Generalized Disjunctive Programming: A Framework for Formulation and Alternative Algorithms for MINLP Optimization

Generalized disjunctive programming (GDP) is an extension of the disjunctive programming paradigm developed by Balas. The GDP formulation involves Boolean and continuous variables that are specified in algebraic constraints, disjunctions and logic propositions, which is an alternative representation to the traditional algebraic mixed-integer programming formulation. After providing a brief review of MINLP optimization, we present an overview of GDP for the case of convex functions emphasizing the quality of continuous relaxations of alternative reformulations that include the big-M and the hull relaxation. We then review disjunctive branch and bound as well as logic-based decomposition methods that circumvent some of the limitations in traditional MINLP optimization. We next consider the case of linear GDP problems to show how a hierarchy of relaxations can be developed by performing sequential intersection of disjunctions. Finally, for the case when the GDP problem involves nonconvex functions, we propose a scheme for tightening the lower bounds for obtaining the global optimum using a combined disjunctive and spatial branch and bound search. We illustrate the application of the theoretical concepts and algorithms on several engineering and OR problems.