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4 Inu AND SUSinu S. FUNJM MUMIERsMSECOND ASILOMAR WORKSHOP ON PROGRESS IN MATHEMATICAL AFOSR-90-0023

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Nk LAUTHORAS)N Margaret H. Wright

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_____On the theory side, now jomplexoity results re presented for variousinterior linear progra ing methods. New i tenior methods withpolynomial complexity w re described for ce tain quadraticprogramming, convex non inear programming, integer programming andlinear complementarity problems. Strategi s have also been developed

for etanin poynoi 1con -rgnce star ing from an infeasiblejint. For all probl m cae~is, prope ties of Newton's method and

Sof the "central path' (the tr~ jectory of inimizers of the logarithmicbarrier function) play a key/role in mu of the analysis. A frequenttheme is the development of 'large-step methods that do not requirethe iterates to follow the central path closely, yet achieve apolynomial complexity bound. Research on the anticipated complexityof certain interior methods may help to explain why the performance othese methods is substantially superior to their worst-case bounds. (~ )

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UNCLASSIFIED UNCLASSIF:ED UNCLASSIFIED SAR

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Final Report

to theAir Force Office of Scientific Research

and theOffice of Naval Research

Second Asilomar Workshop on Progress in Mathematical ProgrammingFebruary 5-7, 1990

Prepared byMargaret H. Wright

AT&T Bell LaboratoriesMurray Hill, New Jersey 07974

Since 1984, excitement about new approaches to linear programming and other optimizationproblems has continued unabated and shows no signs of fading. As part of this activity, the

SIAM-organized workshop on "Progress in Mathematical Programming", was held at Asilomar,

California, February 5-7, 1990, to bring together researchers working in a variety of fields, mostlyrelated to interior methods.

This workshop was the fourth in a series initiated by Neal Glassman and the Office of Naval

Research in 1986. The first waF theld in 1986 at the Naval Postgraduate School in Monterey,California; the second at Asilomar in 1987; and the third in 1988 at Bowdoin College, Maine. The

most recent workshop received generous financial support from the Air Force Office of ScientificResearch and the Office of Naval Research, and was sponsored by the SIAM Activity Group

on Optimization. The technical organizing committee consisted of Nimrod Megiddo, IBM; Kurt

Anstreicher, Yale University and CORE, Belgium; and Margaret Wright, AT&T Bell Laboratories.Sixty-six researchers from ten countries attended the workshop. An atmosphere of intensity

and friendliness was maintained throughout; participants ate all meals together, and then gatheredevery evening for refreshments and further discussion.

No talks were designated as "invited". Rather, participants who wished to give pre-scheduled

talks were asked to submit abstracts in advance, and a subset of these were selected by the orga-nizing committee. The program was deliberately informal, including talks scheduled in advance,

others organized on site, and substantial time reserved for discussion. The broad range of topicscovered by the talks shows the vitality of the field in both theory and practice. The talks werewidely agreed to achieve a remarkably and uniformly high standard of technical content, interest

and presentation.On the theory side, new complexity results were presented for various interior linear pro-

gramming methods. New interior methods with polynomial complexity were described for certain

quadratic programming, convex nonlinear programming, integer programming and linear comple-

mentarity problems. Strategies have also been developed for retaining polynomial convergencestarting from an infeasible point. For all problem categories, properties of Newton's method and

of the "central path" (the trajectory of minimizers of the logarithmic barrier function) play a key

role in much of the analysis. A frequent theme is the development of "large-step" methods that

do not require the iterates to follow the central path closely, yet achieve a polynomial complexity

bound. Research on the anticipated complexity of certain interior methods may help to explain

why the performance of these methods is substantially superior to their worst-case bounds.

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distribUt ion unlimited.

qO 05 25 133

In a more practical vein, the latest computational achievements were described for severalinterior methods, including a primal-dual algorithm based on applying Newton's method to solvethe nonlinear equations associated with the trajectory of the logarithmic barrier function. Otherspeakers discussed issues of sparse linear algebra and numer'cal analysis that arise in solving thespecial linear systems associated with interior methods, such as ordering strategies for the Choleskyfactorization and preconditioning techniques.

"Classical" topics received attention as well-particularly stochastic programming and piece-wise linear programming. Connections between interior methods and analysis of quadratic and

superlinear rates of convergence have been determined that not only shed light on observed com-putational behavior, but also indicate how to choose algorithmic parameters to achieve the bestconvergence rates.

Two directions in applications were described at the workshop: linear programming-basedtechniques have been used with great success to classify malignant and benign tumors in breasttissue; and the Air Force Military Airlift Command has already applied interior methods to solvelinear programs with several hundred thousand variables, and hopes eventually to reach millions

of variables.Although certain themes recurred througi.,ut the meeting, it is impossible to identify a single

trend in work on interior methods. Researchers are increasingly consolidating classical and newwork into a unified body of knowledge. Results and insight3 are still being produced at a rapid

rate, with no indication that the pace is slowing down. Future work seems certain to apply interiormethods to wider and wider classes of combinatorial and general nonlinear problems.

