interpreted as abstrac-
tions of linear maps between vector spaces. However, weak
maps are much more general. The main subject in both chap-
ters is the proof of factorization theorems which allow repre-
senting strong maps as well as weak maps as compositions of
particularly simple maps. The concept of a weak cut is de-
scribed, which was discovered in 1979 by Nguyen and, inde-
pendently, in 1980 by Kung and serves to determine whether
or not the identity map between two matroids, defined on the
same set, is a weak map. A result which characterizes weak cuts
completely is included together with a new proof based on a
suggestion by J. Mason.
The 10th chapter by Hien Q. Nguyen deals with
relations between semimodular functions and matroids. An
integer-valued function f defined on some finite lattice with
f(0) = 0, which is non-decreasing and semimodular, is extended
to some function f defined on a Boolean algebra such that may
be interpreted as a rank function of some matroid.
Apart from the first, each chapter contains a
long list of references. However, a single index of all cited
works would have been very useful.
While the book, Matroid Theory, by D. J. A.
Welsh (1976) is more elementary, the new book by Neil White
considers more difficult problems. So, in my opinion, the book
by Welsh is more appropriate for a first course on matroids,
while this book, Theory ofMatroids, by Neil White is a very good
reference for those familiar with the essential elements of
matroid theory.
- W. Wentzel


Functional Analysis and Control Theory
by Stefan Rolewicz
Reidel, Dordrecht, 1987
ISBN 90-277-2186-6

This is a very nice and clearly written book on
the mathematical theory of linear control systems in the termi-
nology of functional analysis. It should be thought of as an in-
troduction to mathematical control theory for applied mathe-


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0 P T I M A number twenty-five


NOVEMBER 1%98


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PAG 130PTIMAnme wnyfv OEBR18


BOOK REVIEWS =


maticians and systems engineers and is written at a high
mathematical level. It contains many instructive examples and
the theoretical mathematical concepts are well motivated.
The book starts with some fundamental facts
about topology and measure theory. Then an excellent presen-
tation of the needed apparatus of functional analysis follows.
The most essential part of this book is chapter 5 where the
theory of general linear systems is developed. Although this
chapter seems to be rather abstract, it permits the answering of
some fundamental questions, for instance, the meaning of the
Pontryagin-maximum principle for linear systems. Moreover,
it contains important theorems about the existence of universal
time for time depending linear systems. Optimal observability
is defined in a general way, and the duality between optimal
observability and optimal controllability is presented. These
results are then applied to finite dimensional systems de-
scribed by linear ordinary differential equations and to sys-
tems with distributed parameters governed by the diffusion
equation. In this connection the author presents all the funda-
mental facts about differential equations in Banach spaces. In
particular, he treats in detail the results on control of heating
and observability of temperature distributions in a slab under
various boundary conditions.
This book contains several extensions to the
Polish and German editions. An appendix containing neces-
sary and sufficient conditions for weak duality, obtained by the
Dolecki-Kurczyusz method of multifunctions, is included.
D. Pallaschke



Search Problems
by R. Ahlswede and I. Wegener
Wiley, Chichester, 1987
ISBN 0-471-90825-8

This monograph brings together a collection of
results pertaining to problems that can be labeled as search
problems. A very wide spectrum is covered ranging from opti-
mal binary search trees to constructing optimal allocations


through searching for the maximum of a unimodal function
and stochastic approximation. The book is divided into four
parts and 13 chapters.
The material is well organized and well pre-
sented. It is not, however, addressed to the general audience.
As the authors state in the preface, the book is addressed to the
expert. Indeed, a reader familiar with particular topics will find
the chapters devoted to these topics easy to follow. Chapters
treating unfamiliar topics, on the other hand, will require
substantial effort. In some instances the choice of terminology
is unfortunate (a computer scientist may wonder why binary
search refers to something other than the famous algorithm). In
other instances the choice of topic is questionable (a chapter is
devoted to sorting in a book on searching, yet the obvious
connections are not drawn). The book does not cover the
following problems: searching graphs (e.g. depth-first search,
breadth-first search, best-first search, branch-and-bound
search, etc.) searching game trees (e.g. minimax search, alpha-
beta search, etc.), searching combinatorial spaces (e.g. heuristic
search), geometric search (e.g. point location problems), and
database queries (e.g. multidimensional querying).
Despite the above criticisms, the book is a valu-
able contribution to discrete mathematics and the design and
analysis of algorithms.
S. G. Ak



