Title: Optima
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Permanent Link: http://ufdc.ufl.edu/UF00090046/00069
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Title: Optima
Series Title: Optima
Physical Description: Serial
Language: English
Creator: Mathematical Programming Society, University of Florida
Publisher: Mathematical Programming Society, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: January 2003
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Bibliographic ID: UF00090046
Volume ID: VID00069
Source Institution: University of Florida
Holding Location: University of Florida
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emticl Progrmming Society Newsletter
Mathematical Programming Society Newsletter



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Primal Integer Programming
Robert Weismantel
Faculty of Mathematics, University of Magdeburg
Universitatsplatz 2, D-39106 Magdeburg, Germany

September 17, 2002

Primal integer programming is an area of
discrete optimization that -- in a broad sense --
includes the theory and algorithms related to
augmenting feasible integer solutions or
verifying optimality of feasible integer points.
This theory naturally provides an algorithmic
scheme that one might regard as a generic form
of a primal integer programming algorithm:
First detect an integer point in a given domain
of discrete points. Then verify whether this
point is an optimal solution with respect to a
specific objective function, and if not, find
another feasible point in the domain that attains
a better objective function value.
The field of primal integer programming has
emerged in the 1960s beginning with the design
of specific algorithms in combinatorial
optimization and Gomory's all integer
algorithms for the general integer programming
case. While the field stayed active in theory over
the past 40 years, the algorithmic focus has been
nearly exclusively on combinatorial optimization
problems. In fact, augmentation algorithms have
been designed for and applied to a range of
specific linear integer programming problems:
augmenting path methods for solving maximum
flow problems or algorithms for solving the min-
cost flow problem via augmentation along
negative cycles are of this type. Other examples
include the greedy algorithm for solving the
matroid optimization problem, alternating path
algorithms for solving the weighted matching
problem, or methods for optimizing over the
intersection of two matroids. In fact, still many
of the most recent combinatorial algorithms are
of a primal nature; see for instance the
combinatorial methods for minimizing a
submodular function [12] [19] or the
combinatorial algorithm for the independent
path matching problem [20].
The attempt, however, to solve unstructured
integer programs with a primal strategy had lost
its popularity for at least two decades. Which
developments did lead to this situation? What
are the obstacles for a primal strategy in the
general integer case? What is the state of the art?
What are the next steps to be taken? These are
some of the questions that I try to address in the

Let us begin by recalling that all the
algorithmic schemes applied to integer
programming problems without a-priori
knowledge about the structure of the side
constraints resort to the power of linear
programming duality.
Dual type algorithms start solving a linear
programming relaxation of the problem,
typically with the dual simplex method. In the
course of the algorithm one maintains as an
invariant both primal and dual feasibility of the
solution of the relaxation. While the optimal
solution to the relaxation is not integral, one
continues to add cutting planes to the problem
formulation and reoptimizes. Within this
framework, Gomory developed a method of
systematically generating valid inequalities
directly from a given simplex tableau; see [7, 8].
A disadvantage of a dual-type method is,
however, that intermediate stopping does not
automatically yield a primal feasible integral
In contrast to dual methods, primal type
algorithms always preserve integrality of the
solution to the relaxation. They may be
distinguished according to whether the
algorithm preserves dual feasibility or primal
feasibility or none of those. Note that it is
impossible to preserve integrality and both dual
and primal feasibility. The only representative of
the first family of primal type methods is an "all
integer algorithm" that appeared in [9]. We are
not aware of interesting computational
experiments with this method, however.
An alternative to design a primal type
algorithm is to preserve integrality of the
solution to the relaxation and simultaneously
primal feasibility. In order to achieve this, one
typically starts with a primal feasible simplex
tableau. Iteratively, pivot steps are performed on
entries equal to one, only. If such a pivot
element cannot be detected by the standard dual
simplex rule, a special cut row with this property
is generated and then chosen for pivoting. When
at any time the reduced cost coefficients of the
tableau point into the "right direction",
optimality is proved by the simplex criterion.
The cuts one makes use of in this framework
may be Gomory's rounding cuts as in [21], or

