O P T I M AMathematical Programming Society Newsletter

JANUARY2003

No 69

69Primal Integer Programming 2 mindsharpener 8 gallimaufry 10

O P T I M A 6 9 JANUARY 2003 PAGE 2

Primal integer programming is an area ofdiscrete optimization that -- in a broad sense --includes the theory and algorithms related toaugmenting feasible integer solutions orverifying optimality of feasible integer points.This theory naturally provides an algorithmicscheme that one might regard as a generic formof a primal integer programming algorithm:First detect an integer point in a given domainof discrete points. Then verify whether thispoint is an optimal solution with respect to aspecific objective function, and if not, findanother feasible point in the domain that attainsa better objective function value.

The field of primal integer programming hasemerged in the 1960s beginning with the designof specific algorithms in combinatorialoptimization and Gomory's all integeralgorithms for the general integer programmingcase. While the field stayed active in theory overthe past 40 years, the algorithmic focus has beennearly exclusively on combinatorial optimizationproblems. In fact, augmentation algorithms havebeen designed for and applied to a range ofspecific linear integer programming problems:augmenting path methods for solving maximumflow problems or algorithms for solving the min-cost flow problem via augmentation alongnegative cycles are of this type. Other examplesinclude the greedy algorithm for solving thematroid optimization problem, alternating pathalgorithms for solving the weighted matchingproblem, or methods for optimizing over theintersection of two matroids. In fact, still manyof the most recent combinatorial algorithms areof a primal nature; see for instance thecombinatorial methods for minimizing asubmodular function [12] [19] or thecombinatorial algorithm for the independentpath matching problem [20].

The attempt, however, to solve unstructuredinteger programs with a primal strategy had lostits popularity for at least two decades. Whichdevelopments did lead to this situation? Whatare the obstacles for a primal strategy in thegeneral integer case? What is the state of the art?What are the next steps to be taken? These aresome of the questions that I try to address in thefollowing.

Let us begin by recalling that all thealgorithmic schemes applied to integerprogramming problems without a-prioriknowledge about the structure of the sideconstraints resort to the power of linearprogramming duality.

Dual type algorithms start solving a linearprogramming relaxation of the problem,typically with the dual simplex method. In thecourse of the algorithm one maintains as aninvariant both primal and dual feasibility of thesolution of the relaxation. While the optimalsolution to the relaxation is not integral, onecontinues to add cutting planes to the problemformulation and reoptimizes. Within thisframework, Gomory developed a method ofsystematically generating valid inequalitiesdirectly from a given simplex tableau; see [7, 8].A disadvantage of a dual-type method is,however, that intermediate stopping does notautomatically yield a primal feasible integralsolution.

In contrast to dual methods, primal typealgorithms always preserve integrality of thesolution to the relaxation. They may bedistinguished according to whether thealgorithm preserves dual feasibility or primalfeasibility or none of those. Note that it isimpossible to preserve integrality and both dualand primal feasibility. The only representative ofthe first family of primal type methods is an “allinteger algorithm” that appeared in [9]. We arenot aware of interesting computationalexperiments with this method, however.

An alternative to design a primal typealgorithm is to preserve integrality of thesolution to the relaxation and simultaneouslyprimal feasibility. In order to achieve this, onetypically starts with a primal feasible simplextableau. Iteratively, pivot steps are performed onentries equal to one, only. If such a pivotelement cannot be detected by the standard dualsimplex rule, a special cut row with this propertyis generated and then chosen for pivoting. Whenat any time the reduced cost coefficients of thetableau point into the “right direction”,optimality is proved by the simplex criterion.The cuts one makes use of in this frameworkmay be Gomory’s rounding cuts as in [21], or

