A joint decomposition method for global optimization of multiscenario nonconvex mixed-integer nonlinear programs

TL;DR

This paper introduces a joint decomposition approach combining Lagrangian and generalized Benders methods to efficiently solve multiscenario nonconvex MINLP problems to global optimality, avoiding extensive branch and bound searches.

Contribution

The paper presents a novel unified framework that integrates two decomposition techniques for faster global optimization of complex multiscenario nonconvex MINLPs.

Findings

Method solves problems significantly faster than existing solvers.

Reduces the number of relaxed master problems needed.

Eases solution of each relaxed master problem.

Abstract

This paper proposes a joint decomposition method that combines La- grangian decomposition and generalized Benders decomposition, to efficiently solve multiscenario nonconvex mixed-integer nonlinear programming (MINLP) problems to global optimality, without the need for explicit branch and bound search. In this approach, we view the variables coupling the scenario dependent variables and those causing nonconvexity as complicating variables. We systemat- ically solve the Lagrangian decomposition subproblems and the generalized Ben- ders decomposition subproblems in a unified framework. The method requires the solution of a difficult relaxed master problem, but the problem is only solved when necessary. Enhancements to the method are made to reduce the number of the relaxed master problems to be solved and ease the solution of each relaxed master problem. We consider two scenario-based, two-stage stochastic nonconvex MINLP problems that arise from integrated design and operation of process net- works in the case study, and we show that the proposed method can solve the two problems significantly faster than state-of-the-art global optimization solvers.