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GOMATHEMATICALPROGRAMMINGSOCIETYNEWSLETTER

NO52

5 2

EMBER1996

N U M B E R

SEE PAGE TWO

1. Global Optimization Models and Solution Approaches

A large variety of qu antitative d ecision issues, arising in the sciences, engineering an d

economics, can be perceived and m odelled as a constrained op timization pr oblem. Ac-

cording to t his generic description, the best decision often expressed by a real vector

is sought wh ich satisfies all stated feasibility constraints an d m inimizes (or maximizes)

the value of an objective function. Applying standard mathematical programming nota-

tion, we shall consider problems in th e general form

(1) minf(x) subject to x D Rn.

The functionfsymbolizes the objective(s) in th e decision prob lem, and D denotes the

(non-empty ) set of feasible decisions.D is usually defined by a finite nu mber of func-

tions; for the pu rposes of the present discussion, we shall assume that

(2) D ={ x D Rn: l xu; g

j(x) 0 j =1,...,J}.

In (2) l an d u are explicit (finite) boun ds, and gj are given constraint functions. Postulat-

ing now that all fun ctions defined abov e are continu ous, the optimal solution set to

problem (1)-(2) is non -empty.

Most typically, it is assumed th at the d ecision problem m odelled by (1)-(2) has a u nique

locally and , at the sam e time, also globally optimal solution. Un iextremality is often

implied by t he math ematical model structur e (for example, by the strict convexity off,

and the convexity ofD). This paradigm correspond s to the situation in which one, sup-

posed ly, has a su fficiently close initial guess of the feasible region wh ere the op timal

solution x* is located. Hence, the global optim ality of the solution d irectly follows, hav-

ing found the single local optimum offon D. For examp le, linear and convex nonlinear

progr amm ing mod els both, in essence, satisfying the men tioned

Continuous

Global OptimizationSoftware:A Brief Review

AbstractFollowing a concise introduction to multiextremal mathematical programming

problems and global optimization (GO) strategies, a commented list of software

products for analyzing and solving continuous GO problems is presented.

Keywords: Multiextremaloptimization models; continuousglobal optimization; solutionapproaches; software review.

AMS Subject Classification:65K30, 90C05.

conference notes 10

book reviews 12

journals 15

gallimaufry 16

Jnos D. PintrPintr Consulting Services(PCS), and Dalhousie Uni-versity

PCS address: 129Glenforest Drive, Halifax,

NS, Canada B3M 1J2

e-mail: pinter@tuns.cahttp://www.tuns.ca/~pinter/

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un iextremality assump tion in most p ractical cases have been exten-

sively applied in the past decades to formulate and solve an impressive

range of decision problems.

Although very important classes of models naturally belong to theabove category, there is also a broad variety of problems in w hich the

property of uniextremality cannot be simply postulated or verified.

Consider, for instance, the following general problem ty pes:

nonlinear app roximation, including the solution of systems of nonlin-

ear equations and inequalities

mod el fitting to empirical data (calibration, param eterization)

optimized design an d operation of complex black box (oracle) sys-

tems, e.g., in d iverse engineering contexts

configurat ion/ arrangem ent design (e.g., in various data classification,

facility location, resource allocation, or scientific modelling contexts)

Such problems together with numerous other prospective application

areas are d iscussed by Pint r (1996) and in th e extensive list of related

references therein. For furth er ap plications consult, e.g., Pardalos an d

Rosen (1987), Trn an dv

Zilinskas (1989), Floudas and Pardalos (1990,

1992), Grossmann (1996), Bomze, Csendes, Horst and Pardalos (1996),

or sp ecial app lication-related issues of the Journal of Global Op timiza-

tion.

The emerging field of Global Optimization (GO) deals with m athemat i-

cal programm ing pr oblems, in the (possible) presence of multiple local

optim a. Observe that, typically, the nu mber of local (pseud o)solutions is

un know n and it can be quite large. Furtherm ore, the quality of the vari-

ous local and global solutions may d iffer significantly. In the presence

of such structu re often visualized by hilly land scapes correspond ing

to p rojections of the objective function into selected sub spaces (given by

coordinate-pairs of the d ecision variablex) GO problems can be ex-

tremely d ifficult. Hence, most classical nu merical app roaches are, gen-

erally speaking, not d irectly app licable to solve them. For illustra-tion, see Figure 1 wh ich displays a relatively simple compo-

sition of trigonometric functions with imbedded poly-

nom ial argument s, in just tw o variables (denot ed

by x and y).

general mod el form (1)-(2), and note th at the p roblem-classes listed be

low are not n ecessarily distinct; in fact, several of them are h ierarchi-

cally contained by m ore general problem-typ es listed.)

Bilinear and biconvex programming (fis bilinear or biconvex,D isconvex)

Combinator ial optimization (problems w hich have discrete decision

variables infand/ or in gj

can be equivalently reformu lated as GO prob

lems in continu ous variables)

Concave minimization (fis concave,D is convex)

Continuous global optimization (fis continuous,D is compact)

Differential convex (D.C.) optimization (fan d gj

can all be represente

as the d ifference of two correspond ing convex functions)

Fractional programming (fis the ratio of two real functions, and gjar

convex)

Linear and nonlinear complementarity problems (fis the scalar prod

uct of two v ector fun ctions,D is typ ically convex)

Lipschitz optimization (fan d gj

are arbitrary Lipschitz-continuou s

functions) Minimax problems (fis some minim ax objective, the maximu m is co

sidered ov er a discrete set or a convex set,D is convex)

Multilevel optim ization (mod els non-cooperative games, involving h

erarchies of decision-makers, their conflicting criteria are agg regated b

f;D is typically assum ed to be convex)

Multiobjective program ming (e.g., wh en several conflicting linear ob

jectives are to be optimized ov er a polyhed ron)

Multiplicative programming (fis the prod uct of several convex func-

tions, and gj

are convex, or more generally m ultiplicative functions

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Figure 1. A multiextremal function in two variables

Naturally, under su ch circum-

stances, it is essential to use a

prop er global search strategy. Fur-

thermore, instead of exact solu-

tions, most typ ically one has to a c-

cept diverse numerical approxi-

mations to the globally optimal so-

lution (set) and optimu m value.

Following early sporad ic work re-

lated to GO (since the late fifties),

the p resent state-of-the-art is char-

acterized by several dozen mono-

grap hs, a professional journ al and

at least a few thousan d research

articles devoted p rimarily to the

subject. A few illustrative refer-

ences are provided at the end of

this brief review.

The most important GO model-

classes wh ich have been exten-

sively studied includ e the follow-

ing exam ples. (Please recall the

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Network problems (fcan be taken from several function-classes in-

cluding nonconvex ones, and gj

are typ ically linear or convex)

Parametric nonconvex program ming (the feasible regionD a nd / o r

the objectivefmay also depend on a param eter vector)

Quadratic optimization (fis an arbitrary ind efinite quad ratic fun c-

tion; gj

are linear or, in the more g eneral case, can be arbitrary qu adra tic

functions)

Reverse convex programm ing (at least one of the functions gj

ex-

presses a reverse convex constraint)

Separable global optimization (fis an arbitrary non linear in general,

nonconv ex separable function,D is typ ically convex)

Various other nonlinear programming problems, includ ing, e.g.,

nonconv ex stochastic models (in which the d efining fun ctionsf, gj

de-

pend on ran dom factors, possibly in an imp licit, black box man ner)

For detailed descriptions of most of these mod el-types an d th eir connec-

tions consult, e.g., Horst and Pardalos (1995), with nu merou s further

references.

There are several main classes of algorithmic GO app roaches wh ichpossess strong th eoretical convergence properties, and at least in prin-

ciple are straightforward to implement and apply. All such rigorous

GO approaches have an inherent compu tational demand which in-

creases non-polynom ially, as a function of problem-size, even in th e

simplest GO instances. It should be emp hasized at this point th at GO

app roaches are (should b e) typically completed by a traditional local

optim ization phase at least when consid ering also num erical effi-

ciency issues. Global convergence, however, need s to be gu aranteed by

the global-scope algorithm compon ent w hich theoretically should be

used in a comp lete, exhaustive fashion. These remarks in dicate the sig-

nificant d ifficulty of developing robu st and efficient GO software.

Withou t aiming at completeness, several of the most imp ortant GO

strategies are listed below ; for details, consult, for instan ce, the corre-

spon ding w orks from the list of references. (Note that the listing is not

complete, and its items are no t necessarily mutally exclusive; some soft-

ware implementations combine ideas from several approaches.)

