On convex quadratic programs with linear complementarity constraints

The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 54; no. 3; pp. 517 - 554
Main Authors: Bai, Lijie, Mitchell, John E., Pang, Jong-Shi
Format: Journal Article
Language:English
Published: Boston Springer US 01.04.2013
Springer Nature B.V
Subjects:
Bending machines
Computation
Convex and Discrete Geometry
Grants
Logic programming
Lower bounds
Management Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Mixed integer
Operations Research
Operations Research/Decision Theory
Optimization
Quadratic programming
Statistics
Variables
Convex quadratic programming
Logical Benders decomposition
Semidefinite programming
Satisfiability constraints
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:The paper shows that the global resolution of a general convex quadratic program with complementarity constraints (QPCC), possibly infeasible or unbounded, can be accomplished in finite time. The method constructs a minmax mixed integer formulation by introducing finitely many binary variables, one for each complementarity constraint. Based on the primal-dual relationship of a pair of convex quadratic programs and on a logical Benders scheme, an extreme ray/point generation procedure is developed, which relies on valid satisfiability constraints for the integer program. To improve this scheme, we propose a two-stage approach wherein the first stage solves the mixed integer quadratic program with pre-set upper bounds on the complementarity variables, and the second stage solves the program outside this bounded region by the Benders scheme. We report computational results with our method. We also investigate the addition of a penalty term y T Dw to the objective function, where y and w are the complementary variables and D is a nonnegative diagonal matrix. The matrix D can be chosen effectively by solving a semidefinite program, ensuring that the objective function remains convex. The addition of the penalty term can often reduce the overall runtime by at least 50 %. We report preliminary computational testing on a QP relaxation method which can be used to obtain better lower bounds from infeasible points; this method could be incorporated into a branching scheme. By combining the penalty method and the QP relaxation method, more than 90 % of the gap can be closed for some QPCC problems.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-012-9497-4