A hierarchy of relaxations for nonlinear convex generalized disjunctive programming |
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Authors: | Juan P RuizIgnacio E Grossmann |
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Institution: | Carnegie Mellon University, Department of Chemical Engineering, Pittsburgh, PA 15213, United States |
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Abstract: | 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. |
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Keywords: | Combinatorial optimization Convex programming Disjunctive programming Generalized disjunctive programming Tight relaxations |
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