problem specific cuts such as facet-defining cutsfor the TSP in [18]. To the best of ourknowledge, a method of this type for IntegerProgramming has first been suggested by Ben-Israel and Charnes [3]. They employed the factthat for any feasible solution of an integerprogram optimality can be proven by solving asubproblem arising from the non-basic columnsof a primal feasible simplex tableau. Variousspecializations and variants of a primal cuttingplane algorithm were later given by Young [21,22] and Glover [6]. For an overview on thissubject we refer to [11, 5]. Nemhauser andWolsey [17] point out that because of poorcomputational experience, this line of researchhas been very inactive, an exception being thework of Padberg and Hong [18], where strongprimal-type cutting planes are used with somesuccess. According to [17] the reason for thepoor computational performance of generalprimal cutting plane algorithms is that “it isnecessary to produce valid inequalities thatcontain the one-dimensional faces (edges) on apath from the initial point to an optimal point”,see also [13]. This is certainly one explanationwhy this line of research became inactive.

Recent theoretical advances regarding primaltype algorithms and the theory of so-called testsets have also renewed the interest in primalcutting plane algorithms. Letchford and Lodi[15] propose a modern primal cutting planealgorithm for 0/1 programs. The mainingredient is the subproblem of “primalseparation”, see also [4, 16]. They givecomputational results on small randomlygenerated 0/1 multidimensional knapsackproblems with up to twenty five variables. Theircomputational experiments reveal that theproposed algorithm is superior to the originalalgorithm of Young. However, the results do notprovide evidence that primal cutting planealgorithms can become superior to dual cuttingplane methods. Only future research andexperiments can clarify this situation. I havedoubts, however, that primal cutting planealgorithms will become an alternative to the“standard method” of today. One additionalproblem for primal cutting plane algorithmsremains: To this day there is no primal cutting

Primal Integer ProgrammingRobert WeismantelFaculty of Mathematics, University of MagdeburgUniversitatsplatz 2, D-39106 Magdeburg, Germanyweismantel@imo.math.unimagdeburg.de

September 17, 2002

O P T I M A 6 9 JANUARY 2003 PAGE 3

[1] Karen Aardal, Robert E. Bixby,Cor A. J. Hurkens, Arjen K.Lenstra, and Job W. Smeltink.Market split and basisreduction: towards a solutionof the Cornuéjols-Dawandeinstances. INFORMS J.Comput., 12(3):192--202,2000.

[2] Egon Balas and Manfred W.Padberg. On the set-coveringproblem: II. An algorithm forset partitioning. OperationsResearch, 23:74--90, 1975.

[3] Adi Ben-Israel and AbrahamCharnes. On some problemsof diophantine programming.Cahiers du Centre d’Etudes deRecherche Opérationnelle,4:215--280, 1962.

[4] Friedrich Eisenbrand, GiovanniRinaldi, and Paolo Ventura.0/1 optimization and 0/1primal separation areequivalent. In Proceedings ofSODA 02, pages 920--926,2002.

[5] Robert S. Garfinkel and GeorgeL. Nemhauser. IntegerProgramming. Wiley, NewYork, 1972.

[6] Fred Glover. A new foundationfor a simplified primal integerprogramming algorithm.Operations Research, 16:727--740, 1968.

[7] Ralph E. Gomory. Outline ofan algorithm for integersolutions to linear programs.Bulletin of the AmericanMathematical Society, 64:275--278, 1958.

[8] Ralph E. Gomory. Analgorithm for the MixedInteger Problem. TechnicalReport RM-2597-PR, TheRAND Corporation, SantaMonica, CA, 1960.

[9] Ralph E. Gomory. An all-integer integer programmingalgorithm. In J. F. Muth andG. L. Thompson, editors,Industrial Scheduling: (Paperspresented at a Conference on“Factory Scheduling” held at theGraduate School of IndustrialAdministration at CarnegieInstitute of Technology, May10--12, 1961), Prentice-Hallinternational series inmanagement, pages 193--206.Prentice Hall, EnglewoodCliffs, N.J., 1963.

