Global Quadratic Optimization via Conic Relaxation

We present a convex conic relaxation for a problem of maximizing an indefinite quadratic form over a set of convex constraints on the squared variables. We show that for all these problems we get at least 12 37 -relative accuracy of the approximation. In the second part of the paper we derive the conic relaxation by another approach based on the second order optimality conditions. We show that for lp-balls, p ≥ 2, intersected by a linear subspace, it is possible to guarantee (1− 2 p)-relative accuracy of the solution. As a consequence, we prove (1− 1 e lnn)-relative accuracy of the conic relaxation for an indefinite quadratic maximization problem over an n-dimensional unit box with homogeneous linear equality constraints. We discuss the implications of the results for the discussion around the question P = NP . ∗CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium; e-mail: [email protected]