Extended McCormick relaxation rules for handling empty arguments representing infeasibility

Abstract

McCormick’s relaxation technique is one of the most versatile and commonly used methods for computing the convex relaxations necessary for deterministic global optimization. The core of the method is a set of rules for propagating relaxations through basic arithmetic operations. Computationally, each rule operates on four-tuples describing each input argument in terms of a lower bound value, an upper bound value, a convex relaxation value, and a concave relaxation value. We call such tuples McCormick objects. This paper extends McCormick’s rules to accommodate input objects that are empty (i.e., the convex relaxation value lies above the concave, or both relaxation values lie outside the bounds). Empty McCormick objects provide a natural way to represent infeasibility and are readily generated by McCormick-based domain reduction techniques. The standard McCormick rules are strictly undefined for empty inputs and applying them anyway can yield relaxations that are non-convex/concave on infeasible parts of their domains. In contrast, our extended rules always produce relaxations that are well-defined and convex/concave on their entire domain. This capability has important applications in reduced-space global optimization, global dynamic optimization, and domain reduction.