Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

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 formulations. Our first formulation...

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Bibliographic Details
Published in:Journal of global optimization Vol. 81; no. 2; pp. 293 - 321
Main Authors: Gondzio, Jacek, Yıldırım, E. Alper
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2021
Springer
Springer Nature B.V
Subjects:
Computer Science
Integer programming
Investment analysis
Linear programming
Mathematics
Mathematics and Statistics
Mixed integer
Operations Research/Decision Theory
Optimization
Quadratic forms
Real Functions
Tightness
90C20
Global optimization
Nonconvex optimization
90C11
90C26
Mixed integer linear programming
Quadratic programming
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary: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 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 several orders of magnitude.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-021-01017-y