Adap tive partition and search strategies (includ ing, e.g., branch-and -

bound algorithms, Bayesian approaches and interval arithmetic based

method s) (Forg , 1988; Ratschek an d Rokn e, 1988; Mockus, 1989;

Neu maier, 1990; Zhigljavsky, 1991; Han sen, 1992; Horst an d Pardalos,

1995; Hor st an d Tuy, 1996; Pintr, 1996; Kearfott, 1996)

Adap tive stochastic search algorithm s (including rand om search,

simulated a nnealing, evolution and genetic algorithm s) (van Laarhoven

and Aarts, 1987; Zhigljavsky, 1991; Horst and Pardalos, 1995;

Michalew icz, 1996; Pintr , 1996)

Enum erative strategies (for solving combinatorial pr oblems, or certain

structured e.g., concave optimization problems) (Forg, 1988; Horstand Pardalos, 1995; Horst and Tuy, 1996)

Globalized local search method s (applying a grid search or rand om

search type global pha se, and a local search algorithm) (Horst and

Pard alos, 1995; Pintr, 1996)

Heu ristic strategies (deflation, tun neling, filled fun ction methods,

app roximate convex global un derestimation, tabu search, etc.) (Horst

and Pardalos, 1995; Pintr, 1996)

Homotopy (parameter continuation) methods and related approaches

(includ ing fixed p oint meth ods, pivoting algorithm s, etc.) (Horst and

Pardalos, 1995)

Passive (simultaneou s) strategies (un iform grid search, pure rand om

search) (Zhig ljavsky, 1991; Horst an d Pa rd alos, 1995; Pintr , 1996)

Successive approximation (relaxation) method s (cutting p lane, more

general cuts, minoran t construction app roaches, certain nested optimi-

zation and decomp osition strategies) (Forg, 1988; Horst a nd Pardalos

1995; Pintr, 1996)

Trajectory methods (differential equat ion mod el based, path-

following search strategies) (Horst and Pardalos, 1995)

In spite of a considerable pro gress related to the rigorous th eoretical

foundations of GO, software development and standardized use lag

behind . The m ain reason for this is, of course, the inheren t nu merical

difficulty of GO, even in the case of simpler specific instances (such a

the ind efinite quad ratic programm ing pro blem). In general, the diffi-

culty of a global optimization p roblem (GOP) can be expected to in -

crease as some exponen tial fun ction of the prob lem dim ension n. Con

sequently, d imensions 100, 50 or even 10 can be considered as large,

depend ing on the GOP typ e investigated. In the remaind er of this pa-

per, an illustrative list of software prod ucts to solve GOPs is reviewed

2. GO Software:Information Sources and Some General Remarks

For the pu rposes of collecting informat ion for this survey, GO softwar

auth ors have been asked (mainly by sendin g e-mail messages, and by

placing electronic ads at several prominen t math ematical program -

ming sites on the WWW) to submit docum entation related to their

work . The information or lack thereof sum marized below is largely

based on the respon ses received. Add itional information has been col

lected from th e Internet, from several GO books, and from the Journal

of Global Optim ization. Note that thou gh in many research publica-

tions reference is mad e to nu merical examp les, or even to sop histicated

specific app lications, only such work is reported below w hich is under

stood to be a general purp ose and legally distributable program system

For obvious reason s, the pr esent surv ey is far from being complete in

any p ossible sense; rather, it is an attempt to p rovide a realistic picture

of the state-of-the-art, supp orted by in stances of existing software. Th

short review is not inten ded to be either comparative or jud gemental

one simp le reason being that the information r eceived from GO soft-

ware d evelopers is used as is, mostly withou t the p ossibility of actual

software testing. By the sam e token, the accuracy of all information

cannot be guarant eed either. Further research in this direction

includ ing the p reparation of a more comprehensive and d etailed surve

is currently in p rogress.

The software list provided in the next section is simply alph abetical,

withou t categorization. For a more un iform presentation style, abbre-

viations are associated with all software prod ucts listed, even wh ensuch nam es were not given in the docum entation available for this sur

vey (existing names were not chang ed, of course). The descriptions ar

almost formu la-free and extremely concise du e to spa ce restrictions.

For the latter reason, we decided not t o include imp ortant classes of

more specific GO appr oaches and related meth odology. In particular

as reflected by the title pure or mixed integer prog ramm ing and mor

general combinatorial optimization algorithms are not discussed here.

Furtherm ore, althou gh m ost of the available top-of-the-line continu ous

nonlinear (convex) optim ization software can be applied with good

taste and som e luck to analyze GOPs, even the most p rominen t such

systems are excluded from this review. Again, a more detailed sur vey

is planned, app ropriately d iscussing also the program system types

mentioned.

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The hardw are and softwar e platform of the systems reviewed is also

show n wh en such information is available. In order to assist in obtain-

ing ad ditional information, contact person(s), their e-mail add resses, ftp

and/ or WWW sites are listed, whenever known to me. (For brevity,

only a few such p ointers are provid ed in each case.)

The reader is assumed to have at least some basic familiarity with the

GO app roaches mentioned ; for related d iscussions, please consu lt the

references.

3. Short Software Descriptions

BB A GO Algorithm for General Nonconvex Problems

An im plemen tation of a Branch-and-Boun d (B&B) algorithm w hich is

based o n th e difference of convex functions (D.C.) transformation.

Non convexities are identified and categorized as of either special or ge-

neric structure. Special nonconv ex (such as bilinear or u nivariate con-

cave) terms are convex lower bounded using customized bound ingfunctions. For generic nonconvex terms, convex lower bound ing func-

tions are derived by utilizing the parameter (specified by the u ser or

derived based on theory). BB solves general unconstrain ed an d con-

strained p roblems; it requires MINOS and/ or NPSOL for the solution of

linear or convex optimization subp roblems. (Langu ages: C and Fortran.)

Contact: I.P. And roulakis , C.A.

Floudas , C.D. Maranas

, http:/ / titan.princeton.edu / .

ANNEAL Simulated Annealing

ANN EAL is based on th e core SA app roach, includ ing several possibili-

ties for parameter adjustment and a deterministic solution refinement

ph ase. It has been app lied to predict comp lex crystal structur es. Work-

station imp lementation. Contact: W. Bollweg , H. Maurer , H. Kroll

.

ASA CalTech Adapt ive Simulated Annealing

ASA was developed to find the global optimum of a continuous n on-

convex function over a m ultidim ensional interval (box). This algorithm

perm its an annealing schedule for temp erature decreasing exponen -

tially in ann ealing time. The introd uction of re-ann ealing also permits

adap tation to changing sensitivities in the param eter-space. Some other

adap tive options in ASA includ e self-optimize (to find op timal starting

conditions) and qu enching (to method ically find faster performan ce

that m ight be u seful for large param eter-spaces). (Language: C.) Con-

tact: L. Ingber , http:/ / ww w.ingber.com/

#ASA-CODE.

B&BA Family of B&B Algorithms

This obvious acronym (by the p resent au thor) attempts to sum marize

several B&B type algorith ms d eveloped to solve certain stru ctured GO P

classes. These includ e (among others) indefinite quad ratic, quasiconvex-

concave, and general Lipschitz problems. Workstation imp lementa-

tions . (Langu age: C.) Contact: R. H orst , M.

Nast , N . Thoai .

BARON Branch-And-Reduce Optimization Navigator

Combines interval analysis and du ality with enh anced B&B concepts.

The BARON mod ules can han dle structured nonconvex problems u p to

thou sand s of constraints and variables. The library of specialized mod -

ules includes solvers for nu merou s specific GOP-classes. (For other,

more general problems, underestimation routines need to be p rovided

by the u ser.) All modu les can solve also such problems in w hich some

or all of the variables are restricted to integer values. The specialized

mod ules use OSL or MINOS to solve interim subp roblems. Worksta-

tions, UNIX type op erating systems. (Languages: Fortran and GAMS.)

Contact: N .V. Sahinidis, http:/ / archimedes.me.uiuc.edu/ sigma/

baron.html, ftp:/ / aristotle.uiuc.edu .

BGOBayesian Global Opt imizat ion

This program system includ es four versions of Bayesian search, cluster

ing, uniform determ inistic grid, and p ure Monte Carlo search. Boun d

constraints and m ore general constraints can be hand led. Interactive

DOS and UNIX versions are available. (Languag es: Fortran and C.)

Contact: J. Mockus , L. Mockus

.

cGOP Global Opt imizat ion Program

Solves structu red GOPs w hich have an objective fun ction of the form

a

T

x+b

T

y+x

T

Ay+f1(x)+f2(y) with convexf1,f2, and linear constraints. Re-quires the p resence of the commercial codes MINOS and / or CPLEX to

solve linear, mixed-integer linear and convex subp roblems. cGOP has

been used to solve problems involving several hundred variables and

constraints. Versions are available for work stations. (Langu age: C.)

Contact: V. Viswesw aran , C.A. Floudas

, http:/ / titan.princeton.edu / .

CGU Convex Global Underestimator

This approa ch is designed to generate efficient ap proximations to th e

global minimu m of a mu ltiextremal function, by fitting a convex func-

tion to th e set of all know n (calculated ) local minima. This heuristicall

attractive strategy requires only th e sequential solution of au xiliary LP

and som e rather elementar y calculations. CGU has been applied to ca

culate molecular structure pred ictions, up to several dozen var iables.

Implemented on parallel workstations and supercomputers. Contact:

K.A. Dill, A.T. Phillips , J.B. Rosen

.

CRS Controlled Random Search

This is a recently developed v ariant of a pop ular class of rand om searc

based methods which can be applied un der very m ild analytical condi

tions imposed on the GOP. Several other related stochastic search

methods h ave also been d eveloped by this group . Workstation imple-

men tation s. Conta ct: M.M. Ali, A. Trn , S.

Viitanen .

