PTIMA
MATHEMATICAL PROGRAMMING SOCIETY NEWSLETTER
N 30
July 1990
George B.
Dantzig Prize
The Mathematical Programming
Society (MPS) and Society for
Industrial and Applied Mathe
matics (SIAM) are seeking
nominations for their joint
George B. Dantzig Prize
awarded to an individuals) for
original research, which by
virtue of its originality, breadth,
and depth, is having a major
impact on the field of mathe
matical programming. The
contributions eligible for
consideration must be publicly
available and may address any
aspect of mathematical pro
gramming in its broadest sense.
Preference is given to contribu
tions by individuals under 50
years of age.
The Prize will be presented at
the Mathematical Programming
Society's triennial symposium
to be held August 59,1991, in
Amsterdam. Past George B.
Dantzig Prize recipients have
been: M. J. D. Powell and R. T.
Rockefellar in 1982, E. L.
Johnson and M. W. Padberg in
1985,and M. J. Todd in 1988.
The Prize Committee members
are Thomas L. Magnanti, Chair,
Manfred W. Padberg, RI Tyrrell
Rockefellar, and Michael J.
Todd.
CONTINUES, PAGE TWO
1991 Prize Nominations Invited
BealeOrchard ID.R. Fulkerson
Hays Prize
Nominations are being
sought for the Mathematical
Programming Society Beale
OrchardHays Prize for Ex
cellence in Computational
Mathematical Programming.
Purpose:
This award is dedicated to the
memory of Martin Beale and
William OrchardHays, pioneers
in computational mathematical
programming. To be eligible a
paper or a book must meet the
following requirements:
1) It must be on computational
mathematical programming.
The topics to be considered
include:
a) experimental evaluations of
one or more mathematical
programming algorithms.
b) the development of quality
mathematical programming
software (i.e. welldocumented
code capable of obtaining
solutions to some important
class of MP problems) coupled
with documentation of the
applications of the software to
this class of problems (note: the
award would be presented for
the paper which describes this
work and not for the software
itself),
c) the development of a new
computational method that
improves the stateofthe art in
CONTINUES,PACETWO
r_ S
Prize
This is a call for nominations
for the D. Ray Fulkerson Prize
in discrete mathematics that
will be awarded at the XIVth
International Symposium on
Mathematical Programming to
be held in Amsterdam, The
Netherlands, August 59,1991.
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 prizes 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 combina
torics, 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 mathematical quality
and significance".
The nominations for the award
will be presented by the Fulker
son Prize Committee (Martin
CONTINUES,PAGE TWO
A.W. Tucker
Prize
The Mathematical Programming
Society invites nominations for the
A. W. Tucker Prize for an outstand
ing paper authored by a student.
The award will be presented at the
International Symposium on
Mathematical Programming in
Amsterdam, The Netherlands (59
August 1991). All students,
graduate and undergraduate, are
eligible. Nominations of students
who have not yet received the first
university degree are especially
welcome. In advance of the
Symposium an award committee
will screen the nominations and
select at most three finalists. The
finalists will be invited, but not
required, to give oral presentations
at a special session of the Sympo
sium. The award committee will
CONTINUES, PAGE TWO
OPTIMA
NUMBER 30
CONFERENCE NOTES 4
TECHNICAL REPORTS &
WORKING PAPERS 56
BOOK REVIEWS 710
JOURNALS 1011
GALLIMAUFRY
_1____1_1_ 1~
_ _I _~~_
PAGE 2 number thirty JULY 1990
I
George B. Dantzig Prize
Please send nominations to
Thomas L Magnanti, Sloan
School of Management, M.I.T.,
Cambridge, MA 02139, U.S.A.
Nominations are due by Sep
tember 30,1990, and should
provide a brief one or two page
description of the nominee's
outstanding contributions and,
if possible, a current resume
including the nominee's list of
publications.
THOMASL.MAGNANTI
BealeOrchardHays
Prize
computer implementations of
MP algorithms coupled with
documentation of the experi
ment which showed the
improvement, or
d) the development of new
methods for empirical testing of
mathematical programming
techniques (e.g., development of
a new design for computational
experiments, identification of
new performance measures,
methods for reducing the cost of
empirical testing).
2) It must have appeared in the
open literature.
3) If the paper or book is written
in a language other than
English, then an English
translation must also be in
cluded.
4) Papers eligible for the 1991
award must have been pub
lished within the years 1987
through 1990.
These requirements are intended
as guidelines to the screening
committee but are not to be
viewed as binding when work
of exceptional merit comes close
to satisfying them.
Frequency and Amount of the
Award:
Previous recipients of the award
were Michael Saunders (1985)
and Tony J. Van Roy and
Lawrence Wolsey (1988). The
1991 prize of $1500 and a plaque
will be presented in August,
1991, in Amsterdam at the
Awards Session of the Interna
tional Symposium on Mathe
matical Programming sponsored
by the Mathematical Program
ming Society.
Judgement Criteria:
Nominations will be judged on
the following criteria:
1) Magnitude of the contribution
to the advancement of computa
tional and experimental mathe
matical programming.
2) Originality of ideas and
methods.
3) Clarity and excellence of
exposition.
Nominations:
Nominations must be in writing
and include the titles) of the
papers) or book, the authorss,
the place and date of publication
and four copies of the material.
Supporting justification and any
supplementary materials are
welcome but not mandatory.
The awards committee reserves
the right to request further
supporting materials from the
nominees.
Nominations should be mailed
to:
Professor Robert R. Meyer
Computer Sciences Department
1210 W. Dayton Street
The University of Wisconsin
Madison, WI 53706
USA
The deadline for submission of
nominations is November 1,
1990.
ROERT R MEYER
D.R. Fulkerson Prize
Gr6tschel, Chairman, Louis Bill
era, and Paul D. Seymour) to the
Mathematical Programming
Society and the American
Mathematical Society.
Please send your nominations
by January 15, 1991 to:
Prof. Dr. Martin Gratschel
Institute of Mathematics
University of Augsburg
Universititsstr. 8
8900 Augsburg
WEST GERMANY
MARTIN TSCHEL
A.W. Tucker Prize
select the winner and present the
award prior to the conclusion of the
Symposium. The members of the
committee for the 1991 A. W.
Tucker Prize are: Richard W. Cottle,
Stanford University; Thomas M.