problem specific cuts such as facet-defining cuts
for the TSP in [18]. To the best of our
knowledge, a method of this type for Integer
Programming has first been suggested by Ben-
Israel and Charnes [3]. They employed the fact
that for any feasible solution of an integer
program optimality can be proven by solving a
subproblem arising from the non-basic columns
of a primal feasible simplex tableau. Various
specializations and variants of a primal cutting
plane algorithm were later given by Young [21,
22] and Glover [6]. For an overview on this
subject we refer to [11, 5]. Nemhauser and
Wolsey [17] point out that because of poor
computational experience, this line of research
has been very inactive, an exception being the
work of Padberg and Hong [18], where strong
primal-type cutting planes are used with some
success. According to [17] the reason for the
poor computational performance of general
primal cutting plane algorithms is that "it is
necessary to produce valid inequalities that
contain the one-dimensional faces (edges) on a
path from the initial point to an optimal point",
see also [13]. This is certainly one explanation
why this line of research became inactive.
Recent theoretical advances regarding primal
type algorithms and the theory of so-called test
sets have also renewed the interest in primal
cutting plane algorithms. Letchford and Lodi
[15] propose a modern primal cutting plane
algorithm for 0/1 programs. The main
ingredient is the subproblem of "primal
separation", see also [4, 16]. They give
computational results on small randomly
generated 0/1 multidimensional knapsack
problems with up to twenty five variables. Their
computational experiments reveal that the
proposed algorithm is superior to the original
algorithm of Young. However, the results do not
provide evidence that primal cutting plane
algorithms can become superior to dual cutting
plane methods. Only future research and
experiments can clarify this situation. I have
doubts, however, that primal cutting plane
algorithms will become an alternative to the
"standard method" of today. One additional
problem for primal cutting plane algorithms
remains: To this day there is no primal cutting




algorithm for mixed integer programs available,
despite the fact that mixed integer programming
models are the most important models of
discrete optimization.
It is, however, important to remark that even
if primal cutting plane algorithms will not work
in practice, this will not imply that a primal
augmentation approach will not work. An
alternative to primal cutting plane algorithms is
the Integral Basis Method [10]. It uses a
completely different idea for solving the
augmentation subproblem. After turning a
feasible solution into a basic feasible solution of
an appropriate simplex tableau, one manipulates
the set of non-basic columns until either
optimality of the given solution is proved or an
"applicable" improving direction, i.e., an
augmentation vector, is detected. In each
manipulation step, one substitutes one column
by new columns in a way that no feasible
solutions are lost; this is known as a "proper

reformulation" of a tableau according to [10].
The substitution is guided by the concept of
irreducibility of lattice points. In contrast to
generating a basis of a lattice one generates
iteratively a minimal set of lattice points in the
positive orthant that permits a representation of
any feasible solution as a non-negative integer
combination of the selected subset. This
distinguishes the latter approach from the
reformulation method ofAardal et al [1]. We
refer to [10] for the theoretic foundations and
computational results with the Integral Basis
Method. We also remark that the method can be
extended to mixed integer programming
problems, see [14]. A predecessor of the Integral
Basis Method for the special case of set
partitioning problems was invented by Balas and
Padberg about 25 years ago [2].
Computational results with the Integral Basis
Method demonstrate that a couple of iterative
substitutions of non-basic columns with a "bad"

reduced cost coefficient yield a model in which
the linear programming relaxation becomes
significantly stronger than the original linear
programming relaxation. Therefore, the effect of
reformulating a tableau with the help of
irreducible lattice points is not only interesting
per se, but rather suggests a primal-dual
approach for integer programming: perform
Integral Basis Method steps in addition to
cutting planes and branching. Each of the three
operations, i.e., substitution of variables, cutting,
and branching, has an impact on the model and
on the other operations. This is -- in my opinion
-- an area of research to advance our current
ability to solve integer programming models. If
we apply in addition to branching and cutting
the operation of substituting variables, then we
cannot lose performance, but gain a lot. This is
where I see one future direction of research in
integer programming algorithms.


