Logic-Based Decision Support: Mixed Integer Model Formulation

239 Pages·1989·9.503 MB·English

by  Robert G. Jeroslow

LOGIC-BASED DECISION SUPPORT:
Mixed Integer Model Formulation
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ANNALS OF DISCRETE MATHEMATICS 40
General Editor: Peter L. HAMMER
Rutgers University,N ew Brunswick, NJ, U. S.A.
Advisory Editors
C. BERGE, Universite de Paris, France
M. A. HARRISON, University of California, Berkely, CA, U.S.A.
V. KLEE, University of Washington, Seattle, WA, U.S.A.
J.-H. VAN LINT California Institute of Technology, Pasadena, CA, U.S.A.
G.-C.R OTA, Massachusetts Institute of Technology, Cambridge, MA, U.S .A .
This volume is based on lectures delivered at the
First Advanced Research Institute on Discrete Applied Mathematics
supported by the
Air Force Office of Scientific Research
and held at
RUTCOR - Rutgers Center for Operations Research, May 1985.
LOGIC-BASED DECISION SUPPORT
Mixed Integer Model Formulation
Robert G.J EROSLOW t
1989
NORTH-HOLLAND -AMSTERDAM NEWYORK OXFORD *TOKYO
' Elsevier Science Publishers B.V., 1989
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Library of Congress Cataloging-in-Publication Data
Jerosiow, Robert G., 1942-1988
1.9,gic-based decision support.
(ilniials of discrete mathematics ; 40)
Bibliography: p.
1. Decision support systems. 2. Decision-making--
Mathematical models. 3. Logic, Symbolic and
mathematical. I. Title. 11. Series.
T58.62.547 1989 658.4'03 88-31027
lSBN 0-444-87119-5
PRINTED IN THE NETHERLANDS
V
Robert S. Jeroslow
1942 - 1988
vi
ROBERT G. JEROSLOW
1942 - 1988
On August 3 1 this year, towards the middle of the Mathematical Programming Symposium
in Tokyo, Bob Jeroslow had a fatal heart attack. His early, sudden and completely unexpected
death at the age of 46 came as lightning from a clear sky. It was a terrible shock to his friends
and colleagues and in a way made all of us newly aware of our vulnerability and the frailty
of the human condition. Our profession suffered a heavy loss indeed.
Bob started his graduate work in Operations Research (at Columbia first, then at Cornell),
but soon switched to Mathematics and wrote his dissertation in Logic, under Anil Nerode.
He was a fresh Ph.D. in the summer of ’69 on his way to a job in the Math. Department at
the University of Minnesota, when he visited Pittsburgh and we met for the first time. We
had several conversations that I found intellectually stimulating, and I showed him some of
the problems in Discrete Optimization that I was working on and that I felt could benefit
from his background in Logic. We continued our contacts through telephone and corres-
pondence, and the upshot of our interaction was the paper “On the Structure of the Hyper-
cube” (Management Science Research Report No. 198, CMU, August-December 1969).
This joint piece of work, published later as “Canonical Cuts on the Unit Hypercube” in the
SIAM Journal on Applied Mathematics, 23, 1972, seems to have played a role in luring Bob
back to Operations Research, and so in 1972 he joined our group at CMU’s Graduate School
of Industrial Administration as Assistant Professor.
At CMU Bob flung himself wholeheartedly into the ongoing effort of developing a cutting
plane theory for integer and nonconvex programming based on Convex Analysis, that would
use different tools and capture different aspects of the problem than the group theoretic
approach that was prevalent at the time. Among the people outside CMU involved in this
effort, Bob was in contact mainly with Fred Glover. The result of our joint work in this
direction became known as disjunctive programming, or the disjunctive method. It is essen-
tially a theory of optimization over unions of convex polyhedra. Bob wrote several papers
on the subject, some by himself, some with me and some with his former doctoral student,
Charlie Blair. The topics of these papers range from the basic principles of disjunctive pro-
gramming (Annals of Discrete Mathematics, I, 1977; Journal of Optimization Theory and
Its Applications, 30, 1980), to structural properties like the sequential convexifiability of
facial disjunctive sets (SIAM Journal on Control and Optimization, 18, 1980; Discrete
Applied Mathematics, 9, 1984), to methodological contributions like the monoidal cut
strengthening procedure for mixed-integer programs that combines the disjunctive method