Liebling, Swiss Federal Institute of
Technology, Lausanne; Richard A.
Tapia, Rice University; Alan C.
Tucker, State University of New
York at Stony Brook.
Eligibility
The paper may concern any aspect
of mathematical programming; it
may be original research, an
exposition or survey, a report on
computer routines and computing
experiments, or a presentation of a
new and interesting application.
The paper must be solely authored,
and completed after January 1988.
The paper and the work on which it
is based should have been
undertaken and completed in
conjunction with a degree program.
Nominations
Nominations must be made in
writing to the chairman of the
award committee
Richard W. Cottle
Department of Operations
Research
Stanford University
Stanford, California 943054022
by a faculty member at the
institution where the nominee was
studying for a degree when the
paper was completed. Letters of
nomination must be accompanied
by four copies each of: the student's
paper; a separate summary of the
paper's contributions, written by
the nominee, and no more than two
pages in length; and a brief
biographical sketch of the nominee.
The award committee may request
additional information. Nomina
tions and the accompanying
documentation are due on or before
January 5,1991.
RICHARD W. COTTLE
EditorinChief: Peter L. Hammer, Rutcor, Hill Center for the Mathematical Sciences,
Rutgers University, Busch Campus, New Brunswick, NJ 08903.
Sa s o ncs
Daai P oit ..., ,
EditorA Fiaco, George Washington University
Networks
Editors: D.: ligispan, University of Texas, and
F. Glover, University of Colorado
Automated Manufacturing Systems
Editor: J.B. Maizola, Duke University
Tabu Search
Editors: D. de Werra, Technical University Lausanne,
and F. Glover, University of Colorado
Stochastic Programming
Editors: R. Wels and J. Birge
Intransitive Preferences
Editor; W.V. Gehrlein, University of Delaware
This collection represents an important contribution to
decision analysis featuring papers on various aspects of
this theory.
Contributions deal with models ranging from
intransitiviry and the loss of market efficiency,
measurements on finite sets, linear extensions of partial
orders, voting theory, preference reversals, expected
utilities with nonlinear thresholds, individual judgment
statistics for stock market investments, etc.
Topologital Network Design
Editors: J. MacGregor Smith, University of Massachusetts
and P. Winter, University of Copenhagen
This volume presents the refereed proceedings of a
NATO workshop. Papers presented in the workshop
were written by Colbourn, Gavisch, Clover, Hammer,
Hwang, Klingman, Korne, Krarup, Lawler, Lanstra,
Rinnooy Kan, and others. The main subjects discussed at
the meeting include graph theoretical aspects of
topological network design, VLSI design, network
reliability, routing and scheduling, parallel computations,
interactions.of AI and OR, Steiner network and
computational geometry, computer and communication
networks, etc. The volume will present a comprehensive
view of the state of the an in this rapidly developing
area.
Production Planning and Scheduling
Editor: M. Qneyranne, University of British Columbia
The volume presents the state of the art of operations
research applications in production planning and
scheduling. Topics include hierarchical planning and
decomposition approaches, the interface between
planning and scheduling functions, surveys of models.for
production schedutifig, surveys on sequencing theory,
iipleenientation 'of operations research methods in
paectice. Contributors include: Coffman, de Werra.
Glazebrook, Lasserre, Lawler, Magazine, Posner, Thizy,
Van deVelde, Weiss, Yannakakis etc.
AVAILABLE VOLUMES:
22: Ed. B. Rosen, Supercomputers and Largescale
Optimization. 1990
21: Ed. H J. Greenberg & F. Glover, Linkages with
Artificial Intelligence, 1989
20: Ed. B. Shetty, Networks Optimization and
Applications, 1989
19: Ed. P.C. Fishburn & i.H. Lavalle, Choice under
Uncertainly. 1989
18: Ed. F.V. Louveaux a.o., Facility Location Analysis:
Theory and Applications, 1989
17: Ed. A. Kusiak & W.E. Wilhelm, Analysis, Modelling
and Design of Modern Production Systems, 1989
16: Ed. R.L. Keeney a.o., MultiAttribute Decision
Making via O.R.Based Expert Systems, 1989
15: Ed. K.E. Stecke & R. Suri, Flexible Manufacturing
Systems: Operations Research Models and Aoplications
II, 1988
14: Ed. R.R. Meyer & S.A. Zenios, Parallel
Optimization on Novel Computer Archiledures, 1988
13: Ed. B. Simeone a.o., Fortran Codes for Network
Optimization. 1988
12: Ed. R.G. Jeroslow, Approaches to Intelligent
Decision Support, 1988
1011: Ed. T. Ibaraki, Enumerative Approaches to
Combinatorial Optimization, 2 vols. 1987
89: Ed. S.L. Albin & C.M. Harris, Statistical and
Computational Problems in Probability Modelling
2voh. 1987
7: Ed. J. Blazewicz a.o., Scheduling under Resource
Constraint: Deterministic Models, 1986
6: Ed. J.P. Osleeb & S.J. Ratick, Locational Decisions:
Methodology and Applications, 1986
45: Ed. C.L. Monma, Algorithms and Software for
Optimization, 2 vols. 1986
3: Out of print
2: Ed. R.G. Thompson & R.M. Thrall, Normative
Analysis for Policy Decisions, Public and Private. 1985
1; Ed. F. Archetri & F. Maffioli. Stochastics and
Optimization, 1984
Price per volume inl. postage: 5 153.60, or $ 90.00 for members ORSA/TIMS. Please request extensive prospectus for
whole series: vol. 122, 19841989. Proposals for new volumes should be addressed to Peter L. Hammer, EditorinChief.
Flow to order: Please send your order either to your usual agent or directly to our Basel Head Office as mentioned below.
In the United States please address your order to: J.C. Baltzer AG, Scientific Publishing Company,
P.O. Box 8577, Red Bank, NJ 077018577.
PAGE 4 number thirty JULY 1990
Stockholm Optimization Day
The Royal Institute of Technology,
Stockholm, Sweden
June 7,1990
The first Stockholm Optimization Day was
held June 7,1990, at the Royal Institute of
Technology (KTH) under the sponsorship of
the Swedish National Board for Technical
Development. It was organized by P. O.
Lindberg, Director of the Optimization
Group, Department of Mathematics.
