Strengthening of lower bounds in the global optimization of Bilinear and Concave Generalized Disjunctive Programs

This paper is concerned with global optimization of Bilinear and Concave Generalized Disjunctive Programs. A major objective is to propose a procedure to find relaxations that yield strong lower bounds. We first present a general framework for obtaining a hierarchy of linear relaxations for nonconvex Generalized Disjunctive Programs (GDP). This framework combines linear relaxation strategies proposed in the literature for nonconvex MINLPs with the results of the work by Sawaya and Grossmann (2009) for Linear GDPs. We further exploit the theory behind Disjunctive Programming by proposing several rules to guide more efficiently the generation of relaxations by considering the particular structure of the problems. Finally, we show through a set of numerical examples that these new relaxations can substantially strengthen the lower bounds for the global optimum, often leading to a significant reduction of the number of nodes when used within a spatial branch and bound framework.