Eigenvalue Techniques for Convex Objective, Nonconvex Optimization Problems

Consider a minimization problem given by a nonlinear, convex objective function over a nonconvex feasible region. Traditional optimization approaches will frequently encounter a fundamental difficulty when dealing with such problems: 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. As a result (and in particular, because of the nonconvex objective) the lower bound provided by a convex relaxation will typically be extremely poor. Furthermore, we will tend to see very large branch-and-bound (or -cut) trees with little or no improvement over the lower bound. In this work we present theory and implementation for an approach that relies on three ingredients: (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 signficant optimization problems. Altogether, our approach strongly improves lower bounds at a small computational cost, even in very large examples.