Tightening McCormick Relaxations for Nonlinear Programs via Dynamic Multivariate Partitioning

TL;DR

This paper introduces a two-stage method combining constraint programming and dynamic multivariate partitioning to tighten McCormick relaxations in MINLPs, improving computational efficiency and solution quality.

Contribution

It presents a novel dynamic partitioning scheme that creates sparser, tighter relaxations for multi-linear terms, reducing computation time and limiting binary variables.

Findings

Significant reduction in computation time.

Tighter relaxations lead to better bounds.

Effective bound tightening with CP techniques.

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.