[10] Utz-Uwe Haus, MatthiasKöppe, and RobertWeismantel. A primal all-integer algorithm based onirreducible solutions. Toappear in MathematicalProgramming Series B, preprintavailable from URLhttp://www.math.uni-magdeburg.de/~mkoeppe/art/haus-koeppe-weismantel-ibm-theory-rr.ps, 2001.

[11] T. C. Hu. Integer Programmingand Network Flows. Addison –Wesley, Reading, Mass., 1969.

[12] S. Iwata, L. Fleischer, and S.Fujishige. A combinatorialstrongly polynomial algorithmfor minimizing submodularfunctions. Journal of the ACM,48(4):761--777, 2001.

[13] Matthias Köppe. Exact primalalgorithms for general integerand mixed-integer linearprograms. Dissertation,Universität Magdeburg, 2002.

[14] Matthias Köppe. and RobertWeismantel. An algorithm formixed integer optimization.Preprint no. 17, Fakultät fürMathematik, Otto-von-Guericke-UniversitätMagdeburg, 2002.

[15] Adam N. Letchford andAndrea Lodi. Primalseparation algorithms.Technical Report OR/01/5,DEIS, University of Bologna,2001.

[16] Adam N. Letchford andAndrea Lodi. An augment-and-branch-and-cutframework for mixed 0-1programming. In Proceedingsof Aussois V, 2002. to appear.

[17] George L. Nemhauser andLaurence A. Wolsey. Integerand CombinatorialOptimization. Wiley,Chichester, 1988.

[18] Manfred W. Padberg andSaman Hong. On thesymmetric travelling salesmanproblem: a computationalstudy. Math. ProgrammingStud., (12):78--107, 1980.

[19] A. Schrijver. A combinatorialalgorithm minimizingsubmodular functions instrongly polynomial time.Journal of CombinatorialTheory, 8:346--355, 2000.

[20] Bianca Spille and RobertWeismantel. A generalizationof Edmonds matching andmatroid intersectionalgorithms. In Springer LectureNotes in Computer Science2337, pages 9--20, 2002.

[21] Richard D. Young. Asimplified primal (all-integer)integer programmingalgorithm. Operations Research,16(4):750--782, 1968.

[22] Richard D. Young. Theeclectic primal algorithm:Cutting-plane method thataccommodates hybridsubproblem solutiontechniques. Math. Program.,9:294--312, 1975.

References

algorithm for mixed integer programs available,despite the fact that mixed integer programmingmodels are the most important models ofdiscrete optimization.

It is, however, important to remark that evenif primal cutting plane algorithms will not workin practice, this will not imply that a primalaugmentation approach will not work. Analternative to primal cutting plane algorithms isthe Integral Basis Method [10]. It uses acompletely different idea for solving theaugmentation subproblem. After turning afeasible solution into a basic feasible solution ofan appropriate simplex tableau, one manipulatesthe set of non-basic columns until eitheroptimality of the given solution is proved or an“applicable” improving direction, i.e., anaugmentation vector, is detected. In eachmanipulation step, one substitutes one columnby new columns in a way that no feasiblesolutions are lost; this is known as a “proper

reformulation” of a tableau according to [10].The substitution is guided by the concept ofirreducibility of lattice points. In contrast togenerating a basis of a lattice one generatesiteratively a minimal set of lattice points in thepositive orthant that permits a representation ofany feasible solution as a non-negative integercombination of the selected subset. Thisdistinguishes the latter approach from thereformulation method of Aardal et al [1]. Werefer to [10] for the theoretic foundations andcomputational results with the Integral BasisMethod. We also remark that the method can beextended to mixed integer programmingproblems, see [14]. A predecessor of the IntegralBasis Method for the special case of setpartitioning problems was invented by Balas andPadberg about 25 years ago [2].