CURVI Bound-Constrained Global O ptimiz ation

Windward Technologies (WTI) develops advanced numerical and visu

alization software, for solving constrained and unconstrained nonlinea

optim ization problems. On e of their solvers, CURVI is aimed at solvin

bound -constrained nonlinear programs w hich h ave a complicated

possibly m ultiextremal objective fun ction. (Languag e: Fortran.) Con-

tact: T. Aird , http:/ / users.aol.com/ WTI/ .

DE Differentia l Evolut ion Genetic Algorithm for Bound Con-strained GO

DE won third place at the 1st International Contest on Evolutionary

Comp utation on a real-valued function test set. It was the best genetic

algorithm ap proach (the first two places of the contest were w on by

non-GA algorithm s). (Langu ages: Matlab and C.) Contact: R. Storn

, http :/ / http .icsi.berkeley.edu / ~storn/

code.html.

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ESA Edge Searching Algorit hm

An imp lementation of an edge search algorithm for finding th e global

solution of linear reverse convex p rogram s. ESA is based on an efficient

search technique and the use of fathoming criteria on the edges of thepolytop e representing the linear constraints. In addition, the method

incorporates several heuristics, including a cutt ing plane techniqu e

wh ich impro ves the overall performance. Implement ed for several

UNIX platforms; the TPG Test Problem Generator is also a vailable.

(Langu age: Fortran.) Contact: K. Moshirvaziri

, .

GA Geneti c Algorithms

Genetic algorithms as a rule can be applied to GOPs u nd er mild

structur al requirements. Both general and sp ecific information related

to this p opu lar solver class is available from the following sources: A

Comm ented List of Genetic Algorithm Cod es, ftp:/ /

ftp.germany.eu.net/ pub/ research/ softcomp/ ec/ faq/ www / q20_1.htm

GA Archive, http:/ / ww w.aic.nrl.navy.mil/ galist/ src/ . Only a few il-

lustrative examples are listed in the pr esent review.

GAS Geneti c Algorithm

Unconstrained and bound-constrained versions are available. For DOS

and UNIX operating systems. (Language: C++.) Contact: M. Jelasity

, J. Dombi , ftp:/ /

ftp.jate.u-szeged.hu/ pub/ math/ optimization/ GAS/ .

GAucsd Geneti c Algorithm

Developed an d m aintained at t he University of California, San Diego.

GAucsd was w ritten in C u nder Unix but should be easy to port to

other platforms. The package is accompan ied by brief information and

a Users Guide. (Langu age: C.) Contact: nici@ucsd .edu , GAucsd-

request@cs.ucsd.edu, ftp:/ / cs.ucsd.edu/ pub/ GAucsd/ .

GENERATOR Geneti c Algorithm SolverThis method is aimed at solving a variety of (combinator ial and con-

tinuou s multiextremal) scientific and engineering op timization prob-

lems. It is designed to interact with Excel which serves as a user inter-

face. (Platform: Excel.) Contact: New Ligh t Ind ust ries

, http:/ / ww w.iea.com/ ~nli/ .

GC Global Continuation

GC is a continuation ap proach to GO app lying global smoothing in or-

der to d erive a simpler app roximation to the original objective function.

GC is app lied by the author s to distance geometry problems, in the con-

text of molecular chemistry mod elling. IBM SP parallel system imple-

men tation . Contact: J.J.Mor , Z. Wu.

GENOCOP III Genetic A lgorithm for Constrained Problems

Solves general GOPs, in the p resence of add itional constraints and

boun ds (using quad ratic pena lty terms). System param eters, dom ains,

and linear inequalities are inp ut via a d ata file. The objective function

and any non linear constraints are to be given in approp riate C files.

(Langu age: C.) Contact: Z. Michalewicz, http :/ / ww w.coe.un cc.edu/

~zbyszek/ gcreadme.html, ftp:/ / ftp.uncc.edu/ coe/ evol/

genocopIII.tar.Z.

GEODES Minimum-Length Geodesic Computing

Approximating a m inimum-length geodesic on a multidimensional

man ifold, GEODES is differential geometry software. How ever, it has

poten tial also in th e GO context. GEODES includes examp le manifolds

and metrics; it is imp lemented in Elements (a matrix and function ori-

ented scientific modelling environmen t) to comp ute and visualize geo

desics on 2D surfaces plotted in 3-space. Portable to various hard ware

platforms. (Langua ges: C, C++.) Contact: W.L. Anderson

, http:/ / ww w.netcom.com/ ~elements/ ,

http:/ / www.netlib.org/ ode/ geodesic/ .

GLO Global and Local Optimiz er

GLO is a modu lar optimization system developed for black box prob

lems in w hich objective function calculations may ta ke a long time. Its

method ology is based on the coup ling of global (genetic) and local

(variable metric) nonlinear op timization software with scientific app li-

cations software. It has been applied to automated en gineering design

Besides the mod ular op timization control system, GLO also has a

grap hical user interface and includ es a pre-processor. Contact: M.J.

Murph y, http:/ / ww w.llnl.gov/ glo/ 09glo.html, M. Brosius

.

GLOBAL Multist art w ith Stochastic Clustering

GLOBAL can be used for th e solution of the general bou nd -constrainedGOP wh ich has a (measurable) real objective fun ction. The algorithm

a derivative-free imp lementation of the clustering stochastic mu ltistart

method of Boender et al., supplemented with a quasi-Newton local

search routine and with a robust rand om local search method . Avail-

able for UNIX machines, IBM-comp atible mainframes an d PCs. (Lan-

guag es: Fortran and C.) Contact: T. Csendes , http:/ / www .inf.u-szeged.hu/ ~csendes/ , ftp:/ / ftp.jate.u

szeged.hu/ pub/ math/ optimization/ index.html.

GLOBALIZER An Educational Program Syst em for Global O ptmization

Serves for solving un ivariate GOPs. After stating th e problem, the u ser

can choose amon g variou s (rand om search, B&B based, or Bayesian pa

tition based) solver techniques. The softwar e has interactive tutoring

capabilities, prov ides textual and grap hical information. Works on PC

un der MS-DOS. Contact: R.G. Strongin ,

V.P. Gergel, A.V. Tropichev .

GLOPT Constrained Global Opt imizat ion

Solves GOPs w ith a block-separable objective fun ction subject to boun

constraints and block-separable constraints; it find s a nearly globally

optim al point that is near to a tru e local minimizer. GLOPT uses a B&

technique to split the problem recursively into subp roblems that are ei

ther eliminated or redu ced in their size. It includes a new red uction

technique for boxes and n ew w ays for generating feasible points of con

strained nonlinear progra ms. The current implemen tation of GLOPT

uses neither derivatives nor simultaneous information about several

constraints. (Langu age: Fortran .) Contact: A. Neu maier

, S. Dallw ig and H. Schichl.GOPP Global Opt imizat ion of Polynomial Problems usingGrbner Bases

The (local) optim ality cond itions to polynom ial optimization pr oblems

lead to polyn omial equations, und er inequality constraints. App lying

recent Grbner basis techniques, this approach is aimed at finding all

solutions to such system s, hence also finding global optima. (Language

Map le.) Cont act: K. Ha gglof , P.O. Lind berg

, L. Svensson , http:/ /

www.optsyst.math.kth.se.

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GOT Global Optimiz ation Toolbox

GOT combines rand om search and local (convex) optimization. DOS

and HP UX versions are available. (Langu age: Fortran.) Contact: A.V.

Kun tsevich .GSA Generalized Simulated Annealing

GSA is based on the generalized en tropy by Tsallis. The algorithm

obeys detailed balance conditions and , at low temp eratures, it redu ces

to steepest descent. (Note th at the mem bers of the same research grou p

have been involved in th e developm ent of several SA type algorithms.)

Cont act: J.E. Strau b , P.Ama ra

, J. Ma .

IHRImproving Hit- and-Run

IHR is a rand om search based GO algorithm th at can be used to solve

both continuou s and discrete optimization prob lems. IHR generates

random points in the search d omain by choosing a random direction

and selecting a point in that d irection. Versions have been imple-

mented , using different distribu tions for the random d irection, as wellas several ways to r and omly select points along th e search line. The al-

gorithm can also hand le inequ ality constraints an d a h ierarchy of objec-

tive functions. IHR has been u sed to solve GOPs in various d isciplines

such as in en gineering d esign. Contact: Z. Zabinsky

, ftp:/ / ftp.bart.ieng.washington.edu .

IMINBIS Interval Arithmeti c Based GO

This method app lies interval arithmetic techniques to isolate the station-

ary p oints of th e objective function. Next, a to pological characterization

is used to separate minima from m axima and saddle points, followed by

local minimization (sub)searches to select the global solu tion. The

method has been app lied also to noisy problems. Workstation and PC

implem entations, extensive related research. (Langu age: Fortran.) Con-

tact: M.N. Vrahatis , D.G.

Sotiropou los , E.C. Triantaph yllou.

INTBIS Global Solver for Polynomial Systems of Equations

Finds all solutions of polynomial systems of equations, with rigorou sly

guar anteed r esults. The software package INTBIS is ACM-TOMS Algo-

rithm 681; it is available throug h N ETLIB. Distributed w ith the package

are four source code files, samp le input an d ou tpu t files, and a brief

docum entation file. The source files consist of the following : interval

arithmetic, stack managem ent, core INTBIS routin es, and m achine con-

stan ts. (Langu age: Fortran .) Cont act: R.B. Kearfott ,

http:/ / interval.usl.edu/ kearfott.html, ftp:/ / interval.usl.edu/ pub/

interval_math/ intbis/ .

