Parallel Contraction of Variable Space for Global Solution of Nlps

This paper presents a parallel algorithm for obtaining global solutions to general mathematical programming problems with nonconvex constraints involving continuous variables. The proposed algorithm implements an optimization based bound tightening technique (Smith [1996], Ryoo and Sahinidis [1995], Adjiman et al. [2000]) in parallel on the root node of the branch-and-bound tree structure. Upon obtaining the convex relaxation of the original nonconvex nonlinear problem, it may be possible to tighten the bounds on any variable by solving two convex optimization problems. The proposed algorithm is implemented on a Heat Exchanger Network Synthesis problem (Yee and Grossmann [1990]). Computational results demonstrate that variable contraction at the root node can result in a substantial decrease in the number of partitions created when performing the branch-and-reduce global optimization algorithm. Additionally, the solution time may decrease significantly when this variable contraction is done in parallel. The proposed parallel algorithm is implemented in multiple ways for comparison.