A piecewise-quadratic convexification for exactly solving box-constrained quadratic programs
Résumé
This work aims at finding the global minimum of a quadratic function f with box constrained variables. For this goal, we construct a piecewise-quadratic convex relaxation of f defined as the maximum of k ≥ 1 convex functions. We show that when k → ∞ the optimal solution of this relaxation converges to an optimal solution of the SDP relaxation of the initial problem. The resulting convexification is tighter that the one produced by previous related methods that use k = 1. Its integration into a spatial branch-and-bound algorithm brings a second advantage : compared to previous related methods, it can refine the bound at each node by computing new convex functions specifically tailored to act on the considered node. Numerical results suggest that our algorithm can outperform different solvers or other methods based on quadratic convex relaxations.
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