Applied Probability and Queues
by S. Asmussen
Wiley, Chichester, 1987
ISBN 0-471-91173-9

In the Preface the author writes, "This book
treats the mathematics of queueing theory and some related
areas, as well as basic mathematical tools relevant for the study
of such models.... Particular attention has been given to
modern probabilistic points of view, as alternatives to tradi-
tional analytic methods. Within this framework the choice of
the topics is, however, rather traditional."


CONTINUES


NOVEMBER 1988


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0 PT IM A number twenty-five






PAE 4 PT MA ume tey-iv NOVEMBE 198


z13BOOK REVIEWS


The book is subdivided into three parts. In Part
A, bearing the title, Simple Markovian Models, the Markov
chains and the Markov jump processes are described, and the
reader is introduced to queueing theory where elements of
queueing networks are also included.
Part B has the title, Basic Mathematical Tools.
Here renewal theory, regenerative processes and random
walks are discussed.
Part C is devoted to Special Models and Meth-
ods. Here one finds the exposition of the theory of various
special single-server and many-server queues, multi-dimen-
sional Markov processes, and conjugate processes and some
special models playing an important role in applications such
as insurance, dam and storage models. An appendix about
selected background material (point processes, stochastic
ordering, Wald's identity, etc.) closes the book.
The reviewer thinks that it is primarily a post-
graduate level book. Looking at it as such, the mathematical
elegance is what we primarily observe and praise here. If,
however, one wants to use it as a textbook for a graduate
course, then a more elementary knowledge of the theory of
stochastic processes has to be included among the prerequi-
sites.
Andras Prekopa



The Simplex Method
A Probabilistic Analysis
by Karl-Heinz Borgwardt
Springer, Berlin, 1987
ISBN 3-540-17096-0

For an edge-following LP algorithm like the
simplex algorithm the maximum number S(m,n) of basis ex-
changes or pivot steps required to solve a problem in n vari-
ables and m inequality constraints is a reasonable and much
studied measure for the worst-case behavior of the algorithm.
Edge-following LP algorithms differ mainly by the selected
pivot rule, and clearly S(m,n) depends on that choice.
Encouraged by empirical results observed in


practical applications in the 1950s and '60s, there was some
belief that there might be a pivot rule for which S(m,n) is
bounded by a (linear) polynomial in m and n. However, in the
early '70s Klee and Minty proved that for the Dantzig pivot
rule, there are positive constants an and pn, depending on n
only, such that
anm[n/2] n.
Hence, for a very common pivot rule this opti-
mistic assumption about the behavior of the functional S(m,n)
was shown to be incorrect. Later, such negative results were
proved for almost all common pivot rules. It is an open prob-
lem whether a pivot rule with polynomial S(m,n) exists or not.
The proved bad worst-case behavior of most
known pivot rules contrasts the observed performance in
practice, and so it is natural to ask for the average-case behavior
of S(m,n). In a series of papers, Borgwardt was the first to
provide a quite satisfactory analysis of this behavior for a
special pivot rule. Briefly, he proved a polynomial bound for
the expectation value of S(m,n) under some reasonable sto-
chastic assumptions about the input data.
The book under review contains a very de-
tailed description of those results, their proofs and some exten-
sions.
The problems dealt with are of the following
kind: Maximize vTx subject to a1Tx ai,...,ame Rn, and m >n.
For the stochastic considerations it is assumed
that ai,...,am, v are distributed (on R"\ (0) independently, iden-
tically, and symmetrically under rotations. The pivot rule
considered by the author is the Gass-Saaty rule of parametric
linear programming. The main result states that the complete
problem (including Phase I) requires not more than cm/1n-1(n +
1)4 pivot steps on the average, where c is a constant. It should
be noted that the selected pivot rule in this model is by no
means exotic. The rule is, in fact, the standard method for
producing the efficient solutions for an LP with two objectives.
The Gass-Saaty pivot rule has the advantage
that pivots can be described geometrically by means of a line
crossing a system of cones spanned by the vectors al,...,am. The
author calls this cone approach a dual version. It should be
called a polar approach as it deals with the polar of the feasible