Computational results with the Integral BasisMethod demonstrate that a couple of iterativesubstitutions of non-basic columns with a “bad”

reduced cost coefficient yield a model in whichthe linear programming relaxation becomessignificantly stronger than the original linearprogramming relaxation. Therefore, the effect ofreformulating a tableau with the help ofirreducible lattice points is not only interestingper se, but rather suggests a primal-dualapproach for integer programming: performIntegral Basis Method steps in addition tocutting planes and branching. Each of the threeoperations, i.e., substitution of variables, cutting,and branching, has an impact on the model andon the other operations. This is -- in my opinion-- an area of research to advance our currentability to solve integer programming models. Ifwe apply in addition to branching and cuttingthe operation of substituting variables, then wecannot lose performance, but gain a lot. This iswhere I see one future direction of research ininteger programming algorithms.

O P T I M A 6 9 JANUARY 2003 PAGE 4

ISMP 2003 is approaching: The 18thInternational Symposium on MathematicalProgramming takes place August 18-12, 2003 atthe Technical University of Denmark,Copenhagen, in cooperation with the Universityof Copenhagen. It is the main scientific event ofthe Mathematical Programming Society held every3 years on behalf of the Society. The Symposiumattracts more than one thousand researchers fromall areas of mathematical programming. At thesymposium homepage www.ismp2003.dk you willfind all information including the list of plenaryand semiplenary speakers:

Susanne Albers, University of FreiburgKurt Anstreicher, University of IowaSanjeev Arora, Princeton UniversityFrancis Clarke, University of Lyon-1William J. Cook, Georgia Institute of

TechnologySiemion Fajtlowicz, University of HoustonAdrian Lewis, Simon Fraser UniversityTom Luo, McMaster UniversityRenato Monteiro, Georgia Institute of

TechnologyStephen M. Robinson, University of WisconsinMikael Rönnqvist, Linköping UniversityRüdiger Schultz, Gerhard-Mercator

University DuisburgPeter W. Shor, AT&T Labs - ResearchMiklós Simonovits, Hungarian Academy of

SciencesDaniel A. Spielman, MITRobin Thomas, Georgia Institute of

TechnologyLaurence A. Wolsey, Université Catholique de

LouvainA proceedings volume sponsored by Springer

Verlag with a contribution from each speaker willbe published as a special issue of MathematicalProgramming and will be distributed to allparticipants at the symposium.

The scientific program will be complementedby an attractive social program: At ISMP 2003you will have the opportunity to taste thetraditional "City Hall" pancakes served at the CityHall and to participate in a cocktail party in theCelebration Hall at the University of Copenhagen.The conference dinner will take place at thefamous Base Camp restaurant in the former navalarea of the city, recently rebuilt into one of themain cultural centers of the city. Additionalactivities will be arranged in the scenicsurroundings of the campus.

The homepage provides information aboutregistration, travel, and accomodations, and offerspossibilities for web-based abstract submission ofcontributed presentations as well as organizedsessions.

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 April30, 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 oneyear membership to MPS is also available. Seethe homepage for details. In addition to thetraditional congress material as booklet ofabstracts and bag, registration includes thevolume of invited lectures, admission to theopening reception, daily bag lunches, andtransportation for the entire week.

We invite you to participate in the event and tosubmit an individual abstract or propose anorganized session on some special topic of generalinterest. Each such session will consist of threetalks. The submission procedure for both abstractsand sessions is described at the homepage: Justclick on the "How to use this homepage" button.

Program committee:Jørgen Tind (chair), University of CopenhagenJens Clausen (cochair), Technical University of

DenmarkRainer Burkard, Technische Universität,

GrazMartin Grötschel, ZIB, BerlinMichael J. Todd, Cornell UniversityStephen J. Wright, University of Wisconsin

Organizing committee:Jens Clausen (chair), Technical University of

DenmarkJørgen Tind (cochair), University of CopenhagenHans Bruun Nielsen, Technical University of

DenmarkDavid Pisinger, University of CopenhagenMartin Zachariasen, University of Copenhagen

Important dates:April 30, 2003: Deadline for early

registration.May 31, 2003: Deadline for submission

of abstracts.August 18 - 22: ISMP 2003

We look forward to welcoming you inCopenhagen.

