A hierarchy of relaxations for nonlinear convex generalized disjunctive programming

作者：Ruiz Juan P; Grossmann Ignacio E^*

来源：European Journal of Operational Research, 2012, 218(1): 38-47.

DOI：10.1016/j.ejor.2011.10.002

• Combinatorial optimization
• Convex programming
• Disjunctive programming
• Generalized disjunctive programming
• Tight relaxations

We propose 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. We prove that the tightest of these relaxations, allows in theory the solution of the disjunctive convex program as a nonlinear programming problem. We present a methodology to guide the generation of strong relaxations without incurring an exponential increase of the size of the reformulated mixed-integer program. Finally, we apply the theory developed to improve the computational efficiency of solution methods for nonlinear convex generalized disjunctive programs (GDP). This methodology is validated through a set of numerical examples.

• 出版日期2012-4-1