INTOPT_90 Verified (Interval) Global Optimization

Serves to the verified solution of nonlinear systems of equations and un -constrained and bound-and-equality-constrained global optimization.

Based on exhaustive search, driven by a local optimizer, epsilon-infla-

tion, interval Newton m ethod s, and interval exclusion prin ciples; uses

autom atic differentiation. Test results with hu nd reds of test examp les.

The un derlying interv al arithmetic package (ACM TOMS Algorithm

737) is also distribu ted. Workstation and PC implemen tations. (Lan-

guag e: Fortran.) Contact: R.B. Kearfott , http :/ /

interval.usl.edu / kearfott.html, ftp:/ / interval.usl.edu / pub/

interval_math/ intbis/ .

INTGLO, INTGLOB Integral Global Opt imizat ion

These methods solve un constrained and constrained, as well as discret

GOPs by the integral method . They also includ e a discontinu ous pen-

alty fun ction approach for constrained problem s. Problems up to onehu nd red variables have been solved. A set of test problems is also

available, including box or u nconstrained , constrained , concave mini-

mization, discrete variable programs and m ulticriteria program s. For

IBM PCs. (Langu age: Fortra n.) Contact: Q. Zheng

, D. Zhuang .

ISA Inductive Search Algorithm

ISA wo n first place at the 1st International Contest in Evolutionar y

Comp utation o n a real-valued function test-suite. (Languag e: C++.)

Contact: G. Bilchev, information available at http :/ /

solon.cma.univie.ac.at/ ~neum/ glopt/ test_results.html#bilchev.

LGO Continuous and Lipschitz Opt imizat ion

Solves bound -constrained and more general GOPs und er mild struc-

tural requ irements; it can be ap plied also to black box problem s. LGOintegrates several global (adaptive partition and random search based)

and local (derivative-free conjugat e directions type) strategies: these

can be activated in interactive or automatic execution mod es. The PC

version has a menu interface to assist the application developmen t pro

cess, includes a concise information / tutor ing session, and h as visual-

ization capabilities. Available also for w orkstations. LGO has been ap-

plied to p roblems with up to 100 variables (can be configured to encom

pass larger sizes). Accomp anied by a Users Guide and samp le prob-

lems. (Langu age: Fortran .) Cont act: J.D. Pintr ,

http:/ / www.tuns.ca/ ~pinter/ .

LOPS Lipschitz Opt imizat ion Program System

In all appro aches listed below , the objective fun ction is defined over n-

intervals. The Lipschitz-continuity of f or f is also assum ed. Problem-

classes and correspon ding ava ilable versions include: one-dim ensiona

GOPs (sequen tial methods w ith local tun ing, PC version (Language:

C++) one-dimen sional GOPs, parallel solver imp lementations (Lan-

guag e: Alliant FX/ 80, parallel Fortran) mu lti-dim ensional GOPs:

sequential and parallel algorithms using Peano curves (Language:

Alliant FX/ 80, pa rallel Fortran ) Contact: Y.D. Sergeyev

.

MAGESTIC Data Fitt ing by Global Opt imizat ion

Automatic global optimization based on a fast modified Gauss-Newto

app roach combined w ith Monte Carlo search. MAGESTIC handles cal

bration m odel variants (e.g., parameter and error masks for restricted

sub-fitting, implicit equ ation fitting w ithout solving, etc.). Suitable for

use also with Lagrange mu ltipliers for constrained optimization. Uses

Excel as an interface (und er Window s) and for generating grap hics.(Platform: Excel.) Contact: Logix Consu lting ,

http:/ / www .lgx.com/ magestic.html.

MULTISTART Clustering Algorithm

This widely used app roach is based on random search or some other

initial sampling in th e feasible set combined with clustering a nd loca

optim ization launched from the most promising point(s). Imple-

mented on SUN workstations. Several interesting app lications in

combination w ith simulation m odels are related to the analysis of oil

resources. (Language: Fortran.) Contact: S. Buitrago

.

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NETSPEAK General Network Opt imizat ion

This is an algebraic mod elling langu age used to specify, solve, and ana-

lyze general linear, but also possibly nonconvex minimu m cost net-

work flow problems. A wide variety of netw ork and netw ork-relatedtopologies (pure networks, networks with side-constraints and / or side

variables, generalized n etworks) can be mo delled u sing N ETSPEAK.

The language is being developed as a Wind ows ap plication; it features

flexible I/ O, robust program control, and intu itive comman ds. Contact:

B.W. Lamar .

PA Packet Annealing

In PA, the Gibbs d istribution of the ob jective fun ction is deterministi-

cally annealed by tracing the evolu tion of a mu ltiple Gaussian p acket

app roximation. The approach has been applied to analyze complex mo-

lecular conformation models. IBM PC implem entation. Contact: D.

Shallowa y , B.W. Chu rch , M.

Oresic .

PROFIL Interval Branch and Bound MethodBound constrained interval global optimization, with rigorously guar-

anteed resu lts. PROFIL is based on BIAS (Basic Interv al Arithm etic

Subroutines) which pro vides an interface for interval operations. For

PC and a nu mber of UN IX systems. (Langu age: C.) Contact: C. Jansson,

O. Knueppel , http:/ / ww w.ti3.tu-

harburg.de/ Software/ PROFILEnglisch.html, ftp:/ / ti3sun.ti3.tu-

harburg.de/ pub/ profil/ unix/ profopt.tar.Z, ftp:/ / ti3sun.ti3.tu-

harburg.de/ pub/ profil/ pc/ profopt.tgz.

PVGO Parallel Verified Global O ptimiz ation

PVGO is a new parallel method for interval global optim ization. Imp le-

mented on a Con nection Machine CM5. (Language: Pascal-XSC.) Con-

tact: S. Berner .

RSBB Reduced Space Branch-and-Bound MethodRSBB applies variable domain reductions and an underestimating mod-

ule. Two versions run u nd er Unix; one of these algorithms is described

in Chap ters 1 and 2 of Grossmann (1996). (Language: both versions in

C++, they use external Fortran p rocedures.) Contact: T. Epperly,

t.epperly@ic.ac.uk, h ttp:/ / ww w.ps.ic.ac.uk/ ~epperly/ index.html.

SA Simulated Annealing

In add ition to the algorithm itself, this includes on-line interactive dem -

onstration, and additional information on C++ classes , random number

generation, Monte Carlo method s and Forth. A Nelder-Mead simplex

method implem entation is also available. (Langu ages: C, C++, Ada

and Fort h.) Contact: E. Carter , Taygeta Scientific Inc.

, http:/ / ww w.taygeta.com/ annealing/

simanneal.html.SAT Global Opt imizat ion for Satisfiability Problems

Boolean satisfiability (SAT) problems can be d irectly tran sformed into

un constrained GOPs. These, in tu rn, can be solved by sp ecifically tai-

lored solvers. Workstation im plementa tions. Contact: J. Gu

.

SIGMA Stochastic Integration Global Minimization Algorithm

The software package SIGMA is Algorithm 667; app eared in ACM-

TOMS 14 (1988) 366-380. Includ es also th e code of sever al test GOP s.

(Langu age: Fortr an.) Contact: F. Aluffi-Pentin i, V. Parisi and F. Zirilli

, http:/ / ww w.netlib.no/ netlib/ toms/ 667.

SIMANN Simulated Annealing

This program implements the continuous simulated annealing global

optim ization algorithm d escribed in Corana et al., ACM TOMS 13

(1987) No. 3, 262-280. Algorithm mod ifications and man y d etails on it

use can be fou nd in Goffe et al., J. of Econom etrics 60 (1994) No. 1-2, 65

100. (Langu age: Fortra n.) Con tact: B. Goffe,

, http:/ / ww w.netlib.no/ netlib/ opt/

simann.f.

SOLVEX Solver for Nonlinear Optimization Problems

For solving constrained an d u nconstrained n onlinear, multiobjective

and GOPs. SOLVEX algorithm libraries include th e method s listed be-

low: unconstrained m inimization: Hooke-Jeeves direct search, conju-

gate grad ient method , Shor R-algorithm , Powell-Brent meth od; genera

nonlinear programming problem: penalty functions, Lagrange function

method, parameterization method; global optimization: sequential set

covering techniqu e, simulated an nealing, clustering algorithm;

mu lticriteria optim ization: convolution method s, including goal pro-

gramm ing, direct Pareto app roximation. Interactive use enhanced by a

built-in problem ed itor and graph ics capabilities. Contact: M.A.Potapov .

TORUS Stochastic Algorithm for Global Minimization w ithConstraints

It is a Monte Carlo algorithm, combined w ith annealing search pr in-

ciples. The softwar e package TORUS is Algorithm 774; appeared in

ACM-TOMS 21 (1995) 194-213. Includ es also th e code of sever al test

GOPs. Contact: F. M. Rabinowitz, http:/ / ww w.netlib.no/ netlib/ toms

744.