[1] Karen Aardal, Robert E. Bixby,
Cor A. J. Hurkens, Arjen K.
Lenstra, and Job W Smeltink.
Market split and basis
reduction: towards a solution
of the Cornudjols-Dawande
instances. INFORMS J
Comput., 12(3):192--202,

[2] Egon Balas and Manfred W.
Padberg. On the set-covering
problem: II. An algorithm for
set partitioning. Operations
Research, 23:74--90, 1975.

[3] Adi Ben-Israel and Abraham
Charnes. On some problems
of diophantine programming.
Cahiers du Centre d'Etudes de
Recherche Opirationnelle,
4:215--280, 1962.

[4] Friedrich Eisenbrand, Giovanni
Rinaldi, and Paolo Ventura.
0/1 optimization and 0/1
primal separation are
equivalent. In Proceedings of
SODA 02, pages 920--926,

[5] Robert S. Garfinkel and George
L. Nemhauser. Integer
Programming. Wiley, New
York, 1972.

[6] Fred Glover. A new foundation
for a simplified primal integer
programming algorithm.
Operations Research, 16:727--
740, 1968.

[7] Ralph E. Gomory. Outline of
an algorithm for integer
solutions to linear programs.
Bulletin of the American
S. .' 64:275--
278, 1958.

[8] Ralph E. Gomory. An
algorithm for the Mixed
Integer Problem. Technical
Report RM-2597-PR, The
RAND Corporation, Santa
Monica, CA, 1960.

[9] Ralph E. Gomory. An all-
integer integer programming
algorithm. In J. F Muth and
G. L. Thompson, editors,
Industrial Scheduling: (Papers
presented at a Conference on
"Factory Scheduling" held at the
Graduate School ofIndustrial
Administration at Carnegie
Institute of Technology, May
10--12, 1961), Prentice-Hall
international series in
management, pages 193--206.
Prentice Hall, Englewood
Cliffs, N.J., 1963.

[10] Utz-Uwe Haus, Matthias
Koppe, and Robert
Weismantel. A primal all-
integer algorithm based on
irreducible solutions. To
appear in Mathematical
Programming Series B, preprint
available from URL
theory-rr.ps, 2001.

[11] T. C. Hu. Integer Programming
and Network Flows. Addison -
Wesley, Reading, Mass., 1969.

[12] S. Iwata, L. Fleischer, and S.
Fujishige. A combinatorial
strongly polynomial algorithm
for minimizing submodular
functions. Journal of the ACM,
48(4):761--777, 2001.

[13] Matthias K6ppe. Exact primal
algorithms for general integer
and mixed-integer linear
programs. Dissertation,
Universitat Magdeburg, 2002.

[14] Matthias K6ppe. and Robert
Weismantel. An algorithm for
mixed integer optimization.
Preprint no. 17, Fakultat fir
Mathematik, Otto-von-
Magdeburg, 2002.

[15] Adam N. Letchford and
Andrea Lodi. Primal
separation algorithms.
Technical Report OR/01/5,
DEIS, University of Bologna,

[16] Adam N. Letchford and
Andrea Lodi. An augment-
framework for mixed 0-1
programming. In Proceedings
ofAussois V, 2002. to appear.

[17] George L. Nemhauser and
Laurence A. Wolsey. Integer
and Combinatorial
Optimization. Wiley,
Chichester, 1988.

[18] Manfred W. Padberg and
Saman Hong. On the
symmetric travelling salesman
problem: a computational
study. Math. Programming
Stud., (12):78--107, 1980.

[19] A. Schrijver. A combinatorial
algorithm minimizing
submodular functions in
strongly polynomial time.
Journal of Combinatorial
Theory, 8:346--355, 2000.

[20] Bianca Spille and Robert
Weismantel. A generalization
of Edmonds matching and
matroid intersection
algorithms. In Springer Lecture
Notes in Computer Science
2337, pages 9--20, 2002.