Papers in computational and theoretical
mathematical programming were given by
Phillipe Toint (Namur), Athanasios Migdalas
(Linkoping), Walter Murray (Stanford), Don
Hearn (Florida) and Rob Freund (MIT).
Efficient numerical optimization methods for
problems in structural optimization were
given by Krister Svanberg (KTH) and Ulf
Ringertz (Aeronautical Research Institute of
Sweden). P.O. Lindberg gave a summary of
research activities at KTH. Participants
enjoyed dinner and an evening sail on the
Baltic following the conference.
Summer School
Scuola Mathematica
Interuniversitaria
Corso Estivo di Mathematica
"Ottimizzazione Combinatorica"
Cortona, Italy
July 29 August 18,1990
This summer school is being orgainzed by
Achim Bachem and Bruno Simeone. Inter
ested students should contact:
Prof. Giovanni Monegato
Director of the
Scuola Mathematica Interuniversitaria
Politecnico di Torino
Corso Duca degli Abruzzi, 24
10129 Torino
ITALY
ACHIM BACHEM
International Symposium
Applied Mathematical Program
ming and Modelling
APM9D91
Venue: London
January 1416,1991
This international symposium in Europe
supported by the Mathematical Program
ming Society sets the scene for Mathematical
Programming in the 1990s and is a forerun
ner to the 14th Mathematical Programming
Symposium which will be held in Holland in
August 1991. Contributions from America
and Eastern Countries are also invited and
solicited. The symposium is intended to
attract specialists with different backgrounds
such as hardware manufacturers, industrial
research workers, software houses and
academic researchers. The common theme
drawing them together concerns the applica
tion of mathematical programming and
modelling to solve large, difficult and real
problems of industry and commerce.
Contact:
Gautam Mitra
Brunel University
Department of Mathematics and Statistics
Uxbridge, Middlesex UB8 3PH
UNITED KINGDOM
Telephone: Uxbridge (0895) 7400
Email: Mitra@cc.brunel.ac.uk
Second International Conference
on Numerical Optimization and
its Applications
Xi'an Jiaotong University
Xi'an China
June 2427,1991
This conference is being organized by the
Institute for Computing and Applied
Mathematics, Mathematics Department of
Xi'an Jiaotong University and is sponsored by
the National Natural Science Foundation of
China. The focus will be on new results,
algorithms on numerical optimization and
their applications. Specific topics will include
algorithms for Linear and Nonlinear Pro
gramming, Parallel Algorithms for Optimiza
tion, Methods for Global Optimization,
Nonsmooth Optimization, Numerical
Methods for Nonlinear Systems of Equations,
Interval Analysis Methods for Nonlinear
Problems, Optimal Numerical Approxima
tion for Nonlinear Operator Equations and
Applications of Optimization.
The conference committee will also award
prizes for excellent papers contributed by
young researchers under 30 years old.
Before January 31, 1991, anyone interested in
attending the conference or submitting a
paper should contact:
Dr. Chengxian Xu
Institute for Computing and Applied
Mathematics
Xi'an Jiaotong University
P. R. China
Tel: 335011 ext. 744
Telex: 70123 XJTU
11th European Congress on
Operational Research
EURO XI
Aachen
July 1619, 1991
EURO XI will be held at the "RWTH", the
largest European Institute of Technology. It is
located in Aachen (Fed. Rep. of Germany),
three miles each from Belgium andThe
Netherlands. Aachen, in which Charles the
Great, then Emperor of Europe, resided in the
8th Century, is a medieval town. Here the
charm of 1200 years of history merges with
most advanced technological research.
Papers on all subjects related to Operations
Research will be welcome. Special emphasis
will be given, however, to developments
which bear on future technologies.
There will be plenary sessions, semiplenary
sessions devoted to topics of actual interest,
parallel sessions for presenting research
contributions, research reviews or tutorials,
software sessions, and a software fair.
Participants are invited to a) present a paper,
b) present a formal software demonstration
in a software session, c) organize one or
several sessions (two to four papers by
session), or d) present software in the
software fair.
Dates and Deadlines
Dec. 1990: Deadline for abstracts
May 15,1991: Deadline for early registration
Early Registration Fee: DM 300,
Late Registration Fee: DM 450,
Mailing Address: Prof. Dr. Dr. h.c. H.J.
Zimmermann, Operations Research, RWTH
Aachen, Templergraben 64,5100 Aachen
(F.R.G.). FAX: 024180 61 89.
  ~ II
JULY 1990
PAGE 4
number thirty
PAGl___~~CIE 5 number thiry JUL 199
Technical
Reports
WORKING
PAPERS
RUTCOR
Rutgers Center for Operations
Research
Hill Center, Busch Campus
Rutgers University
New Brunswick, New Jersey 08903
G. Ding and C. Wang, "Threshold Digraphs,"
RRR# 5089.
M.S. Chern, "A Note on the Computational
Complexity of Reliability Redundancy Allocation
in a Series System," RRR# 5189.
G. Ding, "Monotone Clutters," RRR# 5289.
G. Ding, "Order the Vertices of a Graph
Linearly," RRR# 5389.
G. Isaak, "A Generalized Blossom Algorithm for
Packing Odd Subtrees," RRR# 5489.
P. Hansen and M. Zheng, "Algorithms for
Determining a Lorenz Point on Trees and
Networks," RRR# 5589.
U.N. Peled and F. Sun, "Enumeration of
Labeled Difference Graphs and an Identity of
Stirling Numbers," RRR# 5689.
B.A. Tesman, "TColorings, List TColorings,
and Set TColorings of Graphs," RRR# 5789.
M.P. de Araga6, "Column Generation Methods
for Probabilistic Logic," RRR# 5889.
U.G. Rothblum, H. Schneider and M.H.
Schneider, "Characterizations of MaxBalanced
Flows," RRR# 190.
G. Isaak, S.r. Kim, T.A. McKee, F.R.
McMorris and F.S. Roberts, "2Competition
Graphs," RRR# 290.
M. Blidia, P. Duchet, F. Maffray, "On the
Orientation of Meyniel Graphs," RRR# 390.
M.H. Rothkopf, "Forecasting When the Forecast
Can Affect the Outcome: An Impossibility
Result," RRR# 490.
X. Lu, "Hamiltonian Games," RRR# 590.
X. Lu, "Claws Contained in All n
Tournaments," RRR# 690.