TRUST Terminal Repeller Unconstrained Subenergy Tunneling

This method formu lates the GOP as the solution of a deterministic dy-

namical system incorporating terminal repellers and a subenergy tun-

neling function. Benchmark tests comparing th is method to other glo-

bal optimization procedures are presented, with favourable results.

The TRUST formulation leads to a simple stopp ing criterion. In add i-

tion, the structure of the equation s enables an implemen tation of the a

gorithm in analog VLSI hard ware (in the sp irit of artificial neural n et-

work s) for further speed enhan cements. Contact: B.C. Cetin, J. Barhen

and W. Bur dick; TRUST is described in JOTA 77 (1993) No. 1.

TVC Toolbox for Verified Computing

Can be app lied to the rigorous solution of nonlinear systems of equa-

tions and to general unconstrained and bound-constrained GOPs. TVC

is based on interval B&B and interval New ton meth ods; it also has au-

tomatic differentiation capabilities. The toolbox can be u sed on PCs,

work stations an d parallel compu ters. (Langu ages: PASCAL-XSC, C++

Test problems have been solved up to one hun dred v ariables. A drive

program an d hund reds of test examples are available from the au thor.Contact: D. Ratz , http :/ /

ww w.uni-karlsruhe.de/ ~iam, http:/ / ourworld.compu serve.com/

homepages/ num erik_software.

UFO Universal Functional O ptimiz ation

Interactive modular system for solving problems and for algorithm de

velopm ent. Several types of GO method s rand om search, continu a-

tion, clustering, and rand om p lus local search can be ap plied. For PC

(Langu age: For tran .) Cont act: L. Luksan , M.

Tuma, M. Siska, J. Vlcek and N. Ramesova, ftp: uivt.cas.cz/ pu b/

msdos/ ufo.

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UNICALC Interval Branch and Bound Algorithm

UNICALC serves for bound -constrained GO; accepts also inequality

and / or equality constraints and decision variables. Contact: A.

Semenov, information available at ftp:/ / ftp.iis.nsk.su/ pub/ ai/ unicalc.VerGO Verified Global Opt imizat ion

VerGO is designed for rigorous bound (and approximate general con-

strained) GO of a twice continuou sly differentiable objective function.

VerGO features include interv al arithmetic, automa tic differentiation,

non-convexity test, mon otonicity test, and local optimization. Tested on

problem s up to over 30 variables. DOS, OS/ 2, Linux and w orkstation

versions. (Langu age: C++.) Contact: R. v an Iwaar den

, http:/ / ww w.cs.hope.edu/ ~rvaniwaa/

VerGO/ VerGO.html.

VTT Interval Arithmeti c Research

The goals of the Interval Arith metic, Constraint Satisfaction and Prob-

ability Project are summ arized as follows: developm ent of portable C++

libraries for interval p rogram ming task s; integration o f the libraries toMicrosoft Excel; application in financial planning softw are pr odu cts

(Platform s: C++, Excel.) Contact: S. De Pascale

, http:/ / ww w.vtt.fi/ tte/ .

GOCONTINUEDHansen , E.R. (1992) Global

Optimization Usin g Interval

Analysis. Marcel D ekker, New

York.

Horst, R. and Tuy, H . (1996) Glob al

Optimization - D eterministic

Approaches. Springer, Berlin /

Heidelberg / New York. (3rd Edn.)

Horst, R. and Pardalos , P.M., eds.

(1995) Handbook of G lobal

Optimization. Kluwer Academic

Publishers, Dordrecht / Boston /

London.

Journal of Global Optimization

(published since 1991, by Kluwer

Academic Publis hers).

Kearfott, R.B. (1996) Rigorous

Global Search: ContinuousProblems. Kluwer Academic

Publishers, Dordrecht / Boston /

London.

Michalewicz, Z. (1996) Genetic

Algorithms + Data Structures =

Evolution Programs. Springer,

Berlin / Heidelberg / N ew York.

(3rd Edn.)

Mockus , J. (1989) Bayesian

Approach to Global Op timization.

Kluwer Academic Publishers,

Dordrecht / Boston / London.

Ne umaier, A. (1990) Interval

Methods for Systems of Equations.

Cambridge University Press,

Cambridge.

4. Acknowledgements

The software review presented hereis based to a significant extent oninformation kindly provided bycolleagues working on GO and/orclosely related areas. I would liketo especially thank ArnoldNeumaier and Simon Streltsov forthe information collected on theirWWW Global Optimization Pages(respectively, http://solon.cma.univie.ac.at/~neum/glopt/, and http://cad.bu.edu/go/). I also wish to thank Faiz Al-Khayyal for his valuable commentson the manuscript.

The space (and time) limitations ofthis review certainly have made itillusory to include all existingsoftware on this rapidly changingarea; omissions are entirelypossible but absolutelyunintentional. It is planned tocontinue this work and to provide amore comprehensive andinformative picture of thestate-of-the-art for the mathematicalprogramming community.Comments and suggestions aremost welcome; they will contributeto an unabridged GO softwarereview in the near future.

References

To avoid a superfluou sly long list-

ing, the reference list is redu ced to

the most top ical journal, and to

several GO monograph s and

handbooks pu blished in the past

ten years.Bomze, I.M., Csend es, T., Horst, R.,

and Pardalos, P.M., eds. (1996)

Developments in Global Optimiza-

tion. Kluwer Academic Publishers,

Dordrecht / Boston / London.

Floudas, C.A. and Pardalos , P.M.

(1990) A Collection of Test

Problems for Constrained Global

Optimization Algorithms. Lecture

Notes in Computer Science 455,

Springer, Berlin / Heidelberg / N ew

York.

Floudas, C.A. and Pardalos , P.M.,

eds. (1992) Recent Advances inGlobal Optimization. Princeton

University Press, Princeton.

Forg, F. (1988) Nonconvex

Programming. Akadmiai Kiad,

Budapest. Grossmann, I.E., ed.

(1996) Global Optimization in

Engineering Design. Kluwer

Academic Publishers, Dordrecht /

Boston / London.

Pardalos , P.M. and Rosen, J.B.

(1987) Constrained Global

Optimization: Algorithms and

Applications. Lecture N otes i n

Computer Science 268, Springer,

Berlin / Heidelberg / N ew York.

Pintr, J.D. (1996) Global Optimi

tion in A ction. Kluw er Academic

Publishers, Dordrecht / Boston /

London.

Ratschek, H. and Rok ne, J.G. (19

New Computer Methods for Glo

Optimization. Ellis Horwood,

Chichester.

Trn, A.A. andv

Zilinskas, A. (198

Global Optimization. Lecture No

in Computer Science 350, Spring

Berlin / Heidelberg / N ew York.van Laarhoven, P.J.M. and Aarts

E.H.L. (1987) Simulated Anneali

Theory and App lications. Kluw e

Academic Publishers, Dordrecht

Boston / London.

Zhigljavsky, A.A. (1991) Theory

Global Random Search. Kluwer

Academic Publishers, Dordrecht

Boston / London.

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The Mathematical Programming Soci-

ety invites nominations for the A.W.Tucker Prize for an outstanding paperauthored by a student. The award willbe presented at the International Sym-posium on Mathematical Program-ming in Lausanne (August 24-29,1997). All students, graduate and un-dergraduate, are eligible. Nominationsof students who have not yet receivedthe first university degree are especiallywelcome. In advance of the Sympo-sium an award committee will screenthe nominations and select at mostthree finalists. The finalists will be in-vited, but not required, to give oralpresentations at a special session of the

Symposium. The award committeewill select the winner and present theaward prior to the conclusion of theSymposium. The members of thecommittee for the 1997 A.W. TuckerPrize are Kurt Anstreicher, Depart-ment of Management Sciences, Uni-versity of Iowa; Rolf Moehring,Fachbereich Mathematik, TechnicalUniversity of Berlin; Jorge Nocedal,EECS Department, NorthwesternUniversity; Jean-Philippe Vial (Chair-man), HEC/Management Studies,University of Geneva; David

Williamson, IBM T.J. Watson Re-search Center, Yorktown Heights.

Eligibility The paper may concern anyaspect of mathematical programming;it may be original research, an exposi-tion or survey, a report on computerroutines and computing experiments,or a presentation of a new and inter-esting application. The paper must besolely authored and completed afterJanuary 1994. The paper and thework on which it is based should havebeen undertaken and completed inconjunction with a degree program.

NominationsNominations must bemade in writing to the chairman ofthe award committee as follows:

Jean-Philippe VialHEC/Management StudiesUniversity of Geneva102, Bd Carl-VogtCH-1211 Geneva 4SwitzerlandFax: 41 22 705 81 04

E-mail: jpvial@uni2a.unige.ch

Call for Nominations: Nominationsare being sought for the Mathemati-cal Programming Society Beale-Orchard-Hays Prize for Excellencein Computational MathematicalProgramming.