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List of Speakers

Second Asilomar Workshop on Progress in Mathematical Programming

February 5-7, 1990

Kurt Anstreicher, On the performance of Karmarkar's algorithm over a sequence of iterations

Paul Boggs, Optimal 3-D methods

George Dantzig, Progress in stochastic programming

Dick den Ilertog, A potential reduction method for smooth convex programming

t-urtis Eaves, Arrangements of linear programs with spheres and hemispheres of objective vectors

Robert Fourer, Algorithmic implications of piecewise-linearity in linear programming applications

Robert Freund, Potential reduction algorithms for solving a linear program from an infeasible"warm start"

Clovis Gonzaga, Large dual updates in potential reduction algorithms for linear programming

Dorit Hochbaum, The complexity of integer nonlinear optimization

Florian Jarre, A homotopy method for convex programming

Bahman Kalantari, A generalized Gordon theorem for homogeneous functions and its implications

Leonid Khachian, Polynomial-time algorithms for doubly stochastic diagonal scaling of positivematrices

Masakazu Kojima, A unified approach to interior point algorithms for linear complementarity

problems

Kenneth Kortanek, Computation in the cc :>-ps" state in limit analysis using the LP primal affine

scaling algorithm

Irvin Lustig, Formulating stochastic programs for interior point methods

Olvi Mangasarian, Pattern separation via linear programming: theory and an application to breastcytology diagnosis

Nimrod Megiddo, Parallel complexity of linear programming

Shinji Mizuno, An O(n 3 L) long step path following algorithm for a linear complementarity problem;

a rank-one updating algorithm for linear programming

Sanjay Mehrotra, On the implementation of primal-dual predictor-corrector algorithms

Walter Murray, Will the real primal-dual algorithm please stand up?

Larry Nazareth, Gradient-based algorithms for linear programming derived from quadratic and

conic models

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James Renegar, lowards a very general theory of condition numbers

Kees Roos, Polynomial-time long-step algorithms based on the use of the logarithmic barrierpenalty function

Ben Rosen, Efficient computation of convex hulls in Rd

Michael Saunders, Preconditioning KKT systems (not AD 2 AT)

David Shanno, Current state of a primal-dual interior code

Gy6rgy Sonnevend. On the complexity of following the central path of linear programs by linearextrapolation

Richard Tapia, On the superlinear and quadratic convergence of primal-dual interior point linearprogramming algorithms

Michael Todd, Anticipated behavior of path-following algorithms for linear programming

Kaoru Tone, An O(v./n L) iteration large-step logarithmic barrier function algorithm for linearprogramming

Pravin Vaidya, A new algorithm for minimizing convex functions over convex sets

Robert Vanderbei, Implementation issues for interior-point methods

Yinyu Ye, Anticipated behavior of interior point algorithms for linear programming

Dong Xiao, On the complexity of the projective interior point method

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LIST OF PARTICIPANTS

Second Asilomar Workshopon Progress in Mathematical Programming

February 4-7, 1990

Professor Ilan AdlerDepartment of Industrial Engineering and Operations Research

University of CaliforniaBerkeley, California 94720email: ilan343cgviolet.berkeley.edu

Professor Kurt M. AnstreicherCORE34 Voie du Roman Pays

1348 Louvair-La-NeuveBelgiumemail: MCORE8%BUCLLN11.BITNET CUNYVM.CUNY.EDU

Professor Earl BarnesIndustrial Engineering and Operations Research

Georgia Institute of TechnologyAtlanta, Georgia 30332email: ebarnes@gtriol.gatech.edu

Professor Robert BixbyDepartment of Mathematical Sciences

Rice UniversityP.O. Box 1892Houston, Texas 77251email: bixby@rice.edu

Professor Lenore BlumDepartment of Mathematics

University of CaliforniaBerkeley, California 94720email: lblum@icsi.berkeley.edu

Dr. Paul T. BoggsNational Institute of Standards and TechnologyBuilding 225, Room A-151Gaithersburg, Maryland 20899email: boggs@ptb.cam.nist.gov

Professor Vijaya ChandruSchool of Industrial EngineeringPurdue UniversityWest Lafayette, Indiana 47907

Professor Richard W. CottleDepartment of Operations ResearchStanford UniversityStanford, California 94305-4022email: cot tleOsierra.stanford.edu

Professor George B. DantzigDepartment of Operations ResearchStanford UniversityStanford, California 94305-4022email: hf.gls~forsythe.stanford.edu