CONTINUES ON FOLLOWING PAGE
Local Attractions
'A i i :' t is a world capital
r irtci ti. n,! ci:. ICIAM91
participants will have opportunities
,ar.r.i th: _r ,:.' r r ":_ '.. ; ":; f v
r',rrnr,t .'UL r'i.j mulumn. ind
more than 100 wellknown historic
sites. l ra. ierr to V.. _i' t,,vi ni
_'r. h ; .IhL" .rt :jta!., n .nlni il
t,, n,it..'ruar, :.,rnpl.,. Al 1i mu
 "..H *!.3.:' M l u r 11. C, L7!
Fu l.nl; /rdJ ,r 'hit'!t H uSC
T" _ i. t a a r_".'.* .! the I CIi. n.:
:.. ', : *. ichich offer free admis
ei. ,r .Pd i,: ,' 'pe:': thL pu ij
?n jr. j week.
Sponsoring Societies
GAMM
MA
SIMAI
*SMAI
SIAM
L l, r~, %:. ,. . INRIA
HFi..., .'r W.~.I
ICIAM 91
July 812,1991 Washington, D.C. USA
Call for Contributed Presentations
Poster and Lecture Formats
Participate in ICIAM 91 by submitting a paper, which you may present
in lecture or poster format. The ICIAM 91 Program Committee is
encouraging contributors to present their papers in poster form to in
crease communication among participants, foster the development of
international friendships, and reduce the need for large numbers of
parallel sessions.
Authors will have approximately 15 minutes for contributed
presentations (lecture format), with an additional five minutes for
questions. Alternatively, they may elect the poster format, which
encourages interactive discussions with individuals interested in their
work using flip charts and other visual aids.
If you desire to present a paper (lecture or poster format), you must
submit a summary not exceeding 100 words on anICIAM 91 contributed
paper/poster presentation form or facsimile. You may also submit an
abstract via email. Macros are available in LaTEX or TEX. To receive
macros via email, contact SIAM at iciam@wharton.upenn.edu. Papers will
be reviewed by the program committee. Everyone who submits a paper
will be notified by mail regarding acceptance.
Deadline date for submission of contributed presentation forms:
September 30, 1990.
Contributed Presentation, Registration, and Announcement Information
To obtain a form and guidelines for submitting a paper, or to receive a list of invited presentations, committee members, future announcements, and ICIAM 91 program/
registration information, please get in touch with SIAM promptly by contacting: ICIAM 91 Conference Manager, c/o SIAM, 3600 University City Science Center,
Philadelphia, PA 191042688 USA. Fax (Telecopy): (215) 3867999; Telephone: (215) 3829800; Email: iciam@wharton.upenn.edu
Accommodations
The conference will take place in
Washington, D.C., at the Sheraton
Washington, a modern air
conditioned hotel with 2,000 guest
rooms, 25 meeting rooms, an
exercise room, a large outdoor
swimming pool, and several
restaurants. The hotel boasts a
multilingual staff fluent in 20
languages, a foreign currency
exchange, electric adapters, and
many other benefits geared toward
international attendees.
The Sheraton is within a few blocks
of the National Zoo, the National
Cathedral, Embassy Row, and
historic Georgetown. The hotel is
also located on the city's "Metro"
subway system, which provides
easy, inexpensive access to most
points of interest, as well as
shopping and dining sites
throughout the city and in nearby
Virginia and Maryland.
For more economical accommoda
tions, there will be dormitory rooms
available at nearby George
Washington University.
I_~ .....  ss` ~ _1 ~ ~
number thirty
JULY 1990
PAGE 5
PAE6nme hit UY19
Technical Reports &
Working Papers
M.H. Rothkopf and R. Engelbrecht
Wiggans, "SealedBids vs. Open Auctions for
Federal Timber: Theory Reconsidered and Data
Reinterpreted," RRR# 790.
Y. Liu, "On the Rectilinear OEmbeddability of
Graphs," RRR# 890.
P.C. Chen, P. Hansen and B. Jaumard, "On
Line and OffLine Vertex Enumeration by
Adjacency Lists," RRR# 990.
O. Gross and U.G. Rothblum,
"Approximations of the Special Radius,
Corresponding Eigenvector and Second Largest
Modulus of an Eigenvalue for a Square
Nonnegative Irreducible Matrices," RRR# 10
90.
Systems of Optimization
Laboratory
Department of Operations
Research
Stanford University
Stanford, CA 943054022
F. Prieto, "Sequential Quadratic Programming
Algorithms for Optimization," SOL 897.
A. Diener, "NearOptimal Operation of a
MultiPlant Manufacturing System with
Central Procurement," SOL 898.
A. Diener, "NearOptimal Operation of a
Single Machine with Continuous Buffer Feed,"
SOL 899.
P.F. de Mazancourt, "A Matrix Factorization
and its Application to LargeScale Linear
Programming," SOL 8910.
A.L. Forsgren, P.E. Gill and W. Murray, "On
the Identification of Local Minimizers in Inertia
Controlling Methods for Quadratic
Programming," SOL 8911.
A.L. Forsgren, P.E. Gill and W. Murray, "A
Modified Newton Method for Unconstrained
Minimization," SOL 8912.
G. Infanger, "(Importance) Sampling within a
Benders' Decomposition Algorithm for
Stochastic Linear Programs," SOL 8913.
B.C. Eaves and U.G. Rothblum, "A Class of
"ONTO" Multifunctions," SOL 8914.
JC. Yao, "Generalized QuasiVariational
Inequality and Implicit Complementarity
Problems," SOL 8915.
JC. Yao, "A Basic Theorem of Complementarity
for the Generalized Variationallike Inequality
Problem," SOL 8916.
R.E. Entriken, "The Parallel Decomposition of
Linear Programs," SOL 8917.
JC. Yao, "On Mean Value Iterations with
Application to Variational Inequality Problems,"
SOL 8918.
JC. Yao, "Fixed Points by Ishikawa Iterations,"
SOL 8919.
S.K. Eldersveld and M.C. Rinard, "A
Vectorization Algorithm for the Solution of
Large, Sparse Triangular Systems of Equations,"
SOL 901.
S.K. Eldersveld and M.A. Saunders, "A
BlockLU Update for LargeScale Linear
Programming," SOL 902.
P.H. McAllister, J.C. Stone and G.B. Dantzig,
"An Interactive Model Management System:
User Interface and System Design," SOL 903.