[21] Richard D. Young. A
simplified primal (all-integer)
integer programming
algorithm. Operations Research,
16(4):750--782, 1968.

[22] Richard D. Young. The
eclectic primal algorithm:
Cutting-plane method that
accommodates hybrid
subproblem solution
techniques. Math. Program.,
9:294--312, 1975.




ISMP 2003

Copenhagen, Denmark

August 18 22, 2003


The Mathematical Programming Society, the
Technical University of Denmark, and the
University of Copenhagen announce:

ISMP 2003 is approaching: The 18th
International Symposium on Mathematical
Programming takes place August 18-12, 2003 at
the Technical University of Denmark,
Copenhagen, in cooperation with the University
of Copenhagen. It is the main scientific event of
the Mathematical Programming Society held every
3 years on behalf of the Society. The Symposium
attracts more than one thousand researchers from
all areas of mathematical programming. At the
symposium homepage www.ismp2003.dk you will
find all information including the list of plenary
and semiplenary speakers:

Susanne Albers,
Kurt Anstreicher,
Sanjeev Arora,
Francis Clarke,
William J. Cook,

Siemion Fajtlowicz,
Adrian Lewis,
Tom Luo,
Renato Monteiro,

Stephen M. Robinson,
Mikael Ronnqvist,
Riidiger Schultz,

Peter W Shor,
Mikl6s Simonovits,

Daniel A. Spielman,
Robin Thomas,

Laurence A. Wolsey,

University of Freiburg
University of Iowa
Princeton University
University of Lyon-1
Georgia Institute of
University of Houston
Simon Fraser University
McMaster University
Georgia Institute of
University of Wisconsin
Linkoping University
University Duisburg
AT&T Labs Research
Hungarian Academy of
Georgia Institute of
University Catholique de

A proceedings volume sponsored by Springer
Verlag with a contribution from each speaker will
be published as a special issue of Mathematical
Programming and will be distributed to all
participants at the symposium.
The scientific program will be complemented
by an attractive social program: At ISMP 2003
you will have the opportunity to taste the
traditional "City Hall" pancakes served at the City
Hall and to participate in a cocktail party in the
Celebration Hall at the University of Copenhagen.
The conference dinner will take place at the
famous Base Camp restaurant in the former naval
area of the city, recently rebuilt into one of the
main cultural centers of the city. Additional
activities will be arranged in the scenic
surroundings of the campus.
The homepage provides information about
registration, travel, and accommodations, and offers
possibilities for web-based abstract submission of
contributed presentations as well as organized

The registration fees for ISMP2003 are:
MPS Member: 2100 DKK before/on April 30,
2003; 2700 DKK after April 30, 2003.
Non-MPS Member: 2600 DKK before/on April
30, 2003; 3100 DKK after April 30, 2003.
Student: 1100 DKK before/on April 30, 2003;
1600 DKK after April 30, 2003.
Special registration pricing including a one
year membership to MPS is also available. See
the homepage for details. In addition to the
traditional congress material as booklet of
abstracts and bag, registration includes the
volume of invited lectures, admission to the
opening reception, daily bag lunches, and
transportation for the entire week.
We invite you to participate in the event and to
submit an individual abstract or propose an
organized session on some special topic of general
interest. Each such session will consist of three
talks. The submission procedure for both abstracts
and sessions is described at the homepage: Just
click on the "How to use this homepage" button.

Program committee:
Jorgen Tind (chair), University of Copenhagen
Jens Clausen (cochair), Technical University of
Rainer Burkard, Technische Universitat,

Martin Grotschel,
Michael J. Todd,
Stephen J. Wright,

Organizing commit
Jens Clausen (chair),

Jorgen Tind (cochair),
Hans Bruun Nielsen,

David Pisinger,
Martin Zachariasen,

Important dates:
April 30, 2003:

May 31, 2003:

August 18 22:

ZIB, Berlin
Cornell University
University of Wisconsin

Technical University of
University of Copenhagen
Technical University of
University of Copenhagen
University of Copenhagen

