Eigenvalue Techniques for Convex Objective, Nonconvex Optimization Problems

Abstract

A fundamental difficulty when dealing with a minimization problem given by a nonlinear, convex objective function over a nonconvex feasible region, is that even if we can efficiently optimize over the convex hull of the feasible region, the optimum will likely lie in the interior of a high dimensional face, “far away” from any feasible point, yielding weak bounds. We present theory and implementation for an approach that relies on (a) the S-lemma, a major tool in convex analysis, (b) efficient projection of quadratics to lower dimensional hyperplanes, and (c) efficient computation of combinatorial bounds for the minimum distance from a given point to the feasible set, in the case of several significant optimization problems. On very large examples, we obtain significant lower bound improvements at a small computational cost.