Global Solutions of Nonconvex Standard Quadratic Programs via Mixed Integer Linear Programming Reformulations

JACEK GONDZIO(J.Gondzioed.ac.uk)
E. ALPER YILDIRIM(alperyildirimku.edu.tr)

Abstract: A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative mixed integer linear programming formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of orders of magnitude.

Keywords: Nonconvex optimization, quadratic programming, mixed integer linear programming, global optimization

Category 1: Nonlinear Optimization (Quadratic Programming )

Category 2: Integer Programming ((Mixed) Integer Linear Programming )

Citation: Technical Report ERGO-18-022, University of Edinburgh, Edinburgh, Scotland, United Kingdom

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Entry Submitted: 10/04/2018
Entry Accepted: 10/04/2018
Entry Last Modified: 10/04/2018

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