Jørgen Tind and Jens Clausen

The Mathematical Programming Society, theTechnical University of Denmark, and theUniversity of Copenhagen announce:

ISMP 2003Copenhagen, DenmarkAugust 18 - 22, 2003www.ismp2003.dk

O P T I M A 6 9 JANUARY 2003 PAGE 5

A. W. Tucker PrizeThe next A.W. Tucker Prize will be awarded at the XVIII MathematicalProgramming Symposium in Copenhagen, August 18-22, 2003 for anoutstanding paper authored by a student. The paper can deal with anyarea of mathematical programming. All students, graduate orundergraduate, are eligible. Nominations of students who have not yetreceived the first university degree are especially welcome. The AwardsCommittee will screen the nominations and select at most three finalists.The finalists will be invited, but not required, to give oral presentations ata special session of the symposium. The Awards Committee will select thewinner and present the award prior to the conclusion of the symposium.

The paper may concern any aspect of mathematical programming; it maybe original research, an exposition or survey, a report on computerroutines and computing experiments, or a presentation of a new andingenious application. The paper must be solely authored and completedsince 2000. The paper and the work on which it is based should havebeen undertaken and completed in conjunction with a degree program.

Nominations must be made in writing to the Chairman of the AwardsCommittee by a faculty member at the institution where the nominee wasstudying for a degree when the paper was completed. Moreover,nominators should send one copy each of: the student's paper; a separatesummary of the paper's contributions, written by the nominee, and nomore than two pages in length; and a brief biographical sketch of thenominee to each of the four members of the Tucker Prize Committee:

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

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

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

Past Winners and Finalists:1988: Andrew Goldberg1991: Michel Goemans. Other finalists: Leslie Hall, Mark Hartmann1994: David Williamson. Other finalists: Dick Den Hertog, Jiming Liu1997: David Karger. Other finalists: Jim Geelen, Luis Nunes Vicente2000: Bertrand Guenin. Other finalists: Kamal Jain, Fabian Chudak

Call for Nominations

Prof. Rainer E. Burkard (Chair)Institute of Mathematics BGraz University of TechnologySteyrergasse 30A-8010 GrazAustriaburkard@tugraz.at

Prof. S. Thomas McCormickFaculty of Commerce and BusinessAdministrationUniversity of British ColumbiaVancouver, B.C., Canada V6T 1Z2Tom.McCormick@commerce.ubc.ca

Prof. Jos F. SturmFaculty of EconomicsDepartment of EconometricsTilburg UniversityP.O. Box 901535000 LE TilburgThe Netherlandsj.f.sturm@uvt.nl

Prof. Leslie E. Trotter, Jr.School of Operations Research andIndustrial EngineeringCornell University235 Rhodes HallIthaca, NY 14853ltrotter@cs.cornell.edu

Nominations are invited for the newlyestablished Lagrange Prize in ContinuousOptimization, awarded jointly by theMathematical Programming Society (MPS) andthe Society for Industrial and AppliedMathematics (SIAM). The Prize will bepresented for the first time at the XVIIIInternational Symposium on MathematicalProgramming in August 2003.

To be eligible, works should form the finalpublication of the main result(s) and should bepublished either (a) as an article in a recognizedjournal, or in a comparable, well-referencedvolume intended to publish final publicationsonly; or (b) as a monograph consisting chiefly of

original results rather than previously publishedmaterial. Extended abstracts andprepublications, and articles published injournals, journal sections or proceedings that areintended to publish non-final papers, are noteligible. The work must have been publishedduring the six calendar years preceding the yearof the award meeting.

Judging of works will be based primarily ontheir mathematical quality, significance, andoriginality. Clarity and excellence of theexposition and the value of the work in practicalapplications may be considered as secondaryattributes.