G.B. Dantzig and Y. Ye, "A Buildup Interior
Method for Linear Programming," SOL 904.
JC. Yao, "A Generalized Complementarity
Problem in Hilbert Space," SOL 905.
ONR Computational Combina
torics URI
Institute for Interdisciplinary
Engineering Studies
304A Potter Engineering Center
Purdue University
West Lafayette, IN 47907
S.S. Abhyankar, T.L. Morin and T.B.
Trafalis, "Efficient Faces of Polytopes: Interior
Point Algorithms, Parameterization of Algebraic
Varieties, and Multiple Objective
Optimization," CC891.
V. Chandru and J.N. Hooker, "Extended
Horn Sets in Propositional Logic," CC892.
G.N. Frederickson and D.J. Guan,
"Nonpreemptive Ensemble Motion Planning on
a Tree," CC893.
G.N. Frederickson and D.J. Guan,
"Preemptive Ensemble Motion Planning on a
Tree," CC894.
V. Chandru, D. Dutta and C.M. Hoffmann,
"Variable Radius Blending Using Dupin
Cyclides," CC895.
R.G. Jeroslow, R.K. Martin, R.L. Rardin and
J. Wang, "Gainfree LeontiefFlow Problems,"
CC896.
G.N. Frederickson, "Using Cellular Graph
Embeddings in Solving all Pairs Shortest Paths
Problems," CC897.
A.K. Gupta and S.E. Hambrusch, "Multiple
Network Embeddings into Hypercubes," CC89
8.
S. Hambrusch and M. Luby, "Parallel
Asynchronous Connected Components in a
Mesh," CC899.
S. Hambrusch and L. TeWinkel, "Parallel
Heuristics for the Steiner Tree Problem in
Images without Sorting or Routing," CC8910.
S. Hambrusch, "An Optimal Parallel
Algorithm for Determining wConnectivity in
Images," CC8911.
D.K. Wagner and H. Wan, "A Polynomial
Time Simplex Method for a Class of
Transshipment Problems," CC8912.
M.C. Fields and G.N. Frederickson, "A
Faster Algorithm for the Maximum Weighted
Tardiness Problem," CC8913.
R. Swaminathan and D.K. Wagner, "A
ForbiddenMinor Characterization of Orientable
GraphTree Pairs," CC8914.
C.R. Coullard, J.G. del Greco and D.K.
Wagner, "Uncovering GeneralizedNetwork
Structures in Matrices," CC8915.
C.R. Coullard, J.G. del Greco and D.K.
Wagner, "Recognizing a Class of Bicircular
Matroids," CC8916.
G.N. Frederickson, "The Information Theory
Bound is Tight for Selection in a Heap," CC89
17.
V. Vinay and V. Chandru, "The Expressibility
of Nondeterministic Auxiliary Stack Automata
and Its Relation to Treesize Bounded
Alternating Auxiliary Pushdown Automata,"
CC8918.
M.J. Kaiser, T.L. Morin and T.B. Trafalis,
"Centers and Invariant Points of Convex
Bodies," CC901.
  ___ ~ ~
JULY 1990
PAGE 6
number thirty
PACE 7
Practical Methods of R R V
Optimization
by R. Fletcher
Wiley, Chichester, 1987
ISBN 0471925475
The book under review is the second
edition of a textbook for senior, undergraduate
and postgraduate students taking courses in
optimization. It aims to present those aspects of
optimization methods which are currently of foremost
importance in solving real life problems.
With an emphasis on practicability, throughout thebook most
attention is given to methods that have proven tobe reliable and
efficient. For each of these methods, the basic features are described
together with heuristics which can be valuable in making the methods
better perform in practice. Also detailed numerical evidence gives an
idea of the relative strength and weakness of each method. Though the
theoretical background is assigned an important role, it is presented not
from the viewpoint of theory for theory's sake but in close connection
with practical aspects of the methods.
Part I (Chapter 1 to Chapter 6) is devoted to unconstrained optimization.
After an illuminating introduction, Chapter 2 discusses the structures
of iterative methods and explains some general schemes of
unconstrained optimization as well as certain desirable features like
convergence, stability and use of quadratic model. Chapter 3 describes
Newtonlike methods. An important place is given to the BFGS formula
which is currently regarded as the best quasiNewton method. Also
numerical experiments arediscussed which provide useful information
abouitthebehaviourofthe methods in practical implementation. Chapter
4 describes conjugate direction methods. It is shown in particular how
conjugate gradient methods are both less efficient and less robust than
quasiNewton methods (therefore would not be preferred in normal
circumstances) but may be the only methods applicable to large
problems. Chapter 5 presents an extensive treatment of restricted step
methods or trust region methods which retain rapid rateofconvergence
of Newton's but are generally applicable and are globally convergent.
Finally, Chapter 6 discusses sums of squares and nonlinear equations.
Since these are encountered in data fitting problems which are the most
frequently solved of all optimization problems, they are thoroughly
treated, in line with the practicability theme of the book. In particular,
the treatment includes the DennisMord theorem characterizing
superlinear convergence in nonlinear systems and its significance.
On the whole, Part I provides the reader with an uptodate knowledge
of the basic theoretical background and standard techniques of
unconstrained optimization.
I I W S that of unconstrained optimization
(e.g. suitable test problems are
lacking), numerical evidence is
given much less attention here than
in Part I. After an introduction
(Chapter 7), linear programming is
presented in Chapter 8in much more
detail than in the first edition, in view
of the most recent developments in polynomial
algorithms for linear programming (Khachian,
Karmarkar, ...). Of course, the presentation is at an
advanced level and stresses features directly related to
practical implementation, such as numerical problemsdue
to magnification of roundoff errors, stability, degeneracy, and
exploiting of sparsity. Also a succinct description is given of the
ellipsoid algorithm and Karmarkar's algorithm. Chapters 9,10 and 13
describe the standard theory of constrained optimization (Lagrange
multiplier, first and second order conditions, convexity) together with
some advanced features (complementary pivoting and the like) and
techniques for general linearly constrained optimization. Emphasis is
placed on active set strategies which are rightly regarded as most intuitive
and flexible. On the other hand, convexity and duality are given a
relatively modest role. The most interesting Chapter of Part I is perhaps
Chapter 11 devoted to nonlinear programming. Although, as the author
says, there is no general agreement on the best approaches and much
research is still tobe done in nonlinear programming, this chapteroffers
an excellent and uptodate account of the situation. Penalty and barrier
functions, multiplier penaltyfunction, and LI exact penalty function are
treated extensively. The SQP method, which can be motivated as a
LagrangeNewton's method (Newton's method applied to find the
stationary point of the Lagrangian function) is discussed in detail. It is
shown that this method has local second order convergence and thus has
the same convergence rate for nonlinear programming as Newton's
method does for unconstrained minimization. New developments,
particularly in the case where only the reduced Hessian matrix is used,
convincethereader of the importanceof this method.Since SQP requires
computing second derivatives, a quasiNewton version is presented
which should be successful on small and medium size problems. The
Maratos effect is also discussed, showing the complexity of the problem.
