A New Theoretical Result for Convex Nonlinear Generalized Disjunctive Programs and its Applications

Abstract

Abstract This paper deals with the theory of reformulations and numerical solution of GDP problems, which are expressed in terms of Boolean and continuous variables, and involve algebraic constraints, disjunctions and propositional logic statements. We describe a framework to generate alternative mixed-integer nonlinear programming formulations for disjunctive convex programs that lead to stronger relaxations. We extend the concept of “basic steps” defined for disjunctive linear programs to the nonlinear case. A basic step is an operation that takes a disjunctive set to another with fewer number of conjuncts. We show that the strength of the relaxations increases as the number of conjuncts decreases, leading to a hierarchy of relaxations. As the main theoretical result of this paper, we prove that the tightest of these relaxations allows the solution of the disjunctive convex program as a nonlinear programming problem. We also present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed-integer program. We apply the theory to improve the computational solution efficiency for nonlinear convex generalized disjunctive programs (GDP). Finally, we briefly discuss how the theory of convex GDP can be applied to nonconvex GDPs that involve for instance bilinear, concave and linear fractional terms.