Full details and prize rules are given athttp://www.mathprog.org/prz/lagrange.htm

To nominate a publication for the prize,please send a copy of the paper and a letter ofnomination by February 28, 2003 to thefollowing address:

Stephen WrightComputer Sciences DepartmentUniversity of Wisconsin1210 W. Dayton StreetMadison, WI 53706,USA.

email: swright@cs.wisc.eduElectronic submissions are preferred.

The Lagrange Prize in Continuous Optimization

O P T I M A 6 9 JANUARY 2003 PAGE 6

Nominations are being sought for the Mathematical Programming SocietyBeale-Orchard-Hays Prize for Excellence in Computational MathematicalProgramming.

EligibilityTo be eligible a paper or a book must meet the following requirements:

1. It must be on computational mathematical programming. The topicsto be considered include:

(a) experimental evaluations of one or more mathematicalprogramming algorithms,

(b) the development of quality mathematical programming software(i.e., well-documented code capable of obtaining solutions tosome important class of mathematical programming problems)coupled with documentation of the application of the softwareto this class of problems (note: the award would be presentedfor the paper that describes this work and not for the softwareitself ),

(c) the development of a new computational method that improvesthe state-of-the-art in computer implementations ofmathematical programming algorithms coupled withdocumentation of the experiment that showed theimprovement, or

(d) the development of new methods for empirical testing ofmathematical programming techniques (e.g., development of anew design for computational experiments, identification ofnew performance measures, methods for reducing the cost ofempirical 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 exceptionalmerit comes close to satisfying them.

Frequency and amount of the awardThe prize will be awarded every three years. The 2003 prize of $1,500 anda medal will be presented in August 2003, at the awards session of the18th International Symposium on Mathematical Programming to be heldin Copenhagen, Denmark.

Judgement criteriaNominations will be judged on the following criteria:

1. Magnitude of the contribution to the advancement of computationaland 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.

NominationsNominations must be in writing and include the title(s) of the paper(s) orbook, the author(s), the place and date of publication, and four copies ofthe material. Supporting justification and any supplementary materials arewelcome but not mandatory. The Screening Committee reserves the rightto request further supporting materials from the nominees. The deadlinefor nominations is March 15, 2003.

Nominations should be mailed to:William CookGeorgia Institute of TechnologySchool of Industrial and Systems EngineeringAtlanta, GA 30332-0205USAemail: bico@isye.gatech.edu

Beale-Orchard-Hays Prize

Call for Nominations

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

Visit http://www.coin-or.org/contest.htm to find out how you canwin an IBM ThinkPad. Prizes will be awarded August 2003 at the2003 ISMP in Copenhagen, Denmark.

Robin Lougee-HeimerIBM TJ Watson Research Centerph: 914-945-3032 fax: 914-945-3434robinlh@us.ibm.comhttp://www.coin-or.org

The COIN-OR Open-Source CodingContest: Win an IBM ThinkPad

O P T I M A 6 9 JANUARY 2003 PAGE 7

We invite research articles on algorithms forlarge-scale nonlinear programming for aforthcoming special issue of MathematicalProgramming, Series B. The goal is to assessrelative strengths and weaknesses of variouscomputational approaches to solving large-scalenonlinear programming problems. Examples ofsuch approaches are first-order vs. second-ordermethods, direct vs. indirect solution to linearequations, methods of estimating Lagrangemultipliers, trust-region vs. step-length methods,and active-set vs. interior-point methods. Allparticipants will be asked to provideperformance profiles for their implementationon a common set of AMPL models, which canbe downloaded from http://mathweb.mathsci.

usna.edu/faculty/bensonhy/nonlinear.

Deadline for submission of full papers:July 31, 2003.

We aim at completing a first review of allpapers by December 31, 2003.

Electronic submissions in the form ofunencoded postscript files are encouraged.Instructions for authors submitting toMathematical Programming can be found athttp://link.springer.de/link/service/journals/1010

7/instr.htm. All submissions will be refereedaccording to the usual standards ofMathematical Programming.

Further information about this issue willappear at http://www.princeton.edu/~

rvdb/MpbNlpIssue.html or can be obtained fromthe guest editors for this volume.