Purpose: This award is dedicated tothe memory of Martin Beale andWilliam Orchard-Hays, pioneers incomputational mathematical pro-gramming. To be eligible a paper ora book must meet the following re-quirements:

1) It must be on computationalmathematical programming. Thetopics to be considered include:

a) experimental evaluations of one or

more mathematical algorithms,

b) the development of quality math-ematical programming software (i.e.well-documented code capable of ob-taining solutions to some importantclass of MP problems) coupled withdocumentation of the applications ofthe software to this class of problems(note: the award would be presentedfor the paper which describes thiswork and not for the software itself),

c) the development of a new compu-tational method that improves the

state-of-the-art in computer imple-mentations of MP algorithmscoupled with documentation of theexperiment which showed the im-provement, or

d) the development of new methodsfor empirical testing of mathematicalprogramming techniques (e.g., devel-opment of a new design for compu-tational experiments, identificationof new performance measures, meth-ods for reducing the cost of empiricaltesting).

2) It must have appeared in the openliterature.

3) If the paper or book is written in alanguage other than English, then anEnglish translation must also be in-cluded.

4) Papers eligible for the 1997 awardmust have been published within theyears 1993 through 1996.

They must be submitted by a facultymember at the institution where thenominee was studying for a degreewhen the paper was completed. Let-ters of nomination must be accompa-nied by a statement that each mem-ber of the committee (including thechairman) was sent the followingdocuments: the students paper; aseparate summary of the papers con-tributions, written by the nominee,and no more than two pages inlength; and a brief biographicalsketch of the nominee.

Deadline Nominations must be sentto the chairman and postmarked nolater than February 15, 1997.

Addresses of the other members ofthe committee:

Prof. Kurt M. AnstreicherDepartment of Management SciencesUniversity of IowaIowa City, IA 52242 USA

E-mail: kanstrei@scout-po.biz.uiowa.edu

Prof. Dr. Rolf H. MoehringFachbereich Mathematik, Sekr. 6-1Technische Universitat BerlinStrasse des 17. Juni 13610623 Berlin Germany

Fax: 49 30 314-25191E-mail: moehring@math.tu-berlin.de

Prof. Jorge NocedalElectrical Engineering and ComputerScienceNorthwestern UniversityEvanston, IL 60208-3118 USAFax: (847) 467-4144

E-mail address:nocedal@eecs.nwu.edu

Dr. David P. WilliamsonIBM T.J. Watson Research LabsP.O. Box 218

Yorktown Heights, NY 10598 USAFax: (914) 945-3434

E-mail: dpw@watson.ibm.com

The above information is reproduced

on the web at the address:

http://dmawww.epfl.ch/roso.mosaic/ismp97/tucker.html

These requirements are intended asguidelines to the screening committbut are not to be viewed as bindingwhen work of exceptional meritcomes close to satisfying them.

Frequency and Amount of the Awar

The prize is awarded every threeyears. The 1997 prize of $1500 andplaque will be presented in August1997, at the Swiss Federal Institute Technology (EPFL), Lausanne Switzerland, at the Awards Session of thInternational Symposium on Math-ematical Programming sponsored bthe Mathematical Programming Soety.

Judgement criteria: Nominations w

be judged on the following criteria:1) Magnitude of the contribution tothe advancement of computationaland experimental mathematical programming.

2) Originality of ideas and methods

3) Clarity and excellence of exposi-tion.

Nominations: Nominations must bin writing and include the title(s) ofthe paper(s) or book, the author(s),the place and date of publication anfour copies of the material. Support

ing justification and any supplementary materials are welcome but notmandatory. The awards committeereserves the right to request furthersupporting materials from the nominees.

Nominations should be mailed to:

Professor Robert J. VanderbeiDept. of Civ. Eng. and OperationsResearchACE-42 Engineering QuadPrinceton UniversityPrinceton, NJ 08544 USA

The deadline for submission of nomnations is January 1, 1997.

This call-for-nomination can beviewed online by visiting:

http://www.sor.princeton.edu/~rvdb/BOH97.html

Nominations for the A.W. Tucker PrizeExtended Deadline

Beale-Orchard-Hays Prize

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call

MPDP-19 Nineteenth Symposium on Mathematical Programming with Data Perturbations

May 22-23, 1997, The George Washington University, Washington, DC

The NINETEENTH Symposium on Mathematical Programming with DataPerturbations will be held at George Washington Universitys Marvin Center on22-23 May 1997. The objective is to bring together practitioners who usemathematical programming optimization models, and deal with questions ofsensitivity analysis, with researchers who are developing techniques applicable tothese problems.

The symposium webpage is :http://rutcor.rutgers.edu:80/~bisrael/MPDP-19.html

CONTRIBUTED papers in mathematical programming are solicited in the followingareas: 1. Sensitivity and stability analysis and their applications; 2. Solution

methods for problems involving implicitly defined functions; 3. Solutionapproximation techniques and error analysis. CLINICAL presentations thatdescribe problems in sensitivity analysis encountered in applications are also

invited.

DEADLINES: 15 March 1997 Registration and submission of tentative title andabstract;1 May 1997 Submission of final abstract for inclusion in the Symposium Program

REGISTRATION FEE: $50 USD payable at the meeting.

To REGISTER and/or SUBMIT ABSTRACT please use the electronic form in the URL:

http://rutcor.rutgers.edu:80/~bisrael/MPDP-19.html#form

Or mail to:MPDP-19c/o Adi Ben-IsraelRUTCOR - Rutgers Center forOperations ResearchP.O. Box 5062Rutgers UniversityNew Brunswick, NJ 08903-5062, USA

Or email/fax to one of the organizerslisted below:

Adi Ben-IsraelRutgers University

Tel: +1-908-445-5631Fax: +1-908-445-5472

Hubertus Th. JongenRWTH-AachenTel: +49-241-804540Fax: +49-241-8888390

Special Issue of Computers &Operations Research:

Travelling Salesman Problem

Computers & Operations Research will publish a sp

cial issue on the Travelling Salesman Problem.P

pers are sought in the broad area of the travelling sale

man problem and its variations which discuss comp

tational and/or algorithmic aspects. In particular,

are soliciting papers on computational study of exact an

or heuristic algorithms, analysis of heuristics, domin

tion analysis and exponential neighbourhoods, problewith special structures, new applications, etc.

All papers submitted for consideration will undergo sta

dard review process. Four copies of the paper, followi

the standard guidelines for Computers & Operations R

search, should be sent by March 1997 to:

Dr. Abraham Punnen

Dept. of Mathematics, Statistics & Computer Scien

University of Brunswick

Saint John, New Brunswick

CANADA E2L 4L5

Financial Support forINTERNATIONAL CONGRESS OF MATHEMATICIANSBerlin, 1998

The ICM98 Organizing Committee has already re-ceived quite a number of requests concerning financial

support for participation at the International Congress

of Mathematicians 1998 in Berlin. The Circular Let-

ter ICM98-CL6 describes how mathematicians from

developing countries can apply for financial help. The

local Organizing Committee is currently making ef-

forts to obtain donations from German industry, gov-

ernment, foundations and individuals to be able to

partially support mathematicians from Eastern Europe

and the independent states of the former Soviet Union.

To secure participation of as many persons as possible,

the local Organizing Committee will only support lo-

cal costs in Berlin. Berlin is very close to Eastern Eu-

rope, and it is expected that applicants will find other

means to cover their travel costs.To handle the applications and manage the financial

support the ICM98 Organizing Committee has set up

a subcommittee, Committee for Support of Mathema-

ticians from Eastern Europe (CSMEE).

CSMEE will distribute application forms for grants,

as described above, for mathematicians from Eastern

Europe in late summer 1997. T hese forms will be

made available through the ICM98 server (http://

elib.zib.de/ICM98) and by e-mail. Applicants will be

asked to provide a brief curriculum vitae (including

academic education, degree, professional employment,

and a list of publications).

Applicants should submit their application form

before January 1, 1998 to CSMEE to one of the following addresses:

Prof. Dr. H. KurkeHumboldt-Universitaet Institut fuer MathematikUnter den Linden 6 D-10099 BerlinGermany

e-mail: kurke@mathematik.hu-berlin.de

Prof. Dr. W. RoemischHumboldt-UniversitaetInstitut fuer Mathematik Ziegelstrasse 13AD-10099 BerlinGermany

e-mail: romisch@mathematik.hu-berlin.de

All applications will be reviewed.

Further questions concerning financial support for

mathematicians from Eastern Europe to attend

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Sponsored by the Dept. of OperationsResearch & the Institute forManagement Science & Engineering,School of Engineering & AppliedScience The George WashingtonUniversity Washington, DC 20052,USA

FOR

PAPERS

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B O O K r e v i e w sCombinatorial Network TheoryEdited by Ding-Zhu Du andD. Frank HsuSeries of Applied Optimization

Kluwer Academic Publishers

Dordrecht, 1996ISBN 0-7923-3777-8

The design of interconnection networks and related theo-

retical research has received a lot of attention in recent

years. A network suitable for parallel computation should

satisfy two major requirements. First, a network should

have computational efficiency to run parallel algorithms.

One of the main overheads incurred during parallel ex-ecution is the time spent on communication between

processors. Thus, the ability of an interconnection net-

work to disseminate information efficiently is one of the

most important properties of a network. Secondly, reli-

ability of a network is another important requirement.

An interconnection network should be fault-tolerant so

that it can work even if some of its links or nodes fail. This

book is devoted to the theoret ical study of dissemination

of information in interconnection networks and their re-

liability.