The last two chapters, 13 and 14, are devoted to other optimization topics
such as integer programming, geometric programming, network
programming, and nonsmooth optimization.
As compared with the first edition of the book, the presentation in this
second edition has been very much improved and updated. No doubt
this is not only an excellent textbook for students but also a very useful
tool for researchers and anyone who has to solve practical problems in
real life by optimization methods.
Part II (Chapter 7to Chapter 13) is devoted to constrained optimization HOANC TUY
which is a subject of greater complexitythan that treated in Part I. In fact,
since the studyof constrained optimization is much less advanced than
CONTINUES um
JULY 1990
PAGE 7 JULY1
PAGE 8 number thill
Classical Principles and Optimization Problems
by B. S. Razumikhin
Reidel, Dordrecht, 1987
ISBN 9027726051
The book is devoted to mathematical programming and optimal control
problems, but the main idea is to use laws and tools of mechanics and
thermodynamics for foundation and derivation of numerical methods.
So there is a new and important connectionbetween physics and modern
methods of optimization and OR. According to theauthor's opinion, the
book is not only written for specialists in the field of optimization but
for a wider group of readers.
In chapter the principle ofvirtualdisplacement (the initial formulation
was given by Bernoulli) is treated. Using the contribution of Lagrange
(analytical statics and analytical dynamics by unifying that principle
with d'Alembert's principle), the author points out that in conservative
field the equilibrium of systems under unilateraland bilateral constraints
is mathematically equivalent to the general problem of mathematical
programming.
By realizations of the detachment principle some optimization methods
(i.e. penalty methods) can be derived (chapter 2). After investigation of
the wellknown energy theorem, interesting consequences follow for
duality theory and for numerical methods. We refer especially to
application of the principles of maximum work and of minimum work
(chapter 3). Chapter 4 covers physical models for systems of linear
equations and inequalities and shows connections to the alternative
theorems and the methods of surplus constraints and surplus variables.
Chapters5 and 6 mention the hodograph method LP and algorithms for
shifting elastic constraints for LP problems. Other essential topics of
mathematical programming are investigated in the next three chapters:
maximum flow in networks (chapter 7), the transportation problem
(chapter 8) and decomposition methods in LP (chapter 9). Moreover, a
general approach to gradient methods is given (chapter 10) and the
aggregation of constraints is stressed (chapter 11).
Chapter 12 is devoted to the laws of thermodynamics. Studying quite
different systems (economical, physical, social and others), processes
exist which are to be considered as transfer or distribution of resources.
Such problems can be found in chapters 1315; we refer especially to
models of economic equilibrium and to von Neumann's model of
economic growth.Thelast threechapters of thebookdeal with analytical
dynamics and optimal control (chapters 1618). The author gives good
insight into some variational principles, emphasizing the historical
developments. As is known, the initial foundation of the calculus of
variations is closely related to the LagrangeHamilton integral extremal
principles of analytical dynamics. In addition to some basic results about
such principles, a variety of tools and methods is presented. Thus the
book is full of interesting ideas for combining physical (mechanical)
principles with both theoretical results and numerical methods of
optimization. It is a good and fundamental contribution for
understandingthebasic ideas of important and often used classes of such
methods. Although there have been some earlier relevant papers, the
book presents a really new look at optimization theory and optimization
methods, demonstrating the value of analogies.
K.H. ELSTER
Probabilistic Analysis of Algorithms
by M. Hofri
Springer, Berlin, 1987
ISBN 3540965785
The judgement of algorithms based on probabilistic criteria about their
efficiency has gained importance during the last decade. This is due
partly to the fact that the growing capacity of computers enables us to
attack (at least try) very highdimensional problems. As long as these
problems stem from realworld applications, our interest is mostly
successoriented, not generalityoriented. That means that we are content
to have a solution for our specific probleminstance, and we do not pay
much attention to the possibility of generalizing our solutionmethod
to a large class (or all) problems of the given type. Hence a bad "worst
casebehavior" may not adequately describe an algorithm because the
method could be very efficient in "most of the cases". In the latter case
itcould berecommended to make use ofthat algorithm quitein contrast
to recommendations in the early days of theoretical computer science.
Another question is how to get probabilistic criteria. One way would be
the random generation of problems and their experimental solutions
followed by statistical evaluations of the observed efficiency of the
employed algorithms. However, the disadvantages are many:
Numerous experiments have to be executed until our statistical results
are reliable. This requires a tremendous amount of computer capacity.
(Didn't wewantto save capacityby studying the efficiency?) Judgements
on the behavior for very high dimensions and asymptotic (more
qualitative) statements are impossible per se, because the capacity is
limited. And our statistical evaluation provides us with numbers but not
with an insight into how the algorithm works. Misinterpretations of the
statistical data are a frequent consequence.
The second way to get probabilistic criteria, which is more complicated
but also more rewarding, is as follows:
Step 1: Define a stochastic model about the distribution of the problem
instances.
Step 2: Characterize the principal behavior of the algorithm on a given
and fixed set of data so that this behavior canbe interpreted as a random
variable.
Step 3: Derive probabilistic statements on that random variable by
accumulating overall possible probleminstances.
The art of carrying out such a probabilistic analysis is what the author
dealswith. Itturnsout immediatelythateverysuchprojectis averyhard
task. This is impressively demonstrated in this book.
  
PAGE 8
number thirty
JULY 1990
R R V I EWS
Afterashortintroductorychapter,muchattentionispaid algorithm or problem analyzable and of recognizing
to the mathematical tools for doing the evaluation step. which algorithms could be attacked successfully.
