Tight Relaxations for Nonconvex Optimization Problems Using the Reformulation-Linearization/Convexification Technique (RLT)

Abstract

This paper provides an expository discussion on applying the Reformulation-Linearization/Convexification Technique (RLT) to design exact and approximate methods for solving nonconvex optimization problems. While the main focus here is on continuous nonconvex programs, we demonstrate that this approach provides a unifying framework that can accommodate discrete problems such as linear zero-one mixed-integer programming problems and general bounded-variable integer programs as well. The basic RLT approach is designed to solve polynomial programming problems having nonconvex polynomial objective functions and constraints. The principal RLT construct for such problems is to first reformulate the problem by adding a suitable set of polynomial constraints and then to linearize this resulting problem through a variable substitution process in order to produce a tight higher dimensional linear programming relaxation. Various additional classes of valid inequalities, filtered through a constraint selection scheme or a separation routine, are proposed to further tighten the developed relaxation while keeping it manageable in size. This relaxation can be embedded in a suitable branch-and-bound scheme in order to solve the original problem to global optimality via a sequence of linear programming problems. Because of the tight convexification effect of this technique, a limited enumeration or even the use of a local search method applied to the initial relaxation’s solution usually serves as an effective heuristic strategy. We also discuss extensions of this approach to handle polynomial programs having rational exponents, as well as general factorable programming problems, and we provide recommendations for using RLT to solve even more general unstructured classes of nonconvex programming problems. Some special case applications to solve squared-Euclidean, Euclidean, or l p -distance location-allocation problems are discussed to illustrate the RLT methodology.