Caley graphs provide suitable candidates for interconnec-

tion networks. One of their advantages is their symmet-

ric structure, which makes them easy to build and sim-

plifies routing and communication algorithms. Examples

of prominent interconnection networks which are Caley

graphs include the H ypercube, the Butterfly and the

Cube-Connected-Cycles network. In the first two chap-

ters of the book Caley graphs are considered. Since Caley

graphs are defined on groups, group theory provides a

powerful tool in the study of Caley graphs.

An extensive study of the connectivity of Caley graphs

defined on finite Abelian groups is presented in Chapter

1. Connectivity can be viewed as a measure of the reli-

ability of a network. Along with the presentation of new

results, it is shown how some well-known results from the

additive group theory were rediscovered and applied to

graph-theoretical problems such as that of network con-

nectivity. Unfortunately, this chapter contains many

textual mistakes.

State of the Art in GlobalOptimization

Computational Methods andApplicationsEdited by C.A. Floudas andP.M. PardalosNonconvex Optimization and Its

Applications 7

Kluwer Academic Publishers

Dordrecht 1996ISBN 0-7923-3838-3

This is the seventh volume of an excellent and muneeded series in Nonconvex Optimization and Its A

plications put together by leaders in this field. The vo

ume contains 36 invited and thoroughly refereed pape

that were presented at the conference on State of the A

in Global Optimization: Computational Methods a

Applications. The conference was organized by C.

Floudas and P.M. Pardalos and held at Princeton Un

versity, April 28-30, 1995. The conference pape

spanned the gamut of theory, computational impleme

tations and applications of global optimization.

Among other fine contributions, one finds the followin

R. Horst and N. van Thoai propose two types of n

finite branch and bound algorithms for global minim

zation of separable concave functions under linear co

straints with totally unimodular matrices. The key obs

vation is that the underlying problems can be viewed

integer programs. Finiteness can, therefore, be achiev

by integral branching.

K.G. Ramakrishnan, M.G.C. Resende and P.M

Pardalos report on computational experience with

branch and bound algorithm for Quadratic Assignme

Problems. All problems of dimension n 15 of QAPL

are solved.

R.B. Kearfott reports practical experience with an i

terval branch and bound algorithm for equality-costrained optimization.

T . Van Voorhis and F. Al-Khayyal use range reducti

techniques to accelerate a branch and bound algorith

for quadratically constrained quadratic programs.

J.P. Shectman and N.V. Sahinidis prove that exhau

tiveness and branching on the incumbent whenev

possible in tandem ensure finiteness of rectangular-su

division-based branch and bound algorithms for glob

minimization if separable concave programs.

The study of edge and vertex connectivity in Caley graphscontinues in Chapter 2. In contrast to Chapter 1, most

of the proofs in this chapter are obtained by using atoms

of graphs.

De Bruijn and Kautz digraphs and their generalizations

are other popular candidates for interconnection net-

works. Their popularity is explained by the fact that these

networks have a highly symmetric structure and give

optimal or nearly optimal solutions of the problem of

minimizing the diameter and maximizing connectivity for

a graph with a given number of nodes and degree. Con-

nectivity and diameter are important parameters for both

reliability and the ability of a network to disseminate

information efficiently. Chapter 3 presents an extensivestudy of diameter, connectivity, line-connectivity, super-

line connectivity and the Hamiltonian property of de

Bruijn and Kautz digraphs and their generalizations.

In Chapter 4 various properties (in particular, link con-

nectivity) of extended double loop networks (EDLN) are

studied. A large class of EDLN includes such well-known

networks as generalized de Bruijn networks, the Imase-

Itoh networks and double loop networks. This study

shows that certain EDLN networks are suitable candidates

for good interconnection networks.

Dissemination of information in interconnection net-

works is considered in Chapter 5. The three main prob-

lems of dissemination of information are broadcasting,

accumulation and gossiping. These problems are consid-

ered for all well-known interconnection networks under

several communication modes. The solutions for these

problems are given mainly in terms of lower and upper

bounds on communication t ime. This chapter provides

the reader with many basic proof techniques and ideas in

this area, gives a good survey of known results and for-

mulates many open problems.

The book is aimed at graduate students and researchers

and provides the reader with a deep insight into combi-

natorial network theory. Some of its chapters (especially

Chapter 5) would also be suitable for undergraduate stu-dents.

-E.A. STHR

B O O K

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H . Tuy shows that many important location problems(Webers problem with attraction and repulsion, con-

strained multisource and multifacility problems and oth-

ers) can be formulated as d.c. optimization problems in

low-dimensional spaces and describes algorithms for their

solution.

Additional articles in the volume are authored by S.

Zlobec; E. Novak and K. Ritter; S. Shi, Q. Zheng and D.

Zhuang; B. Ramachandran and J.F. Pekny; G. Isac; P.

Maponi, M.C. Recchioni and F. Zirilli; G.H. Staus, L.T.

Biegler and B.E. Ydstie; V. Visweswaran, C.A. Floudas,

M.G. Ierapetritou and E.N. Pistikopoulos; J. Barhen and

V. Protopopescu; D. MacLagan, T. Sturge and W.

Baritompa; K. Holmqvist and A. Migdalas; D.W. Bulgerand G.R. Wood; W. Edmonson, K. Srinivasan, C. Wang

and J. Principe; E. Falkenauer; I.P. Androulakeis, V.

Visweswaran and C.A. Floudas; T. Q ian, Y. Ye and P.M.

Pardalos; A. Torn and S. Viitanen; K.I.M. McKinnon,

C. Millar and M. Mongeau; E. Haddad; J. Shi and Y.

Yoshitsugu; I. Garcia and G.T. H erman; P. Sussner, P.M.

Pardalos and G.X. Ritter; J.A. Filar, P.S. Gaertner and

M.A. Janssen; W.F. Eddy and A. Mockus; L. Mockus and

G.V. Reklaitis; A. Lucia and J. Xu; J.R. Banga and W.D .

Seider; M. Turkay and I.E. Grossmann; F. Friedler, J.B.

Varga, E. Feher and L.T. Fan; E.S. Fraga.

The above papers cover the theory and algorithms of

deterministic global optimization, stochastic global op-

timization, branch and bound methods, interval arith-

metic methods, d.c. programming, duality, concave pro-

gramming, bilevel programming, integral optimization,

decomposition methods, logic-based algorithms, and

trust algorithms.

The book is also very rich in applications in resource

allocation, computer vision, chemical process design,

control and optimization, chemical and phase equilib-

rium, facility location, climate change dynamic visualiza-

tion, batch process scheduling, and process synthesis.

One, therefore, cannot help but to fully agree with the

editors that the book will be a valuable source of infor-mation to faculty, students and researchers in optimiza-

tion, engineering, mathematics, computer sciences and

related areas.

-NIKOLAOS SAHIN IDIS

Postoptimal Analyses,Parametric Programming andRelated Topics

Tomas Galde Gruyter

Berlin, 1995ISBN 3-11-014060-8.

This book is the second edition of a monograph first writ-

ten in the years 1968-1969 in Czech, translated in 1973

to German and in 1979 to English. Each chapter is divided

into two parts. The first is based on illustrative examples;

the second part is an abridged mathematical presentation.

Each chapter ends by a selected bibliography.

This presentation in two parts has the great advantage that

the book is easy to read and understand for a wide audi-

ence in the first part and is mathematically complete in the

second part. It is also a good, quite complete reference on

the subject. H owever, it has two main disadvantages. The

first is probably linked to the origin of the book: there

remain some typographical errors and inconsistencies in the

notations. Furthermore, the references at the end of each

chapter often date before 1970. The second disadvantage

is an omission: the consideration of the sensitivity analysis

results in case of degeneracy. This part is quite poor and

recently proposed new results should be introduced.

Chapter one is devoted to fixed basic concepts and nota-

tions in linear programming. It recalls all the very well-

known concepts in linear programming such as basic vari-

ables, reduced cost, dual value, etc. It also recalls the basis

of the Simplex method for solving linear problems. The dual

problem is also presented with the dual Simplex method

and finally the concepts of primal or dual degenerate so-

lutions are presented.

Chapter two is devoted to suboptimal solutions, redundant

constraints and degeneracy. For suboptimal solution, the

effect on the objective level and on the activities level of

producing a nonoptimal activity level is derived from the

optimal Simplex tableau. A redundant constraint is definedas a constraint that does not influence the feasible region.

A weakly redundant constraint has a point in common with

the feasible region. The strongly redundant one does not.

This simply means that the associate slack cannot be

equalised to zero. This can be easily detected and the cor-

responding constraint can be deleted. A primal degenerate

solution is obtained when some basic variables are equal to

zero. This can lead to a degenerate step in the Simplex

algorithm: the base changes but the algorithm remains at

the same vertex.

Chapter three is devoted to sensitivity analysis with respeto changing right-hand side without basis exchange. T

classical sensitivity analysis with a single component of t

right-hand side is introduced through a simple examp

The same basis remains optimal if the basic variables rem

nonnegative. One can deduce a range for the maxim

variation of this component. The effect on the optimal ba

variables is given by the corresponding column of the

verse of the basic matrix. The effect on the objective fun

tion is given by the value of the corresponding dual va

able. Then sensitivity analysis with respect to several co

ponents of the right-hand side depending on a scalar p

rameter is examined. Finally, the multiparametric sensit

ity analysis case is considered, i.e. changing several compnents of the right-hand side depending on several para

eters.

