Perspective reformulations of mixed integer nonlinear programs with indicator variables

We study mixed integer nonlinear programs (MINLP)s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume fixed values, and, when...

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Bibliographic Details
Published in:Mathematical programming Vol. 124; no. 1-2; pp. 183 - 205
Main Authors: Günlük, Oktay, Linderoth, Jeff
Format: Journal Article Conference Proceeding
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.07.2010
Springer
Springer Nature B.V
Subjects:
Applied sciences
Approximation
Calculus of Variations and Optimal Control; Optimization
Collection
Combinatorics
Exact sciences and technology
Flows in networks. Combinatorial problems
Full Length Paper
Hulls
Hulls (structures)
Indicators
Integer programming
Linear programming
Logistics
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Mixed integer
Nonlinearity
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Operations research
Optimization techniques
Portfolio theory
Studies
Theoretical
Variables
Variance analysis
Perspective functions
90C30
Mixed-integer nonlinear programming
90C11
Lower bound
Convex set
conic programming
Constraint satisfaction
Mixed integer programming
Non linear programming
Quadratic programming
Location problem
Convex programming
Facility location
Integer programming
Separation method
Upper bound
Convex hull
Plant layout
Portfolio management
Analytical function
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We study mixed integer nonlinear programs (MINLP)s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume fixed values, and, when it is “turned on”, forces them to belong to a convex set. Many practical MINLPs contain integer variables of this type. We first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints, and a convex hull description of this set is derived. We then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions. Further, we show that for many classes of problems, the convex hull can be expressed via conic quadratic constraints, and thus relaxations can be solved via second-order cone programming. Our work is closely related with the earlier work of Ceria and Soares (Math Program 86:595–614, 1999) as well as recent work by Frangioni and Gentile (Math Program 106:225–236, 2006) and, Aktürk et al. (Oper Res Lett 37:187–191, 2009). Finally, we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role. In particular, we present computational results for three applications—quadratic facility location, network design with congestion, and portfolio optimization with buy-in thresholds—that show the power of the reformulation technique.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-010-0360-z