Extending the QCR method to general mixed-integer programs

• Published in: Mathematical programming Vol. 131; no. 1-2; pp. 381 - 401
• Main Authors: Billionnet, Alain, Elloumi, Sourour, Lambert, Amélie
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.02.2012; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Eigenvalues; Equivalence; Exact sciences and technology; Full Length Paper; Inequalities; Inequality; Integer programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical models; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Mixed integer; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Optimization techniques; Quadratic programming; Semidefinite programming; Solvers; Studies; Theoretical; General integer programming; 90C22 Semidefinite programming; 90C11 Mixed integer programming; 90C20 Quadratic programming; Mixed-integer programming; Convex reformulation; Quadratic programming; Experiments; Semi-definite programming; 90C26 Nonconvex programming; Non convex programming; Semi definite programming; Convex programming; Relaxation; Inequality constraint; Relaxation method; Mathematical programming; Branch and bound method; Minimization; Mixed integer programming; Mixed method; Standards; Integer programming; Computation time; Implicit enumeration method; Equality constraint; Objective function; Quadratic function
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-010-0381-7

Let ( MQP ) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of ( MQP ), i.e. we reformulate ( MQP ) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of ( MQP ). Computational experiences are carried out with instances of ( MQP ) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in 1 h of CPU time by a standard solver.