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Solving Linear Programs with Complementarity Constraints using Branch-and-Cut

Bin Yu(binyu610***at***gmail.com)
John E. Mitchell(mitchj***at***rpi.edu)
Jong-Shi Pang(jongship***at***usc.edu)

Abstract: 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.

Keywords: linear programs with complementarity constraints, MPECs, branch and cut

Category 1: Complementarity and Variational Inequalities

Category 2: Combinatorial Optimization (Branch and Cut Algorithms )

Citation: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180. October 28, 2016.

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Entry Submitted: 10/28/2016
Entry Accepted: 10/28/2016
Entry Last Modified: 10/28/2016

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