Dr. Dick den HertogDepartment of Technical Mathematics and InformaticsDelft University of TechnologyJulianalaan 1322628 BL DelftThe Netherlandsemail: WIOR012%HDETUD1.TUDELFT.NLOCUNYVM.CUNY.EDU

Professor B. Curtis EavesDepartment of Operations ResearchStanford UniversityStanford, California 94305-4022

Professor Robert FourerDepartment of Industrial EngineeringNorthwestern UniversityEvanston, Illinois 60208email: 4er@iems.nwu.edu

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Professor Robert FreundSloan School of ManagementMassachusetts Institute of TechnologyCambridge, Massachusetts 02139email: RFREU ND%SLOAN.BITNET(gmitvma.mit.edu

Dr. David M. GayAT&T Bell Laboratories, Room 2C-463600 Mountain AvenueMurray Hill, New Jersey 07974email: dmg@research.att.com

Professor Philip E. GillDepartment of Mathematics, Code C-012University of California, San DiegoLa Jolla, California 92093email: peg~optimal.ucsd.edu

Dr. Neal GlassmanAir Force Office of Scientific ResearchBuilding 410Boiling Air Force BaseWashington, DC 20332

Professor Jean-Louis GoffinFaculty of ManagementMcGill University1001 Sherbrooke Street WestMontr6al H3A 1G5Qu6bec, Canadaemail: MA56@MCGILLA.BITNET

Professor Donald GoldfarbIndustrial Engineering and Operations ResearchColumbia UniversityNew York, New York 10027email: goldfarb(cunixd.cc.columbia.edu

Professor Allan GoldsteinDepartment of MathematicsUniversity of WashingtonSeattle, Washington 98195email: gold@math.washington.edu

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Professor Clovis C. GonzagaCOPPEFederal University of Rio de JaneiroCx. Postal 6851121945 Rio de Janeiro, RJBrazilemail: clov@lncc.bitnet

Professor Dorit HochbaumSchool of Business Administration350 Barrows HallUniversity of California, BerkeleyBerkeley, California 94720email: dorit @ernie.Berkeley.edu

Dr. Florian JarreInstitut fir Angewandte MathematikUniversitt Wiirzburg, Am HublandD-8700 WiirzburgWest Germanyemail: angm026@vax.rz.uni-wuerzburg.dbp.de

Professor Bahman KalantariDepartment of Computer ScienceRutgers UniversityBusch CampusNew Brunswick, New Jersey 08903email: kalantar(@cs.rutgers.edu

Professor Leonid KhachianDepartment of Computer ScienceRutgers UniversityBusch CampusNew Brunswick, New Jersey 08903

Professor Masakazu KojimaDepartment of Information ScienceTokyo Institute of TechnologyOh-Okayama, Meguro-kuTokyo 152, Japanemail: kojima@titisna.is.titech.ac.jp

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Professor Kenneth 0. KortanekCollege of Business AdministrationUniversity of IowaIowa City. Iowa 52242

Dr. Jeffrey C. LagariasAT&T Bell Laboratories, Room 2C-373600 Mountain AvenueMurray lill, New Jersey 07974email: jcl9research.att.com

Professor Irvin LustigCivil Engineering and Operations ResearchPrinceton UniversityPrinceton, New Jersey 08544email: Irvin.Lustig@BASIE.Princeton.edu

Professor Olvi L. Mangasarian

Computer Sciences DepartmentUniversity of WisconsinMadison, Wisconsin 53706einail: olvi@cs.wisc.edu

Professor Roy MarstenIndustrial Engineering and Operations ResearchGeorgia Institute of TechnologyAtlanta, Georgia 30332email: rmarsten@OgtriO1.gatech.edu

Professor Garth P. McCormickDepartment of Operations Research

George Washington UniversityWashington, DC 20052

Dr. Nimrod MegiddoIBM Almaden Research Center, K53-802

650 Harry RoadSan Jose, California 95120

email: megiddogibm.com

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Professor Sanjay MehrotraDepartment of Industrial EngineeringNorthwestern UniversityEvanston, Illinois 60201email: mehrotra@iems.nwu.edu

Professor Shinji MizunoDepartment of Industrial Engineering and Management

Tokyo Institute of TechnologyOh-Okayama, Meguro-kuTokyo 152, Japanemail: mizunogme.titech.ac.jp

Dr. Clyde MonmaBell Communications Research, Room 2Q-346445 South StreetMorristown, New Jersey 07962email: clydegflash.beflcore.com

Dr. Renato D. C. MonteiroAT&T Bell Laboratories, Room 1F-437Crawfords Corner Road

Hlolmdel, New Jersey 07733email: renu@homxc.att.com

Professor Walter MurrayDepartment of Operations ResearchStanford UniversityStanford, California 94305-4022

email: walter~sol-walter.stanford.edu

Dr. Charles MylanderOffice of Naval Research, Code 111800 North Quincy StreetArlington, Virginia 22217-5000