Deadline for early
Deadline for submission
of abstracts.
ISMP 2003

We look forward to welcoming you in

Jorgen Tind and Jens Clausen



Call for Nominations

A. W Tucker Prize

The next A.W Tucker Prize will be awarded at the XVIII Mathematical
Programming Symposium in Copenhagen, August 18-22, 2003 for an
outstanding paper authored by a student. The paper can deal with any
area of mathematical programming. All students, graduate or
undergraduate, are eligible. Nominations of students who have not yet
received the first university degree are especially welcome. The Awards
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 symposium. The Awards Committee will select the
winner and present the award prior to the conclusion of the symposium.

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
ingenious application. The paper must be solely authored and completed
since 2000. The paper and the work on which it is based should have
been undertaken and completed in conjunction with a degree program.

Nominations must be made in writing to the Chairman of the Awards
Committee by a faculty member at the institution where the nominee was
studying for a degree when the paper was completed. Moreover,
nominators should send one copy 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 to each of the four members of the Tucker Prize Committee:

Prof. Rainer E. Burkard (Chair)
Institute of Mathematics B
Graz University of Technology
Steyrergasse 30
A-8010 Graz

Prof. S. Thomas McCormick
Faculty of Commerce and Business
University of British Columbia
Vancouver, B.C., Canada V6T 1Z2
Tom McC'rmick@comrmerce bc ll.

Prof. Jos F Sturm
Faculty of Economics
Department of Econometrics
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands

Prof. Leslie E. Trotter, Jr.
School of Operations Research and
Industrial Engineering
Cornell University
235 Rhodes Hall
Ithaca, NY 14853

The Awards Committee may request additional information. The
deadline for nominations is February 1, 2003.

Nominations and the accompanying documentation must be written in
a language acceptable to the Awards Committee.

The winner will receive an award of $750 (U.S.) and a certificate. The
other finalists will also receive certificates. The Society will also pay partial
travel expenses for each finalist to attend the symposium. These
reimbursements will be limited in accordance with the amount of
endowment income available. A limit in the range from $500 to $750
(U.S.) is likely. The institutions from which the nominations originate will
be encouraged to assist any nominee selected as a finalist with additional
travel expense reimbursement.

Past Winners and Finalists:
1988: Andrew Goldberg
1991: Michel Goemans. Other finalists: Leslie Hall, Mark Hartmann
1994: David Williamson. Other finalists: Dick Den Hertog, Jiming Liu
1997: David Karger. Other finalists: Jim Geelen, Luis Nunes Vicente
2000: Bertrand Guenin. Other finalists: Kamal Jain, Fabian Chudak

The Lagrange Prize in Continuous Optimization

Nominations are invited for the newly
established Lagrange Prize in Continuous
Optimization, awarded jointly by the
Mathematical Programming Society (MPS) and
the Society for Industrial and Applied
Mathematics (SIAM). The Prize will be
presented for the first time at the XVIII
International Symposium on Mathematical
Programming in August 2003.
To be eligible, works should form the final
publication of the main results) and should be
published either (a) as an article in a recognized
journal, or in a comparable, well-referenced
volume intended to publish final publications
only; or (b) as a monograph consisting chiefly of

original results rather than previously published
material. Extended abstracts and
prepublications, and articles published in
journals, journal sections or proceedings that are
intended to publish non-final papers, are not
eligible. The work must have been published
during the six calendar years preceding the year
of the award meeting.
Judging of works will be based primarily on
their mathematical quality, significance, and
originality. Clarity and excellence of the
exposition and the value of the work in practical
applications may be considered as secondary

Full details and prize rules are given at
To nominate a publication for the prize,
please send a copy of the paper and a letter of
nomination by February 28, 2003 to the
following address:
Stephen Wright
Computer Sciences Department
University of Wisconsin
1210 W. Dayton Street
Madison, WI 53706,
email: swright@cs.wisc.edu
Electronic submissions are preferred.

