Concave tents: a new tool for constructing concave reformulations of a large class of nonconvex optimization problems

Abstract

Optimizing a nonlinear function over nonconvex sets is challenging since solving convex relaxations may lead to substantial relaxation gaps and infeasible solutions that must be "rounded" to feasible ones, often with uncontrollable losses in objective function performance. For this reason, these convex hulls are especially useful if the objective function is linear or even concave, since concave optimization is invariant to taking the convex hull of the feasible set. We propose the notion of concave tents, which are concave approximations of the original objective function that agree with this objective function on the feasible set, and allow for concave reformulations of the problem. Concave tents, therefore, are special cases of concave extensions. In this text, we derive such concave tents for a large class of objective functions as the optimal value functions of conic optimization problems. Hence, evaluating our concave tents requires solving a conic problem. We can find supergradients by solving the conic dual problem, so that differentiation is of the same complexity as evaluation. For feasible sets that are contained in the extreme points of their convex hull, we construct these concave tents in the original space of variables. For general feasible sets, we propose a double lifting strategy, where the original optimization problem is lifted into a higher-dimensional space in which the concave tent can be constructed easily. We investigate the relation of the so-constructed concave tents to concave envelopes and naive concave tents based on concave quadratic updates. Based on these ideas, we propose a primal heuristic for a class of robust discrete quadratic optimization problems that can be used instead of classical rounding techniques. Numerical experiments suggest that our techniques can be beneficial as an upper-bounding procedure in a branch-and-bound solution scheme.