Guest Editors:David F. Shannoshanno@rutcor.rutgers.eduRUTCORRutgers University640 Bartholomew RoadPiscataway, NJ 08854

Robert J. Vanderbeirvdb@princeton.eduORFEPrinceton UniversityPrinceton, NJ 08544

Call for Papers

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

The 3rd annual McMaster OptimizationConference (MOPTA 03) will be held at thecampus of McMaster University. It will behosted by the Advanced Optimization Lab atthe Department of Computing and Softwareand it is co-sponsored by the Fields Institute andMITACS.

SCOPEThe conference aims to bring together a

diverse group of people from both discrete andcontinuous optimization, working on boththeoretical and applied aspects. We aim to bringtogether researchers from both the theoreticaland applied communities who do not usuallyget the chance to interact in the framework of amedium-scale event.

Distinguished researchers will give one-hourlong invited talks on topics of wide interest.Invited speakers include:

Laurent El Ghaoui, University of California, Berkeley, CALisa K. Fleischer, Carnegie Mellon University, Pittsburg, PAMinyue Fu, University of Newcastle, NSW, AustraliaMasakazu Kojima, Tokyo Institute of Technology, Tokyo, JapanGeorge Nemhauser, Georgia Institute of Technology, Atlanta, GAArkadi Nemrovski, TECHNION, Haifa, IsraelStratos Pistikopoulos, Imperial College, London, UKMargaret H. Wright, Courant Institute, New York University, NY

CONTRIBUTED TALKSEach accepted paper will be allotted a 25

minute talk. Authors wishing to speak should

submit an abstract via the conference Web pagein ASCII or LaTex source, toterlaky@mcmaster.ca by April 30, 2003. Pleaseuse "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 CommitteeTamás Terlaky, terlaky@mcmaster.ca (Chair,McMaster University)

Further information is available athttp://www.cas.mcmaster.ca/~mopta/

Prof. Tamás Terlaky, Canada Research Chair in OptimizationDepartment of Computing and Software,Office: ITC 110McMaster University, 1280 Main Street WestHamilton, Ontario, Canada, L8S 4L7Phone: +1-905 525-9140 ext. 27780, FAX: +1-905 524-0340

3rd Annual McMaster Optimization Conference:Theory and Applications(MOPTA 03)July 30 - August 1, 2003, McMaster University Hamilton, Ontario, Canadahttp://www.cas.mcmaster.ca/~mopta

O P T I M A 6 9 JANUARY 2003 PAGE 8

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

Figure 1

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

http://www.pro.or.jp/~fuji/java/puzzle/numline/index-eng.html

Problems

Interested readers may enjoy trying to solve the following problems:

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

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

Digit Tiles Revisited

The previous installment of Mindsharpener concerned with arranging digit tiles. Figures 3 and 4display 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, d times thenumber of white squares touched by the white squares of digit d.

mindsharpener We invite OPTIMA readers to submit solutions to the problems to Robert Bosch(bobb@cs.oberlin.edu). The most attractive solutions will be presented in aforthcoming issue.

Slither LinkRobert A. BoschDecember 5, 2002

O P T I M A 6 9 JANUARY 2003 PAGE 9

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

maximize sum(t1 in 0..10, t2 in 0..10) value[t1,t2]*x[t1,t2]

subject to {

forall (t1 in 0..10) sum (t2 in 0..10) x[t1,t2] = 1;

forall (t2 in 0..10) sum (t1 in 0..10) x[t1,t2] = 1;

forall (t in 0..10) x[t,t] = 0;

forall (t1, t2 in 0..10: t1<>t2) x[t1,t2] + x[t2,t1] <= 1;

};

For the maximization problem, no additional subtour elimination constraints were needed (inaddition to the 2-city ones). For the minimization problem, only one additional subtour eliminationconstraint was needed.