S P T A69




Call for Nominations

Beale-Orchard-Hays Prize

Nominations are being sought for the Mathematical Programming Society
Beale-Orchard-Hays Prize for Excellence in Computational Mathematical

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., well-documented code capable of obtaining solutions to
some important class of mathematical programming problems)
coupled with documentation of the application of the software
to this class of problems (note: the award would be presented
for the paper that describes this work and not for the software
(c) the development of a new computational method that improves
the state-of-the-art in computer implementations of
mathematical programming algorithms coupled with
documentation of the experiment that 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. Documentation must be written in a language acceptable to the
Screening Committee.
4. It must have been published during the three calendar years
preceding the year in which the prize is awarded.
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
The prize will be awarded every three years. The 2003 prize of $1,500 and
a medal will be presented in August 2003, at the awards session of the
18th International Symposium on Mathematical Programming to be held
in Copenhagen, Denmark.

Judgement criteria
Nominations will be judged on the following criteria:
1. Magnitude of the contribution to the advancement of computational
and experimental mathematical programming.
2. Originality of ideas and methods.
3. Degree to which unification or simplification of existing
methodologies is achieved.
4. Clarity and excellence of exposition.

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 Screening Committee reserves the right
to request further supporting materials from the nominees. The deadline
for nominations is March 15, 2003.

Nominations should be mailed to:
William Cook
Georgia Institute of Technology
School of Industrial and Systems Engineering
Atlanta, GA 30332-0205
email: bico@isye.gatech.edu

The COIN-OR Open-Source Coding
Contest: Win an IBM ThinkPad

At the 2000 International Symposium for Mathematical Programming
(ISMP), COIN-OR (http://www.coin-or.org) was launched by IBM
Research with the goal of promoting "open-source" software for
operations-research professionals. Our lofty mission was to explore an
alternative means for developing, managing, and distributing OR
software so that OR professionals could benefit from peer-reviewed,
archived, openly-disseminated software (much in the same way we
already benefit from theory). In November 2002, a milestone
toward the project's long-term objectives was reached when the
INFORMS board unanimously accepted a proposal to become the
new host of the COIN-OR initiative. To celebrate, we're throwing a

Visit http://www.coin-or.org/contest.htm to find out how you can
win an IBM ThinkPad. Prizes will be awarded August 2003 at the
2003 ISMP in Copenhagen, Denmark.

Robin Lougee-Heimer
IBM TJ Watson Research Center
ph: 914-945-3032 fax: 914-945-3434

10 P T-iS-,-- 6 91



Call for Papers

Mathematical Programming Series B
Special Issue on Large-Scale Nonlinear Programming

We invite research articles on algorithms for
large-scale nonlinear programming for a
forthcoming special issue of Mathematical
Programming, Series B. The goal is to assess
relative strengths and weaknesses of various
computational approaches to solving large-scale
nonlinear programming problems. Examples of
such approaches are first-order vs. second-order
methods, direct vs. indirect solution to linear
equations, methods of estimating Lagrange
multipliers, trust-region vs. step-length methods,
and active-set vs. interior-point methods. All
participants will be asked to provide
performance profiles for their implementation
on a common set of AMPL models, which can
be downloaded from http://mathweb.mathsci.

Deadline for submission of full papers:
July 31, 2003.
We aim at completing a first review of all
papers by December 31, 2003.
Electronic submissions in the form of
unencoded postscript files are encouraged.
Instructions for authors submitting to
Mathematical Programming can be found at
7/instr.htm. All submissions will be refereed
according to the usual standards of
Mathematical Programming.
Further information about this issue will
appear at http://www.princeton.edu/-
rvdb/MpbNlpIssue.html or can be obtained from
the guest editors for this volume.

