Product Disaggregation in Global Optimization and Relaxations of Rational Programs

Synopsis

In this chapter, we consider the product of a single continuous variable and the sum of a number of continuous variables. We show that “product disaggregation” (distributing the product over the sum) leads to tighter linear programming relaxations, much like variable disaggregation does in mixedinteger linear programming. We also derive closed-form expressions characterizing the exact region over which these relaxations improve when the bounds of participating variables are reduced.

In a concrete application of product disaggregation, we develop and analyze linear programming relaxations of rational programs. In the process of doing so, we prove that the task of bounding general linear fractional functions of 0–1 variables is NP-hard. Then, we present computational experience to demonstrate that product disaggregation is a useful reformulation technique for global optimization problems. In particular, we apply product disaggregation to develop eight different mixed-integer convex programming reformulations of 0–1 hyperbolic programs. We obtain analytical results on the relative tightness of these formulations and propose a branch-and-bound algorithm for 0–1 hyperbolic programs. The algorithm is used to solve a discrete p-choice facility location problem for locating ten restaurants in the city of Edmonton.