Figures 5 and 6 display conjectured-to-be maximum value and conjectured-to-be minimum valuearrangements on an unrestricted board. The Figure 5 arrangement was obtained by solving thefollowing OPL Studio constraint programming formulation to optimality:

maximize sum(i in 1..3, j in 1..3) hvalue[x[i,j],x[i,j+1]]

+ sum(i in 1..2, j in 1..4) vvalue[x[i,j],x[i+1,j]]

subject to{

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

};

The variable x[i,j] equals the number of the digit tile that is placed in the ith row and jth columnof the board (which has three rows and four columns). Instead of value[t1,t2], we havehvalue[t1,t2], the contribution to total value that comes from placing digit tile t1 just to the left ofdigit tile t2, and vvalue[t1,t2], the contribution to total value that comes from placing t1 rightabove t2. Note that the variables appear as subscripts in the objective function. At first glance, theconstraints look wrong---each one contains two equals signs! But they make sure that each digit tile isplaced 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 programmingformulation to optimality:

minimize sum(i in 1..6, j in 1..2) hvalue[x[i,j],x[i,j+1]]

+ sum(i in 1..5, j in 1..3) vvalue[x[i,j],x[i+1,j]]

subject to {

forall(t in 0..9) sum(i in 1..6, j in 1..3) (x[i,j] = t) = 1;

sum(i in 1..6) (x[i,2] = 10) = 0;

};

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

O P T I M A 6 9 JANUARY 2003 PAGE 10

Application for Membership

I wish to enroll as a member of the Society.

My subscription is for my personal use and not for the benefit of any library or institution.

� I will pay my membership dues on receipt of your invoice.

� I wish to pay by credit card (Master/Euro or Visa).

CREDIT CARD NO. EXPIRATION DATE

FAMILY NAME

MAILING ADDRESS

TELEPHONE NO. TELEFAX NO.

EMAIL

SIGNATURE

Mail to:

Mathematical Programming Society3600 University City Sciences CenterPhiladelphia, PA 19104-2688 USA

Cheques or money orders should be madepayable to The Mathematical ProgrammingSociety, Inc. Dues for 2003, includingsubscription to the journal MathematicalProgramming, are US $80.Student applications: Dues are one-half theabove rate. Have a faculty member verify yourstudent status and send application with duesto above address.

Faculty verifying status

Institution

�

gallimaufry

Professor Terry Rockafellar

(University of Washington) has accepted an

part time appointment, starting spring, 2003,

as Adjunct Research Professor with the Center

for Applied Optimization in the ISE

Department at the University of Florida.

Jan Karel Lenstrahas in October 2002 joined ISE, Georgia

Institute of Technology as chaired Full

Professor (John P. Hunter, Jr. Chair).

Karen Aardalhas in October 2002 joined ISE,

Georgia Institute of Technology,

as Associate Professor.

O P T I M AMATHEMATICAL PROGRAMMING SOCIETY

UNIVERSITY OF

FLORIDACenter for Applied Optimization371 WeilPO Box 116595Gainesville, FL 32611-6595 USA

FIRST CLASS MAIL

EDITOR:

Jens ClausenInformatics and Mathematical Modelling,Technical University of DenmarkBuilding 305 room 218DTU, 2800 LyngbyTlf: +45 45 25 33 87 (direct) Fax: +45 45 88 26 73e-mail: jc@imm.dtu.dk

CO-EDITORS:Robert BoschDept. of MathematicsOberlin CollegeOberlin, Ohio 44074 USAe-mail: bobb@cs.oberlin.edu

Alberto CapraraDEIS Universita di Bologna,Viale Risorgimento 2,I - 40136 Bologna, Italye-mail: acaprara@deis.unibo.it

FOUNDING EDITOR:

Donald W. Hearn

DESIGNER:

Christina Loosli

PUBLISHED BY THE

MATHEMATICAL PROGRAMMING SOCIETY &

GATOREngineering® PUBLICATION SERVICES

University of Florida

Journal contents are subject to change by thepublisher.