Solving linear programs with complementarity constraints using branch-and-cut

A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bi...

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Bibliographic Details
Published in:Mathematical programming computation Vol. 11; no. 2; pp. 267 - 310
Main Authors: Yu, Bin, Mitchell, John E., Pang, Jong-Shi
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2019
Springer Nature B.V
Subjects:
Algorithms
Computation
Full Length Paper
Game theory
Integer programming
Linear programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Operations Research/Decision Theory
Optimization
Theory of Computation
Linear programs with complementarity constraints
MPECs
90C57
90C33
65K10
90C26
Branch-and-cut
ISSN:1867-2949, 1867-2957
Online Access:Get full text
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Summary:A linear program with linear complementarity constraints (LPCC) requires the minimization of a linear objective over a set of linear constraints together with additional linear complementarity constraints. This class has emerged as a modeling paradigm for a broad collection of problems, including bilevel programs, Stackelberg games, inverse quadratic programs, and problems involving equilibrium constraints. The presence of the complementarity constraints results in a nonconvex optimization problem. We develop a branch-and-cut algorithm to find a global optimum for this class of optimization problems, where we branch directly on complementarities. We develop branching rules and feasibility recovery procedures and demonstrate their computational effectiveness in a comparison with CPLEX. The implementation builds on CPLEX through the use of callback routines. The computational results show that our approach is a strong alternative to constructing an integer programming formulation using big- M terms to represent bounds for variables, with testing conducted on general LPCCs as well as on instances generated from bilevel programs with convex quadratic lower level problems.
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ISSN:1867-2949
1867-2957
DOI:10.1007/s12532-018-0149-2