mixed integer nonlinear programming

www.it-ebooks.info For further volumes: The IMA Volumes in Mathematics and its Applications Volume 154 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 of Minnesota in 1982. The primary mission of the IMA is to foster research of a truly interdisciplinary nature, establishing links between mathematics of the highest caliber and important scientific and technological problems from other disciplines and industries. To this end, the IMA organizes a wide 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 we believe are of particular value to the broader scientific community. 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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 www.it-ebooks.info FOREWORD This IMA Volume in Mathematics and its Applications MIXED INTEGER NONLINEAR PROGRAMMING contains expository and research papers 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 would like to thank Jon Lee (Industrial and Operations Engineer- ing, University of Michigan) and Sven Leyffer (Mathematics and Computer Science, Argonne National Laboratory) for their superb role as program or- 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 Many engineering, operations, and scientific applications include a mixture 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 and discrete variables. MINLP is one of the most flexible model- ing paradigms available for optimization; but because its scope is so broad, 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 mathematical programmers — are interested in solving large-scale MINLP 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- integer linear programming (MILP) and nonlinear programming (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 at the IMA, with the goal of synthesizing these advances and inspiring new ideas in order to transform MINLP. The workshop attracted more 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 x f(x) subject to x ∈F, (1) where f : R n → 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. Even though mixed-integer optimization prob- 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 : R n → R m , and a set of indices, I⊂{1, ,n},of integer variables: F = {x ∈ R n | c(x) ≤ 0, and x i ∈ 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 problems that can be directly modeled as convex MINLPs, the subject has relevance to methods that create convex MINLP subproblems. Algorithms and software for convex mixed-integer nonlinear programs (P. Bonami, M. Kilin¸c, and J. Linderoth) discusses the state of the art for 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: y 2 ≤ z,wherey is vector variable and z is a scalar variable. Subgradient- based outer approximation for mixed-integer second-order cone program- ming (S. Drewes and S. Ulbrich) demonstrates how such constraints can be handled by using outer-approximation techniques. A main difficulty, which the authors address using subgradients, is that at the point (y,z)=(0, 0), the function y 2 is not differentiable. Many convex MINLPs have “off/on” decisions that force a continuous variable either to be 0 or to be in a convex set. Perspective reformula- tion and applications (O. G¨unl¨uk and J. Linderoth) describes an effective reformulation technique that is applicable to such situations. The perspec- 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 programs involve con- tinuous variable together with Boolean variables which model logical propo- 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 general disjunctions 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 two approaches: disjunctive branch-and-bound, which branches on the dis- 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, spatial branching is performed on an original problem variable. 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. Part III. Nonlinear programming. For several important and practical approaches to solving MINLPs, the most important part is the fast and accurate solution of NLP subproblems. NLPs arise both as nodes in branch- and-bound trees and as subproblems for fixed integer or Boolean variables. The papers in this section discuss two complementary techniques for solving NLPs: active-set methods in the form of sequential quadratic programming (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 programming approximations and are particularly 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) suggests a primal-dual regularization that penalizes the con- 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 each node represents an arithmetic operation, incoming edges represent op- erations, and outgoing edges represent the result of the 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- cusses how expression graphs allow gradients and Hessians to be computed www.it-ebooks.info [...]... techniques for solving Mixed Integer Linear Programs and incorporating these techniques into software, significant improvements have been made in the ability to solve these problems Key words Mixed Integer Nonlinear Programming; Branch and Bound 1 Introduction Mixed- Integer Nonlinear Programs (MINLP)s are optimization problems where some of the variables are constrained to take integer values and the... Complexity On the complexity of nonlinear mixed- integer optimization 533 Matthias K¨ppe o Theory and applications of n-fold integer programming 559 Shmuel Onn www.it-ebooks.info CONTENTS xvii Part IX: Applications MINLP Application for ACH interiors restructuring 597 Erica Klampfl and Yakov Fradkin A benchmark library of mixed- integer optimal control problems... of nonlinear mixed- integer optimization (M K¨ppe) o is a survey on the computational complexity of MINLP It includes incomputability results that arise from number theory and logic, fully polynomialtime approximation schemes in fixed dimension, and polynomial-time algorithms for special cases Theory and applications of n-fold integer programming (S Onn) is an overview of the theory of n-fold integer programming, ... optimizing over integer variables with the handling of nonlinear functions Even if we restrict our model to contain only linear functions, MINLP reduces to a Mixed- Integer Linear Program (MILP), which is an NP-Hard problem [55] On the other hand, if we restrict our model to have no integer variable but allow for general nonlinear functions in the objective or the constraints, then MINLP reduces to a Nonlinear. .. problems A global-optimization algorithm for mixed- integer nonlinear programs having separable nonconvexity (C D’Ambrosio, J Lee, and A W¨chter) a introduces a method for MINLPs that have all of their nonconvexity in separable form The approach aims to retain and exploit existing convexity in the formulation Global optimization of mixed- integer signomial programming problems (A Lundell and T Westerlund)... www.it-ebooks.info PART I: Convex MINLP www.it-ebooks.info www.it-ebooks.info ALGORITHMS AND SOFTWARE FOR CONVEX MIXED INTEGER NONLINEAR PROGRAMS PIERRE BONAMI∗ , MUSTAFA KILINC† , AND JEFF LINDEROTH‡ ¸ Abstract This paper provides a survey of recent progress and software for solving convex Mixed Integer Nonlinear Programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality... vii Part I: Convex MINLP Algorithms and software for convex mixed integer nonlinear programs 1 Pierre Bonami, Mustafa Kilin¸, and Jeff Linderoth c Subgradient based outer approximation for mixed integer second order cone programming 41 Sarah Drewes and Stefan Ulbrich... and Andreas W¨chter a Global optimization of mixed- integer signomial programming problems 349 Andreas Lundell and Tapio Westerlund Part VI: Mixed- Integer Quadraticaly Constrained Optimization The MILP road to MIQCP 373 Samuel Burer and Anureet Saxena Linear programming relaxations of quadratically constrained... Gill and Elizabeth Wong Using interior-point methods within an outer approximation framework for mixed integer nonlinear programming 225 Hande Y Benson Part IV: Expression Graphs Using expression graphs in optimization algorithms 247 David M Gay Symmetry in mathematical programming 263 Leo Liberti xv www.it-ebooks.info xvi CONTENTS Part... surveys results in mixed- integer quadratically constrained programming Strong convex relaxations and valid inequalities are the basis of efficient, practical techniques for global optimization Some of the relaxations and inequalities are derived from the algebraic formulation, while others are based on disjunctive programming Much of the inspiration derives from MILP methodology Linear programming relaxations . Mathematics and its Applications MIXED INTEGER NONLINEAR PROGRAMMING contains expository and research papers based on a highly successful IMA Hot Topics Workshop Mixed- Integer Nonlinear Optimization:. n-fold integer programming (S. Onn) is an overview of the theory of n-fold integer programming, which enables the polynomial-time solution of fundamental linear and nonlinear inte- ger programming. software for convex mixed integer nonlinear programs 1 Pierre Bonami, Mustafa Kilin¸c, and Jeff Linderoth Subgradient based outer approximation for mixed integer second order cone programming 41 Sarah