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www.it-ebooks.info The IMA Volumes in Mathematics and its Applications Volume 154 Forfurthervolumes: http://www.springer.com/series/811 www.it-ebooks.info Institute for Mathematics and its Applications (IMA) The Institute for Mathematics and its Applications was estab- lished by a grant from the National Science Foundation to the University ofMinnesotain1982. TheprimarymissionoftheIMAistofosterresearch of a truly interdisciplinary nature, establishing links between mathematics of the highest caliber and important scientific and technological problems fromotherdisciplinesandindustries. Tothisend,theIMAorganizesawide variety of programs, ranging from short intense workshops in areas of ex- ceptional interest and opportunity to extensive thematic programs lasting a year. IMA Volumes are used to communicate results of these programs that webelieve are ofparticularvalue to the broaderscientific community. ThefulllistofIMAbookscanbefoundattheWebsiteoftheInstitute for Mathematics and its Applications: http://www.ima.umn.edu/springer/volumes.html. Presentation materials from the IMA talks are available at http://www.ima.umn.edu/talks/. Video library is at http://www.ima.umn.edu/videos/. Fadil Santosa, Director of the IMA * * * * * * * * * * IMA ANNUAL PROGRAMS 1982–1983 Statistical and Continuum Approaches to Phase Transition 1983–1984 Mathematical Models for the Economics of Decentralized Resource Allocation 1984–1985 Continuum Physics and Partial Differential Equations 1985–1986 Stochastic Differential Equations and Their Applications 1986–1987 Scientific Computation 1987–1988 Applied Combinatorics 1988–1989 Nonlinear Waves 1989–1990 Dynamical Systems and Their Applications 1990–1991 Phase Transitions and Free Boundaries 1991–1992 Applied Linear Algebra Continued at the back www.it-ebooks.info Jon Lee • Sven Leyffer Editors Mixed Integer Nonlinear Programming www.it-ebooks.info Editors Jon Lee Sven Leyffer Industrial and Operations Engineering Mathematics and Computer Science University of Michigan Argonne National Laboratory 1205 Beal Avenue Argonne, Illinois 60439 Ann Arbor, Michigan 48109 USA USA ISSN 0940-6573 ISBN 9 78-1-4614-1926-6 e - IS B N 978-1-4614-1927-3 DOI10.1007/978-1-4614-1927-3 Sprin ger New York Dordrecht Heidelberg London Library of Congress Control Number: 2011942482 Mathematics Subject Classification (2010): 05C25, 20B25, 49J15, 49M15, 49M37, 49N90, 65K05, 90C10, 90C11, 90C22, 90C25, 90C26, 90C27, 90C30, 90C35, 90C51, 90C55, 90C57, 90C60, 90C90, 93C95 (cid:164) Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) www.it-ebooks.info FOREWORD This IMA Volume in Mathematics and its Applications MIXED INTEGER NONLINEAR PROGRAMMING contains expository and researchpapers based on a highly successful IMA Hot Topics Workshop “Mixed-Integer Nonlinear Optimization: Algorith- mic Advances and Applications”. We are grateful to all the participants for making this occasion a very productive and stimulating one. We wouldlike to thank JonLee (Industrial and Operations Engineer- ing,UniversityofMichigan)andSvenLeyffer(MathematicsandComputer Science,ArgonneNationalLaboratory)fortheirsuperbroleasprogramor- ganizers and editors of this volume. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Fadil Santosa, Director of the IMA Markus Keel, Deputy Director of the IMA v www.it-ebooks.info www.it-ebooks.info PREFACE Manyengineering,operations,andscientificapplicationsincludeamixture of discrete and continuous decision variables and nonlinear relationships involving the decision variables that have a pronounced effect on the set of feasible and optimal solutions. Mixed-integer nonlinear programming (MINLP) problems combine the numerical difficulties of handling nonlin- ear functions with the challenge of optimizing in the context of nonconvex functions anddiscrete variables. MINLP is one of the most flexible model- ingparadigmsavailableforoptimization;butbecauseitsscopeissobroad, in the most general cases it is hopelessly intractable. Nonetheless, an ex- panding body of researchers and practitioners — including chemical en- gineers, operations researchers,industrial engineers, mechanical engineers, economists, statisticians, computer scientists, operations managers, and mathematicalprogrammers—areinterestedinsolvinglarge-scaleMINLP instances. Of course, the wealth of applications that can be accurately mod- eled by using MINLP is not yet matched by the capability of available optimization solvers. Yet, the two key components of MINLP — mixed- integerlinear programming(MILP)andnonlinearprogramming(NLP) — have experienced tremendous progress over the past 15 years. By cleverly incorporating many theoretical advances in MILP research, powerful aca- demic, open-source, and commercial solvers have paved the way for MILP to emerge as a viable, widely used decision-making tool. Similarly, new paradigms and better theoretical understanding have created faster and more reliable NLP solvers that work well, even under adverse conditions such as failure of constraint qualifications. In the fall of 2008, a Hot-Topics Workshop on MINLP was organized attheIMA,withthegoalofsynthesizingtheseadvancesandinspiringnew ideas in order to transformMINLP.The workshopattractedmore than 75 attendees, over 20 talks, and over 20 posters. The present volume collects 22 invited articles, organized into nine sections on the diverse aspects of MINLP. The volume includes survey articles, new research material, and novel applications of MINLP. In its most general and abstract form, a MINLP can be expressed as minimize f(x) subject to x∈F, (1) x where f : Rn → R is a function and the feasible set F contains both non- linear and discrete structure. We note that we do not generally assume smoothness of f or convexity of the functions involved. Different realiza- tions of the objective function f and the