with the algebraic approach (European Journet of’ Operational Research, 4, 1980).
While at CMU, Bob also made some interesting contributions to complexity theory. In
one of them (Discrete Mathematics, 4, 1973) he extended the Klee-Minty result about the
simplex method requiring exponentially many steps on certain problem classes, to a non-
R. JEROSL 0 W Vii
standard variant of the simplex method which uses the pivot column choice rule of maxi-
mizing the improvement of the objective function value.
In 1978, already a full professor, Bob moved to Georgie Tech. During the late seventies
and early elghties he wrote a number of important papers with Charlie Blair on the value
function of an integer program (Discrete Mathematics, 19, 1977 and 25, 1979;Mathematical
Programming, 23, 1982; Discrete Mathematics, 9, 1984 and 10, 1985). These papers deal
with issues like subadditive duality and sensitivity analysis in integer programming.
Starting around 1982, Bob got interested in problems of integer programming represent-
ability (Mathematical Programming Study 22, 1984;D iscrete Applied Mathematics, 17, 1987;
European Journal of Operational Research, 12, 1988). Here he drew on earlier work by Bob
Meyer, as well as on my 1974 technical report on the properties of the convex hull of feasible
points of a disjunctive set. That report contained a linear representation ofthe convex hull of a
union of polyhedra in ahigher dimensional space. At the time, this representation did not seem
important because its dimension grows exponentially with the number of polyhedra in the
union. However, in 1982, Bob recognized the crucial fact that if applied selectively to a subset
of the constraints instead of the full set, this representation provides one of the chiefvehicles
towards obtaining formulations whose linear programming relaxations are tight. Bob coined
the term “sharp” for representations in which the inequalities by themselves define the integer
hull, and obtained several important results concerning such representations. He became con-
vinced that introducing appropriately chosen new variables is in many situations a more
efficient way of sharpening a formulation than generating cutting planes in the original
Variables. We had many lively discussions on this matter and were planning on writing a
joint paper on the subject. Other people with whom he interacted on this topic include
Charlie Blair, Kipp Martin, Ron Rardin, and his student Jim Lowe.
After a while, around, 1985, Bob’s preoccupation with representability focused on the
integer programming representation of logical inference; and, more generally, on the appli-
cation of mathematical programming techniques to artificial intelligence, expert systems etc.
(Computers and Operations Research, 19, 1986;D ecision Support Systems, 1988 ;Annals of
Discrete Mathematics, 1988). Here was finally an area upon which Bob could bring to bear
the full arsenal of his training in Logic, combined with his knowledge of the polyhedral
method. His pathbreaking work in this new and exciting area, much of which he presented
in his Lecture Notes for the first ARIDAM at RUTCOR, published in the present volume,
may ultimately prove to be the most influential part of his entire professional legacy.
Besides being an outstanding mathematician, Bob had exceptional pedagogical skills: his
students used to rave about him. He was a very earnest person, scrupulously conscientious
about his commitments and obligations, generous with his time for students and colleagues
alike. He would sometimes worry without a good reason and get excited, or become suspi-
cious; on those occasions he needed somebody, a friend, to calm him down. But ifhe needed
friendship, he also offered it: he was loyal and reliable. Beyond personal relations, Bob was
a warmhearted, sensitive human being, who cared about issues of fairness and justice, and
was never indifferent to the plight of people he knew about. We will all badly miss him.
Egon Balas
Viii
for
.
Richard J Dufb
an applied mathematician in the grand style,
a gentle man
ix
"...Science as well as technology,
will in the near and in the farther future
increasingly turn from problems of intensity,
substance, and energy, to problems of structure,
.. .. .. "
organization, idormatioL and control.
J. von Neumann, 1949,
in his attribution of
the views of N. Wiener