Global solution of non-convex quadratically constrained quadratic programs

Abstract

The class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QCQPs with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let $\left(P\right)$ be a QCQP. Our approach to solve $\left(P\right)$ is first to build an equivalent mixed-integer quadratic problem $\left(P^{\ast }\right)$. This equivalent problem $\left(P^{\ast }\right)$ has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints $y=x{x}^{\mathrm{T}}$, where x are the initial variables of problem $\left(P\right)$. We then propose to solve $\left(P^{\ast }\right)$ by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO.

Keywords:

• general mixed-integer quadratic programming
• global optimization
• spatial branch-and-bound
• quadratic convex relaxation
• experiments

Acknowledgements

The authors are grateful to Alain Billionnet for all our collaborations and for a careful reading of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental data for this article can be accessed at https://doi.org/10.1080/10556788.2017.1350675.

Additional information

Funding

This research was partially funded by the Fondation Mathématique Jacques Hadamard (FMJH) through the Gaspard Monge Program for Optimization and operations research (PGMO).