Guest Editors:
David F Shanno
Rutgers University
640 Bartholomew Road
Piscataway, NJ 08854

Robert J. Vanderbei
Princeton University
Princeton, NJ 08544

3rd Annual McMaster Optimization Conference:

Theory and Applications

(MOPTA 03)
July 30 August 1, 2003,
McMaster University
Hamilton, Ontario, Canada

The 3rd annual McMaster Optimization
Conference (MOPTA 03) will be held at the
campus of McMaster University. It will be
hosted by the Advanced Optimization Lab at
the Department of Computing and Software
and it is co-sponsored by the Fields Institute and

The conference aims to bring together a
diverse group of people from both discrete and
continuous optimization, working on both
theoretical and applied aspects. We aim to bring
together researchers from both the theoretical
and applied communities who do not usually
get the chance to interact in the framework of a
medium-scale event.
Distinguished researchers will give one-hour
long invited talks on topics of wide interest.
Invited speakers include:

Laurent El Ghaoui,
University of California, ' CA
Lisa K. Fleischer,
Carnegie l' University Pittsburg, PA
Minyue Fu,
University ofNewcastle, NSWAustralia
Masakazu Kojima,
Tokyo Institute of Technology, Tokyo, Japan
George Nemhauser,
Georgia Institute of Technology, Atlanta, GA
Arkadi Nemrovski,
TECHNION Haifa, Israel
Stratos Pistikopoulos,
Imperial ,..,,. London, UK
Margaret H. Wright,
Courant Institute, New York University NY

Each accepted paper will be allotted a 25
minute talk. Authors wishing to speak should

submit an abstract via the conference Web page
in ASCII or LaTex source, to
terlaky@mcmaster.ca by April 30, 2003. Please
use "MOPTA 03" in the email subject line.
Notification of acceptance / Program available:
May 31, 2003. Deadline for early registration:
June 30, 2003.
On behalf of the Organizing Committee
Tamas Terlaky, terlaky@mcmaster.ca (Chair,
McMaster University)
Further information is available at
Prof. Tamas Terlaky,
Canada Research Chair in Optimization
Department of Computing and Software,
Office: ITC 110
McMaster University, 1280 Main Street West
Hamilton, Ontario, Canada, L8S 4L7
Phone: +1-905 525-9140 ext. 27780,
FAX: +1-905 524-0340




We invite OPTIMA readers to submit solutions to the problems to Robert Bosch

forthcoming issue.

Slither Link
Robert A. Bosch
December 5, 2002

1 1 3 2 2 3
1 1 2 2
3 1 2 2

1 2 1 0
0 1 2 1
1 1 1 2 0 2
3 I 2 2
1 2 1 4]

1 1 2 2
1 2 3 1 2 1
* P @ P S

Figure 2

In a slither link puzzle, the goal is to find a cycle that consists of horizontal and vertical line
segments and satisfies the adjacency conditions: for each square s and for every number a, if square s
has the number a in it, then s must be adjacent to precisely a segments of the cycle. See Figure 1
for an example.

3 3 3 3 3
2 1 2 1
2 2 2 2
2 2 3 2 2 3
3 332 332
a a e J 3 a
Puzale SJJIulk)In

Figure 1

Slither link puzzles are available (as freeware and as shareware) for PDAs. Also, Hirofumi
Fujiwara has a very nice slither link website:


Interested readers may enjoy trying to solve the following problems:

1. Devise an integer programming formulation or a constraint programming formulation for
solving slither link puzzles.

2. Solve the slither link puzzle displayed in Figure 2. This puzzle was devised by Hirofumi
Fujiwara. It is the most difficult 10 x 10 puzzle at his site.

Digit Tiles Revisited

The previous installment of M/ lilrPffeH er concerned with arranging digit tiles. Figures 3 and 4
display the maximum value and minimum value arrangements of the tiles on a 5 x 30 board.
(Recall that to compute the value of an arrangement, we add up, over all digits d, a times the
number of white squares touched by the white squares of digit d.

