Tightening McCormick Relaxations for Nonlinear Programs via Dynamic Multivariate Partitioning

Abstract: In this work, we propose a two-stage approach to strengthen piecewise McCormick relaxations for mixed-integer nonlinear programs (MINLP) with multi-linear terms. In the first stage, we exploit Constraint Programing (CP) techniques to contract the variable bounds. In the second stage we partition the variables domains using a dynamic multivariate partitioning scheme. Instead of equally partitioning the domains of variables appearing in multi-linear terms, we construct sparser partitions yet tighter relax- ations by iteratively partitioning the variable domains in regions of interest. This approach decouples the number of partitions from the size of the variable domains, leads to a significant reduction in computation time, and limits the number of binary variables that are introduced by the partitioning. We demonstrate the performance of our algorithm on well-known benchmark problems from MINLPLIB and discuss the computational benefits of CP-based bound tightening procedures.