feasible set F give rise to key classes of MINLPs addressed by papers in this collection. vii www.it-ebooks.info viii PREFACE Part I. Convex MINLP. Eventhoughmixed-integeroptimizationprob- lems are nonconvex as a result of the presence of discrete variables, the term convex MINLP is commonly used to refer to a class of MINLPs for which a convex program results when any explicit restrictions of discrete- ness on variables are relaxed (i.e., removed). In its simplest definition, for a convex MINLP, we may assume that the objective function f in (1) is a convex function and that the feasible set F is described by a set of convex nonlinear function, c : Rn → Rm, and a set of indices, I ⊂ {1,...,n}, of integer variables: F ={x∈Rn |c(x)≤0, andxi ∈Z,∀i∈I}. (2) Typically, we also demand some smoothness of the functions involved. Sometimes it is useful to expand the definition of convex MINLP to sim- ply require that the functions be convex on the feasible region. Besides problemsthatcanbe directly modeledasconvexMINLPs,the subjecthas relevance to methods that create convex MINLP subproblems. Algorithms and software for convex mixed-integer nonlinear programs (P. Bonami, M. Kilinc¸, and J. Linderoth) discussesthe stateofthe artfor algorithms and software aimed at convex MINLPs. Important elements of successful methods include a tree search(to handle the discrete variables), NLP subproblems to tighten linearizations,and MILP master problems to collect and exploit the linearizations. A special type of convex constraint is a second-order cone constraint: (cid:7)y(cid:7)2 ≤z,wherey isvectorvariableandz isascalarvariable. Subgradient- based outer approximation for mixed-integer second-order cone program- ming(S.DrewesandS.Ulbrich) demonstrateshowsuchconstraintscanbe handledbyusingouter-approximationtechniques. Amaindifficulty,which the authors address using subgradients,is that at the point (y,z)=(0,0), the function (cid:7)y(cid:7)2 is not differentiable. ManyconvexMINLPshave“off/on”decisionsthatforce acontinuous variable either to be 0 or to be in a convex set. Perspective reformula- tion and applications (O. Gu¨nlu¨k and J. Linderoth) describes an effective reformulationtechniquethatisapplicabletosuchsituations. Theperspec- tive g(x,t) = tc(x/t) of a convex function c(x) is itself convex, and this property can be used to construct tight reformulations. The perspective reformulation is closely related to the subject of the next section: disjunc- tive programming. Part II. Disjunctive programming. Disjunctive programsinvolvecon- tinuousvariabletogetherwithBooleanvariableswhichmodellogicalpropo- sitions directly rather than by means of an algebraic formulation. Generalized disjunctive programming: A framework for formulation and alternative algorithms for MINLP optimization (I.E. Grossmann and J.P. Ruiz) addresses generalized disjunctive programs (GDPs), which are MINLPs that involve generaldisjunctions and nonlinear terms. GDPs can www.it-ebooks.info PREFACE ix be formulated as MINLPs either through the “big-M” formulation, or by using the perspective of the nonlinear functions. The authors describe twoapproaches: disjunctivebranch-and-bound,whichbranchesonthedis- junctions, and and logic-based outer approximation, which constructs a disjunctive MILP master problem. Under the assumption that the problem functions are factorable (i.e., the functions can be computed in a finite number of simple steps by us- ing unary and binary operators), a MINLP can be reformulated as an equivalent MINLP where the only nonlinear constraints are equations in- volving two or three variables. The paper Disjunctive cuts for nonconvex MINLP (P. Belotti) describes a procedure for generating disjunctive cuts. First,spatialbranchingisperformedonanoriginalproblemvariable. Next, bound reduction is applied to the two resulting relaxations, and linear relaxations are created from a small number of outer approximations of each nonlinear expression. Then a cut-generation LP is used to produce a new cut. PartIII.Nonlinearprogramming. Forseveralimportantandpractical approaches to solving MINLPs, the most important part is the fast and accuratesolutionofNLPsubproblems. NLPsarisebothasnodesinbranch- and-boundtreesandassubproblemsforfixedintegerorBooleanvariables. Thepapersinthissectiondiscusstwocomplementarytechniquesforsolving NLPs: active-setmethodsintheformofsequentialquadraticprogramming (SQP) methods and interior-point methods (IPMs). Sequential quadratic programming methods (P.E. Gill and E. Wong) is a survey of a key NLP approach, sequential quadratic programming (SQP), that is especially relevant to MINLP. SQP methods solve NLPs by a sequence of quadratic programmingapproximationsand areparticularly well-suited to warm starts and re-solves that occur in MINLP. IPMs are an alternative to SQP methods. However, standard IPMs can stall if started near a solution, or even fail on infeasible NLPs, mak- ing them less suitable for MINLP. Using interior-point methods within an outer approximation framework for mixed-integer nonlinear programming (H.Y.Benson)suggestsaprimal-dualregularizationthatpenalizesthecon- straints and bounds the slack variables to overcome the difficulties caused by warm starts and infeasible subproblems. Part IV. Expression graphs. Expression graphs are a convenient way to represent functions. An expression graph is a directed graph in which eachnoderepresentsanarithmeticoperation,incomingedgesrepresentop- erations,andoutgoingedges representthe resultofthe operation. Expres- sion graphs can be manipulated to obtain derivative information, perform problem simplifications through presolve operations, or obtain relaxations of nonconvex constraints. Using expression graphs in optimization algorithms (D.M. Gay) dis- cusseshowexpressiongraphsallowgradientsandHessianstobe computed www.it-ebooks.info

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