These tools stem from different fields of mathematics and their To summarize, this is a bookof very high standard, suited for
applicationrequiresacertain amountof knowledgeinthesefields. graduatestudentswith abroad educationinmathematics. Itis also
In chapter 2 the author deals with Bernoulli numbers and extremely useful for people doing research in that field by providing
polynomials, generating functions for probabilities and moments,
Lagrange Expansions, Poisson transforms, Laplace transforms, Mellin
transforms and summation formula. Further topics are the symbolic
operator methods, asymptotics from generating functions (mainly
complex functionstheory). Following this is a paragraph with selected
results (the most useful ones) from probability theory. So far we have
a large collection of tools, tricks and ideas which could be useful for Step
3. However, there is still no direct and inevitable connection to
probabilistic analysis of algorithms. The following three chapters (3, 4,
5) accomplish this purpose.
Chapter 3 deals with algorithms for determining the maximal value
among n numbers and for sorting them. Here and in the following
chapters we get a hint of what is to be done in Step 2; and Step 3 is given
in detail. Chapter 4 describes algorithms for communication networks.
Here the interest is directed towards the capacity or the average number
of useful messages such a network can carry under a varietyof different
modes of communication requirements. Further issues are collision
resolution, message delay, etc. This chapter is, in my view, much more
complicated than the others.
Chapter 5 reports on probabilistic evaluation of bin packing heuristics,
particularly the nextfit and nextfit decreasing packing methods. Here
results on the average number of bins, on the number of pieces packed
into the specific bins and on the total load in thesebins arederived under
uniform distribution on [0,1] respectively [0, a] with a < 1, where the bin
capacity is constantly 1. An appendix provides the reader with many
additional formulas for the evaluation.
The three main chapters are meant to give typical examples for
probabilistic analyses. It cannot be the aim of such a book to give a
complete overview. In the latter case it would have been necessary to
dispense with the derivations. But here the strong emphasis on the
techniques sometimes leads to a situation where the main theorems and
results get into the background.
Every paragraph is followed by many interesting, instructive and
challenging exercises. (This is true of the whole book.) It takes a great
deal of mathematical effort for the reader to verify all the technical
derivations. Simultaneously, the reader is provided with a significant
number of evaluation concepts and techniques. It may be relevant to
mention that I could solve along open problem by use of one trick which
I had found in the book.
As in the closely related book of Kemp, Fundamentals ofthe Average Case
Analysis of Algorithms, the emphasis is put on the evaluation side and
hence the reader has the impression of having a "textbook of
mathematical tricks" in hand. Thus there is a definite need for a book
which covers the difficulties and ideas of Step 2, i.e. of making an
them with many technical tricks. And it is very interesting for
mathematicians in general who want to learn which and how many fields
of mathematics enter this theory. It is another proof that mathematics
is a unit. It is not all a survey and it emphasizes the evaluation aspect
very strongly. For people willing to invest a lot of effort in reading and
in followingthe ideas of theauthor, I candefinitelyrecommend this book.
KARL HEINZ BORGWARDT
Algorithmics: Theory and Practice
by B. Brassard and P. Bratley
Prentice Hall, New Jersey, 1988
ISBN 0130232432
After reading or browsing through this book, many teachers will feel
stimulated to teach a course on algorithms and their analysis. I am also
certainthatmanystudents will be excited enough to go out and program
some of the algorithms analysed or suggested as exercises in the book.
The first chapter introduces most of the concepts that are needed:
algorithms, average and worst case analyses and their importance, as
well as some of theexamples and the important data structures that crop
up throughout the book. In the second chapter the reader gets down to
the seriousbusiness of analysing the efficiency of algorithms. Asymptotic
notation is introduced in a somewhat nonstandard form, and then
several sorting algorithms including heapsort, Euclid's algorithm and
set merging algorithms are analysed. This leads naturally to a clear
section on the study of recurrence relations. These two important
chapters are not simple, and as the authors suggest this is a book for
advancedundergraduateorgraduate courses. However, there are many
examples, sections and ideas that could be used somewhat earlier in
discrete mathematics and algorithm courses.
The book continues with chapters on greedy algorithms, divide and
conquer and dynamic programming. That on greedy algorithms looks
at tree and shortest path problems on graphs, scheduling problems, such
scheduling unit processing time jobs with deadlines, for which the
greedy algorithm is optimal, and terminates with a section on greedy
heuristics, which is disappointingly brief given their wide application.
The section on divide andconquer deals with manyimportantexamples
of recursion and topdown decomposition or simplification. Binary
search, quicksort and median finding algorithms, and the recurring
problem of multiplication of large integers are tackled. Dynamic
programming is presented in contrast as a bottomup approach with
chained matrix multiplication, shortest paths and optimal search trees
as the main examples.
CONTINUES
I 111a rrFlff*,VIZI&
~  ~  
number thirty
JULY 1990
PAGE 9
PAE1 ubrtit UY19
The authors suggest that a one semester course should consist of the
above five chapters and a selection from the remainder, entitled
respectively: exploring graphs; preconditioning and precomputation:
probabilistic algorithms: transformations of the domain and
introduction to complexity. Each is fascinating, though my preference
goes to the first three.Thechapter on graphs, which would also consider
as basic and would introduce much earlier, treats the obvious but
important topics of traversing trees and the connectivity questions that
can therebybe treated. Finallyit examines the concepts of backtracking
and branch and bound using the 8 queens problem as a fascinating
example.
Preconditioning is presented via the problems of repeated evaluation
of a polynomial and string searching problems. The long chapter on
probabilistic algorithms is compulsive reading, presenting clearly the
different possible approaches. One section deals with randomness in
numerical problems as in simulation or numerical integration. Another
treats Monte Carlo algorithms that with a small probability give wrong
answer such as those for primality testing or testing the equality of two
sets. A third deals with Las Vegas algorithms that occasionally fail to give
a solution, with, as examples, the problems of finding square roots
modulo an integer, and of factorisation. Searching and hashing
algorithms make up another section in which the random choices lead
to an algorithm whose worst case expected running time on any given
instance is improved.
The penultimate chapter is mainly devoted to the fast Fourier transform
and the multiplication of large integers, and the final chapter on
complexity introduces decision trees, the concept of reduction for matrix
graphical and polynomial problems, and NPCompleteness.
Altogether there is a world of material in this book, and mathematical
programmers from all walks of life should find topics to amuse, interest
or stimulate them.
