Convex Programming Methods for Global Optimization

Abstract

We describe four approaches to solving nonconvex global optimization problems by convex nonlinear programming methods. It is assumed that the problem becomes convex when selected variables are fixed. The selected variables must be discrete, or else discretized if they are continuous. We first survey some existing methods: disjunctive programming with convex relaxations, logic-based outer approximation, and logic-based Benders decomposition. We then introduce a branch-and-bound method with convex quasi-relaxations (BBCQ) that can be effective when the discrete variables take a large number of real values. The BBCQ method generalizes work of Bollapragada, Ghattas and Hooker on structural design problems. It applies when the constraint functions are concave in the discrete variables and have a weak homogeneity property in the continuous variables.