Professor Larry NazarethDepartment of Pure and Applied MathematicsWashington State UniversityPullman, Washington 99164-2930

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Mr. Toshihito NomaDept. of Information SciencesTokyo Institute of TechnologyNleguro-ku, Tokyo 152Japanemail: nomagtitisna.is.titech.ac.jprosen @umn-cs.cs.umn.edu

Dr. Roman PolyakIBM T.J. Watson Research CenterP.O. Box 218Yorktown Heights, New York 10598-0218

Ms. Dulce PonceleonComputer Science DepartmentStanford UniversityStanford, California 94305email: dulcegna-net.stanford.edu

Professor James RenegarSchool of Operations Research and Industrial Engineering

Upson HallCornell UniversityIthaca, New York 14853-7501email: renegar@orie.cornell.edu

Dr. Mauricio G. C. ResendeAT&T Bl Laboratories, Room 2D-152600 Mountain AvenueMurray Hill, New Jersey 07974email: mgcrg research.att.com

Professor Cornelius RoosDepartment of Technical Mathematics and InformaticsDelft University of TechnologyJulianalaan 1322628 BL DelftThe Netherlandsemail: WIORO12%HDETUD1.TUDELFT.NLOCUNYVM.CUNY.EDU

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Professor J. Ben RosenComputer Science DepartmentUniversity of MinnesotaMinneapolis, Minnesota 55455email: rosencgumn-cs.cs.umn.edu

Professor Uriel G. RothblumRUTCORHill Center for Mathematical SciencesRutgers UniversityNew Brunswick, New Jersey 08903email: rot hblum@cancer.rutgers.edu

Professor Michael A. SaundersDepartment of Operations ResearchStanford UniversityStanford, California 94305-4022email: mike~sol-michael.stanford.edu

Professor David F. ShannoDepartment of Civil Engineering and Operations ResearchPrinceton UniversityPrinceton, New Jersey 08544

Professor C. M. ShettySchool of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332-0205

Professor Ariela SoferDepartment of Operations ResearchGeorge Mason University4400 University DriveFairfax, Virginia 22030ASOFEROGMUVAX.BITNET

Professor Gy6rgy SonnevendInstitut fir Angewandte Mathematik und StatistikUniversitat Wiirzburg, Am HublandD-8700 WiirzburgWest Germanyemail: angm0480vax.rz.uni-wuerzburg.dbp.de

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Dr. Richard E. StoneAT&T Bell Laboratories, Room 3K-328Crawfords Corner RoadHolmdel, New Jersey 07733email: Richard.E.Stone~att.com

Professor Kunio TanabeInstitute of Statistical Mathematics4-6-7 Minamizatu, MinatokuTokyo 106Japan

Professor Richard A. TapiaDepartment of Mathematical SciencesRice UniversityP.O. Box 1892Houston, Texas 77251-1892email: ratcgrice.edu

Professor Tamas TerlakyDepartment of Technical Mathematics and InformaticsDelft University of TechnologyJulianalaan 1322628 BL DelftThe Netherlands

Professor Michael J. ToddSchool of Operations Research and Industrial EngineeringUpson HallCornell UniversityIthaca, New York 14853-7501email: miketoddggvax.cs.cornell.edu

Dr. John A. TomlinIBM Almaden Research Center, K53-802650 Harry RoadSan Jose, California 95120-6099email: tomlincibm.com

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Professor Kaoru ToneGraduate School for Policy ScienceSaitama UniversityUrawa, Saitama 338Japanemail: D53330JPNKUDPC.BITNET

Professor Theodore B. TrafalisSchool of Industrial EngineeringPurdue UniversityWest Lafayette, Indiana 47907email: trafalisgecn.purdue.edu

Professor Kathryn TurnerDepartment of MathematicsUtah State UniversityLogan, Utah 84322-3900email: kturner~cc.usu.edu

Professor Pravin VaidyaDepartment of Computer ScienceUniversity of Illinois at Urbana-Champaign1304 West Springfield AvenueUrbana, Illinois 61801

Dr. Robert VanderbeiAT&T Bell Laboratories, Room 2C-115600 Mountain AvenueMurray Hill, New Jersey 07974email: rvdb( research.att.com

Captain Keith A. WareHQ MAC/XPYRScott Air Force Base, Illinois 62225

Dr. Margaret H. WrightAT&T Bell Laboratories, Room 2C-462600 Mountain AvenueMurray Hill, New Jersey 07974email: mhw0research.att.com

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Mr. Dong XiaoIndustrial Engineering and Operations Research

Columbia UniversityNew York, New York 10027

Professor Yinyu Ye

Department of Management Sciences

University of IowaIowa City, Iowa 52242

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