An Adaptive, Multivariate Partitioning Algorithm for Global Optimization of Nonconvex Programs

TL;DR

This paper introduces an adaptive multivariate partitioning algorithm that efficiently solves nonconvex MINLPs to global optimality by leveraging disjunctive relaxations and adaptive domain partitioning, outperforming traditional methods on benchmark problems.

Contribution

The paper presents a novel adaptive partitioning algorithm that combines disjunctive formulations with domain partitioning to improve global optimization of nonconvex MINLPs, with proven convergence guarantees.

Findings

Successfully solves large-scale, previously intractable instances.

Reduces optimality gap in challenging pooling problem.

Outperforms existing global solvers on benchmark problems.

Abstract

In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of piecewise polyhedral relaxation approaches via disjunctive formulations to solve MINLPs to global optimality in contrast to the conventional spatial branch-and-bound approaches. In order to maintain relatively small-scale mixed-integer linear programs at every iteration of the algorithm, we adaptively partition the variable domains appearing in the multi-linear terms. We also provide proofs on convergence guarantees of the proposed algorithm to a global solution. Further, we discuss a few algorithmic enhancements based on the sequential bound-tightening procedure as a presolve step, where we observe the importance of solving piecewise relaxations compared to basic convex relaxations to speed-up the convergence of the algorithm to global optimality. We demonstrate the effectiveness of our disjunctive formulations and the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare with state-of-the-art global optimization solvers. With this novel approach, we solve several large-scale instances which are, in some cases, intractable by the global optimization solver. We also shrink the best known optimality gap for one of the hard, generalized pooling problem instance.