Chapter four concerns linear parametric programming w

respect to change in the right-hand side. This implies

general a basis exchange. The critical values of the param

eters are defined as the values for which the optimal ba

changes. First, the case of a single change in the right-ha

side is considered. Then, a change in several componen

of the right-hand side is considered. The case where the

components change linearly with a single parameter a

the case where they change multilinearly with several p

rameters are both considered.

At the end of this chapter, the problem of sensitivity ana

sis under primal degeneracy is briefly considered. In fa

the sensitivity analysis with the right-hand side consists

determining the critical interval where the same basis

mains optimal. If the optimal solution is degenerate, se

eral optimal bases exist. The critical interval is defined he

as the union of all the critical intervals associated with t

different optimal bases. Note that the analysis concerni

the shadow price that can be different left and right is n

very developed here. For a more complete analysis, see,

example, M.P. Williams, Model Building in Mathema

cal Programming, John Wiley, 1990. Also, the assertion

Gal (see page 175) that commercial LP software offeri

sensitivity analysis and shadow prices yield false results wh

primal degeneracy occurs, is not correct. There is o

exception that I know: XPRESS-MP of Dash Associat

which gives the two (left and right) correct values.

Chapter five concerns sensitivity analysis with respect

changing cost coefficients without basis exchange. Note t

very bad choice made for notations: the market price

denoted by c and the unit cost by p (see page 211). The ca

of a single change for a nonbasic objective coefficient a

for a basic variable objective coefficient are considere

Then the case of changing several cost coefficients depen

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ing on a scalar or multiple parameters is considered. In each

case the critical region of the parameter (i.e. the region in

which the same basis is optimal) is computed.

Chapter six is devoted to linear parametric programming

with respect to change in cost coefficients with basis ex-

change. Since the partial derivative of the optimal value of

the objective function with cost coefficient is the value ofthe primal variables, it is sufficient to compute the succes-

sive optimal solutions when the objective coefficients vary.

This can be done by changing from basis to adjacent basis,

simply by a primal Simplex step. Then the problem of

finding the minimal and maximal value of the parameter

such that the problem has a bounded optimal solution is

considered. It is asserted (see example 6-2, page 239) that

the example has no solution for zero value of the param-

eter. This is false. There is not finite optimal solution. The

geometric interpretation (slope change) of cost coefficient

change is then given. Then the case of changes depending

on several parameters is considered. The chapter ends with

the problem of determining the optimal objective costcoefficients where these coefficients can vary homoge-

neously with a parameter. The consideration of degeneracy

is omitted in this chapter.

Chapter seven is dedicated to sensitivity analysis to simul-

taneous changing of the right-hand side and of the cost

coefficients. The objective coefficients and the right-hand

side are written here as linear functions of the same param-

eter. First, the case of changing of the right-hand side and

cost coefficient with a scalar parameter is considered. Then

the case of the dependence on a vector of parameters is

considered.

Chapter eight is dedicated to sensitivity analysis with re-

spect to the elements of the technological matrix often noted

A in linear problems. The coefficient at the intersection of

line i and column j denotes the consumption of the ith pro-

duction factor per unit of the jth production and can be

affected by a change in technology. Two small examples

of one column of A depending on a single parameter illus-

trate the complications which arise from such changes: the

optimal value of the objective function varies nonlinearly

with this parameter! This chapter is certainly one of the most

original as such results are rarely presented in books and

most of the time not published. I agree totally with the

author when he says that, despite the importance of this

sensitivity analysis with respect to matrix coefficients, papers

have been rarely published. He cites as examples his CORE

discussion papers 7013 and 7018 dating from 1970!

However, I regret that the author directly goes to the case

of one column that entirely varies and does not consider

the partial derivatives of the optimal value of the objective

function with respect to a single coefficient of matrix A. One

can derive from the (more complicated) formula of the

abridged mathematical presentation of chapter 8 the pretty

nice result: this partial derivative is equal to the opposite

of the product of the dual optimal value associated with the

line of the coefficient by the primal optimal value associ-

ated with the column of the coefficient. The degenerate case

is also totally omitted. The result however exists: it is forth-

coming in Mathematical Programming under the title

Generalised derivative of the optimal solution of the ob-

jective function of a linear problem with respect to matrix

coefficient by D. De Wolf and Y. Smeers.

Chapter nine is dedicated to multicriteria linear program-ming, i.e. the problem of maximising several conflicting

linear goals over a linear feasible region. One can define in

this framework an efficient solution as a nondominated

solution in Pareto sense: i.e. when trying to improve the

value of one goal, the value of at least one of the remaining

objective functions becomes worse. A method for determin-

ing the set of all efficient solutions is proposed. The reason

why multicriteria problems are considered in this book is

the following: the so-called Efficiency Theorem states that

there is a one-to-one correspondence between efficient

solutions and optimal solutions for the homogeneous

multiparametric problem that is defined in chapter six.

Therefore parametric programming can help to computeefficient solutions for the mult icriteria linear problem.

Finally, the non-essential objective functions are defined

as objective functions that do not affect the set of efficient

solutions. By an efficiency test, one can identify these non-

essential objectives and thus reduce the number of objec-

tive functions to consider.

Chapter ten indicates possible applications of sensitivity

analysis and linear parametric programming in decision

making. Firstly, if for some reason, one wants to change the

value of some basic variables, it is sufficient to change the

corresponding right-hand side. Changing the value of some

nonbasic value is equivalent to considering a suboptimal

solution. Secondly, the problem of the inconsistency of the

constraint is considered. For practical applications with

thousands of constraints, it often occurs that the solution

set is empty (one artificial variable remains positive in phase

one). A simple way of removing inconsistency is to change

the value of the right-hand side. Thirdly, the problem of

redundancy of some linear inequalities is considered. Re-

call that a redundant constraint is a constraint that does not

affect the feasible region. One way for detecting such con-

straints using sensitivity analysis is to notice that the range

of variation of the right-hand side of such constraints var-

ies from minus infinity to plus infinity. In conclusion, sen-

sitivity analysis helps to compute efficient solutions in

multicriteria programming (first application), to remove

inconsistency (second application) and to detect redundant

constraints (third application).

-DANIEL DE WOLF

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A P P L I C A T I O N F O R M E M B E R S H I P

JOU

RNA

LVolume 75, No. 1

A. von Arnim, R. Schrader and

Y. Wang, The permutahedron of

N-sparse posets

Z.-Q. Luo, J.-S. Pang, D. Ralph

and S.-Q. Wu,Exact penaliz a-

tion and stat ionarity conditions

of mathematical programs w ith

equilibrium constraints

Tsuchiya, T. and R.D.C.

Monteiro, Superlinear conver-

gence of the a ffine scaling

algorithm

Volume 75, No. 2

R.J.B. Wets, Challenges in

stochastic programming

E.L. Plambe ck, B.-R. Fu, S.M.

Robinson and R. Suri, Sample-

path optimiz ation of convex

stochastic performance func-

tions

S.S. Nielsen and S.A. Zenios, A

stochast ic programming model

for funding si ngle premium

deferred annuities

K. Marti,Differentiation

formulas for probability

functions: The transformat ion

method

P. Kall and J. Maye r, SLP-IOR:

An interacti ve model ma nage-

ment system for stochastic linear

programs

G. Infanger and D .P. Morton,

Cut sharing for mult ist age

stochastic l inear programs w ith

interst age d ependency

J.L. Higl e and S. Sen ,Duality

and st atistical tests of optimality

for tw o st age stochastic pro-

grams

K. Frauendorfer,Barycentric

scenario t rees in convex mult i-

stage stochastic programming

N.C.P. Edirising he and W.T.

Ziemba,Implementing bounds-

based approximations in convex-

concave tw o-stage stochastic

programming

J.R. Birge, C.J. Do noh ue, D .F.Holmes and O.G. Svintsitski,A

parallel implementation of the

nested decomposition algorithm

for multistage stochastic linear

programs

a t h e m a t i c a l

rogramming

Volume 75, No. 3

M. Constantino,A cutt ing plane

approach to capacitated lot -

sizing w ith start-up costs

H. Yamashita and H. Yabe,

Superlinear and quadratic

convergence of some primalduainterior point methods for

constrained opt imizat ion

J.-P. Crouz eix and J.A. Ferland,

Criteria for d ifferenti able

generaliz ed monot one maps

T. De Luca, F. Facchinei and C.

Kanzow,A semismooth equatio

approach to the solution of

nonlinear complementarit y

problems

D. Goeleven, G.E. Stavroulakis

and P.D. Panagiotopoulos,

Solvability theory for a class of

hemivariational inequalitiesinvolving copositive plus

matrices.Applications in robotic

E.C. Sew ell, Binary integer

programs w ith tw o variables pe

inequality

M.D. Grigoriadis and L.G.

Khachiyan,Approximate

minimum-cost multicommodity

flow s in Oe -2 KNM time

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sHarvey Greenberg has developed a Mathem

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for the next OPTIMA is Feb. 15, 1997.