0.2 + 7-6 + + 7+ 3-0 + + + 9-9 +-10O+ G-9 + 2-0 + 1-2 = 352

Figure 3

10 P T-I M A- 6 91



Er I

8-3 + 3-A + l + + 4-i6 + 2-4 + 0 5 + 7-3 + 13 + 9.- = 18t

Figure 4

One way to obtain these solutions is to model the problem as an asymmetric TSP. There are 11
cities. Cities 0 through 9 correspond to the ten digit tiles, and city 10 corresponds to a blank
"dummy" tile. Let value [tl,t2] equal the contribution to total value that comes from placing digit
tile t1 just to the left of digit tile t,. (For example, value[7,1] = (7+1)*2 = 16.) Let x[ tl,t2] equal 1 if
digit tile t1 is placed just to the left of digit tile t2, and 0 if not. Then, in OPL Studio, the model can
be written as follows:

maximize sum(tl in 0..10,
subject to i

all (tl in
all (t2 in
all (t in
all (tl, t

0..10) sum (t2
0..10) sum (t
..10) x[t,t]
in 0 .10: tl<

in 0..10) value[ tl,t2]*x[ tl,t2]

in 0. .10) x[ tl,t2] = 1;
I in 0. .10) x[ tl,t2] = 1;
= 0;
:>t2) x[ tl,t2] + x[ t2,tl] <= 1;

1.7 + 54 + 14
+ 7,8 +-915+8,16+ U-5
+46 +6,124 2-7 =494

Fig rr 5

For the maximization problem, no additional subtour elimination constraints were needed (in
addition to the 2-city ones). For the minimization problem, only one additional subtour elimination
constraint was needed.
Figures 5 and 6 display conjectured-to-be maximum value and conjectured-to-be minimum value
arrangements on an unrestricted board. The Figure 5 arrangement was obtained by solving the
following OPL Studio constraint programming formulation to optimality:

maximize sum(i in 1..3, j in 1..3) hvaluex[ i,j] ,x[i,j+l]]
+ sum(i in 1..2, j in 1..4) walue[ x[i,j] ,x[i+l,j]]
subject toe

ill(t in 0..9) sum(i in 1..3, j in 1..4) (x[i,j]

.) = 1;

The variable x[ i, j] equals the number of the digit tile that is placed in the ith row andjth column
of the board (which has three rows and four columns). Instead of value[ tl, t2] we have
value[ tl, t2] the contribution to total value that comes from placing digit tile t1 just to the left of
digit tile t,, and walue[ tl, t2] the contribution to total value that comes from placing t1 right
above t2. Note that the variables appear as subscripts in the objective function. At first glance, the
constraints look wrong---each one contains two equals signs! But they make sure that each digit tile is
placed exactly once. They do this by counting the total number of times that the x[ i, j] 's equal t,
and then asserting that the total must equal one.
The Figure 6 arrangement was obtained by solving a similar OPL Studio constraint programming
formulation to optimality:

minimize sum(i in 1..6, j in 1..2) value[ x[ i,j] ,x[ i,j+1]]
+ sum(i in 1..5, j in 1..3) walue[ x[i,j],x[i+l,j]]
subject to {
forall(t in 0..9) sum(i in 1..6, j in 1..3) ([ i,j]
sum(i in 1..6) ([ i,2] = 10) = 0;

.) = 1;

The last constraint ensures that the middle column of the arrangement---here, it was assumed that the
board has six rows and three columns---does not contain any copies of tile 10 (the dummy tile).

5-1 + 7-2
+ 0-7 + 3-3
+ 2.4
+ B-I +9-1
+ 4-3
+ 8-1 = 76

Figure 0





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Center for Applied Optimization
371 Weil
PO Box 116595
Gainesville, FL 32611-6595 USA


Jens Clausen
Informatics and Mathematical Modelling,
Technical University of Denmark
Building 305 room 218
DTU, 2800 Lyngby
Tlf: +45 45 25 33 87 (direct)
Fax: +45 45 88 26 73
e-mail: jc@imm.dtu.dk

Robert Bosch
Dept. of Mathematics
Oberlin College
Oberlin, Ohio 44074 USA
e-mail: bobb@cs.oberlin.edu

Alberto Caprara
DEIS Universita di Bologna,
Viale Risorgimento 2,
I 40136 Bologna, Italy
e-mail: acaprara@deis.unibo.it

Donald W. Hearn

Christina Loosli

University of Florida

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