LAURENCE A. WOLSEY
Vol.46, No.3
J.K. Lenstra, D.B. Shmoys and t. Tardos,
"Approximation Algorithms for Scheduling
Unrelated Parallel Machines."
A.R. Conn and G. Cornujols, "A Projection
Method for the Uncapacitated Facility Location
Problem."
C. McDiarmid, "On the Improvement per
Iteration in Karmarkar's Algorithm for Linear
Programming."
P.M. Pardalos and N. Kovoor, "An Algorithm for
a Singly Constrained Class of Quadratic
Programs Subject to Upper and Lower Bounds."
M.S. Gowda and T.I. Seidman, "Generalized
Linear Complementarity Problems."
K. Ito and K. Kunisch, "The Augmented
Lagrangian Method for Equality and Inequality
Constraints in Hilbert Spaces."
M.E. Dyer and A.M. Frieze, "On Patching
Algorithms for Random Asymmetric Travelling
Salesman Problems."
G.L. Nemhauser and L.A. Wolsey, "A Recursive
Procedure to Generate All Cuts for 01 Mixed
Integer Programs."
S. Dafermos, "Exchange Price Equilibria and
Variational Inequalities."
B. Luderer and R. Risiger, "On Shapiro's Results
in Quasidifferential Calculus."
Vol.47, No.1
Y. Ye and M. Todd, "Containing and Shrinking
Ellipsoids in the PathFollowing Algorithm."
W. Cook, "CuttingPlane Proofs in Polynomial
Space."
M. Padberg and G. Rinaldi, "An Efficient
Algorithm for the Minimum Capacity Cut
Problem."
A. Iusem and A. De Pierro, "On the Convergence
Properties of Hildreth's Quadratic Programming
Algorithm."
Y. Yuan, "On A Subproblem of Trust Region
Algorithms for Constrained Optimization."
R. Durier, "On Pareto Optima, the FermatWeber
Problem, and Polyhedral Gauges."
J. Orlin and J. Vande Vate, "Solving the Linear
Matroid Parity Problem as a Sequence of
Matroid Intersection Problems."
A. Shapiro, "On Differential Stability in
Stochastic Programming."
C. Tiahrt and A. Poore, "A Bifurcation Analysis
of the Nonlinear Parametric Programming
Problem."
X.S. Zhang and D.G. Liu, "A Note on the
Continuity of Solutions of Parametric Linear
Programs."
Vol.47, No.2
W. Cook, R. Kannan and A. Schrijver, "Chvdtal
Closures for Mixed Integer Programming
Problems."
P. Vaidya, "An Algorithm for Linear Program
ming which Requires 0(((m+n)n'+(m+n)'sn)L)
Arithmetic Operations."
A. Jourani and L. Thibault, "Approximate
Subdifferential and Metric Regularity: The Finite
Dimensional Case."
_ s~
~I ___ ~
PAGE 10
number thirty
JULY 1990
JULY 1990
M. Padberg and G. Rinaldi, "Facet Identification
for the Symmetric Traveling Salesman Polytope."
F. Granot and J. SkorinKapov, "Some Proximity
and Sensitivity Results in Quadratic Integer
Programming."
K. Holmberg, "On the Convergence of Cross
Decomposition."
A. Dax, "A New Theorem of the Alternative."
Vol.47, No.3
J.V. Burke, J.J. Mord and G. Toraldo, "Conver
gence Properties of Trust Region Methods for
Linear and Convex Constraints."
K.M. Anstreicher, "A Standard Form Variant,
and Safeguarded Linesearch, for the Modified
Karmarker Algorithm."
D. Goldfarb and J. Hao, "A Primal Simplex
Algorithm that Solves the Maximum Flow
Problem in at Most nm Pivots and O(n2m) time."
M. Gr6tschel and Y. Wakabayashi, "Facets of the
Clique Partitioning Polytope."
R.R. Merkovsky and D.E. Ward, "General
Constraint Qualifications in Nondifferentiable
Programming."
H.I. Gassman, "MSLiP: A Computer Code for the
Multistage Stochastic Linear Programming
Problem."
M.J. Best and N. Chakravarti, "Active Set
Algorithms for Isotonic Regression; A Unifying
Framework."
J. Lee, "A Spectral Approach to Polyhedral Di
mension."
Vol.48, No.1
E. Gelman and J. Mandel, "On Multilevel
Iterative Methods for Optimization Problems."
S.M. Gorelick, "Large Scale Nonlinear Determin
istic and Stochastic Optimization: Formulations
Involving Simulation of Subsurface Contamina
tion."
C.T. Kelley and E.W. Sachs, "Approximate
QuasiNewton Methods.
W.W. Symes, "Velocity Inversion:A Case Study
in InfiniteDimensional Optimization."
M.H. Schneider, "Matrix Scaling, Entropy
Minimization, and Conjugate Duality (II):The
Dual Problem."
Ph.L. Toint and D. Tuyttens, "On Large Scale
Nonlinear Network Optimization."
Application for Membership
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PAGE 11
~ ~~~
  ~ ~~
number thirty
8~1bs=MMWW1K,, "CrNra I
T HE mathematical programming community was saddened
T by the news in April of the death of Stella Dafermos (Brown).
Stella, a leading researcher in the transportation science community,
made many notable contributions to equilibrium models and
algorithms. !JCraig Tovey (Georgia Tech) has been awarded the
Jacob Wolfowitz Prize for his paper "Simulated Simulated
Annealing" which appeared in volume 8 of the American Journal of
Mathematical and Management Sciences. fORSA and TIMS awarded
the 1990 John von Neumann Prize to Richard M. Karp (Berkeley)
for his contributions to computational theory and algorithms in
operations research and management science. !Deadline for the
next OPTIMA is October 1, 1990.
Books for review should be
sent to the Book Review Editor,
Prof. Dr. Achim Bachem,
Mathematiches Institute der
Universitiit zu K6ln,
Weyertal 8690, D5000 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
PUBUCATION SERVICES OF THE
COLLEGE OF ENGINEERING,
UNIVERSITY OF FLORIDA.
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P T I M A
MATHEMATICAL PROGRAMMING SOCIETY
303 Weil Hall
College of Engineering
University of Florida
Gainesville, Florida 32611 USA
FIRST CLASS MAIL
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 ~ 
PAGE 12
number thirty
JULY 1990
