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The IMA Volumes in Mathematics
and its Applications
Volume 154
Forfurthervolumes:
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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
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ceptional interest and opportunity to extensive thematic programs lasting
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that webelieve are ofparticularvalue to the broaderscientific community.
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IMA ANNUAL PROGRAMS
1982–1983 Statistical and Continuum Approaches to Phase Transition
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Continued at the back
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Jon Lee • Sven Leyffer
Editors
Mixed Integer Nonlinear
Programming
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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
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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
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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.
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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
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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
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