Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

• Published in: Mathematical programming Vol. 158; no. 1-2; pp. 235 - 266
• Main Authors: Billionnet, Alain, Elloumi, Sourour, Lambert, Amélie
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2016; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Branch & bound algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computer Science; Full Length Paper; Integer programming; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Operations Research; Optimization techniques; Studies; Theoretical; Variables; 90C20; Semidefinite programming; Branch-and-bound algorithm; 90C11; Integer quadratic programming; Equivalent convex reformulation; semidefinite programming; branch-and-bound algorithm
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-015-0921-2

We propose a solution approach for the general problem ( QP ) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem ( QP ) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of ( QP ). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.