A computational study for bilevel quadratic programs using semidefinite relaxations

Abstract

In this paper, we deal with bilevel quadratic programming problems with binary decision variables in the leader problem and convex quadratic programs in the follower problem. For this purpose, we transform the bilevel problems into equivalent quadratic single level formulations by replacing the follower problem with the equivalent Karush Kuhn Tucker (KKT) conditions. Then, we use the single level formulations to obtain mixed integer linear programming (MILP) models and semidefinite programming (SDP) relaxations. Thus, we compute optimal solutions and upper bounds using linear programming (LP) and SDP relaxations. Our numerical results indicate that the SDP relaxations are considerably tighter than the LP ones. Consequently, the SDP relaxations allow finding tight feasible solutions for the problem. Especially, when the number of variables in the leader problem is larger than in the follower problem. Moreover, they are solved at a significantly lower computational cost for large scale instances.