PAGE 14


OPTIMA number twenty-five


NOVEMBER 1988





PAGE~~ 15 0 P nme wnt-ieNVEBR18


BOOK REVIEWS -


polyhedron. The algorithm is not the dual one for the primal
problem in the sense of LP theory. The expected number of
I i Ut A can be expressed as an integral over spherical measures
of such cones. Actually, the evaluation of this integral is very
involved and it is the essential part of the stochastic analysis.
Two chapters of the book and the appendix are devoted to the
evaluation of such integrals. The interested reader is provided
,' i th all the details of the computation. This will be appreciated
by those who try to become familiar with the methods of this
active area of research.
Several ramifications of the problem are also
presented, including problems with nonnegativity con-
straints, the restriction to Phase II, asymptotic results and
special distributions. The reader who is not interested in all the


technical details will find the main results and ideas in the first
chapter. It also includes an interesting survey of the results
obtained by other researchers. Alternative stochastic models,
especially the sign-invariance model, are discussed.
The book is written very clearly. The reader is
well guided through the difficult technical parts. For this
purpose the numerous illustrations of the geometry involved
prove very helpful. I recommend this book to anyone inter-
ested in questions related to the performance of LP algorithms.
In addition, because of its geometric approach, it will also be of
considerable interest to those working in stochastic or convex
geometry.
- P. Kleinschmidt


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~-~-- -^I-- I--- ~---~-- -~-~-s --- ~s~s


PAGE 15


OP TIM A number twenty-five


NOVEMBER 1988





PAG 160PTIMAnubrtet-iv OEBR18


Gallimaufry


The MPS Symposium in Tokyo had one extremely sad note.
Robert G. Jeroslow (Georgia Tech) died of an apparent heart attack while attending the
meeting. Symposium attendees expressed their written condolences in a volume which was
assembled and sent to his son Avi.
An international conference on parallel computing was held in
Verona, Italy, September 28-30. Proceedings will be edited by D. J. Evans (Loughborough
Univ., U. K.) and C. Nordai Sutti (Univ. di Verona)...J. K. Lenstra has agreed to serve as MPS
liaison with the International Statistical Institute which handles the Society business affairs.
Members may contact him if they have problems with ISI. Address is Centre for Mathematics
and Computer Science, P. O. Box 4079,1009 AB Amsterdam, The Netherlands; telephone: +31-
20-592-4087; email: jkl@cwi.nl...The MPS Executive Committee has announced solicitations for
a site for the 1994 Symposium...A Call for Papers dealing with topics on Local Optimization
has been announced for a special issue of Discrete Applied Mathematics by Editor Donna
Crystal Llewellyn, ISyE, Georgia Tech, Atlanta GA 30332-0205; telephone: 404-894-2340; email:
dllewell@gtri01.bitnet. The Eleventh Symposium on Mathematical Programming with Data
Perturbations will be held 25-26 May, 1989 at the George Washington University. Contact
Anthony Fiacco (202) 994-7511.
Deadline for the next OPTIMA is February 1, 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 Kiln,
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.


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0 PT IM Anumber twenty-five


NOVEMBER 1988




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