Using general triangle inequalities within quadratic convex reformulation method

Abstract

We consider the exact solution of Problem (P) which consists in minimizing a quadratic function subject to quadratic constraints. We start with an explicit description of new general triangle inequalities that are derived from the ranges of the variables of (P). We show that they extend the triangle inequalities, introduced for the binary case, to variables that belong to a generic interval. We also prove that these inequalities cut feasible solutions of McCormick envelopes, and we relate them to the literature. We then introduce (SDP), a strong semidefinite relaxation of (P), that we call ‘Shor's plus RLT plus Triangle’, which includes both the McCormick envelopes and the general triangle inequalities. We further show how to compute a convex relaxation whose optimal value reaches the value of (SDP). In order to handle these inequalities in the solution of (SDP), we solve it by a heuristic that also serves as a separation algorithm. We then solve (P) to global optimality with a branch-and-bound based on . Finally, we show that our method outperforms the compared solvers.