Global Nonconvex Optimization with Integer Variables

TL;DR

This paper extends a novel nonconvex optimization technique to integer variables, demonstrating competitive performance and often outperforming leading solvers like BARON and SCIP on challenging problems.

Contribution

The paper introduces an extended branch-and-bound algorithm using the Relaxation Perspectification Technique for integer and mixed-integer nonconvex problems.

Findings

The method often outperforms BARON in computational time and optimality.

The approach successfully solves problems that SCIP and BARON cannot within time limits.

Strong performance demonstrated on real-world and benchmark nonconvex problems.

Abstract

Nonconvex optimization refers to the process of solving problems whose objective or constraints are nonconvex. Historically, this type of problems have been very difficult to solve to global optimality, with traditional solvers often relying on approximate solutions. Bertsimas et al. introduce a novel approach for solving continuous nonconvex optimization problems to provable optimality, called the Relaxation Perspectification Technique - Branch and Bound (RPT-BB). In this paper, we extend the RPT-BB approach to the binary, mixed-binary, integer, and mixed-integer variable domains. We outline a novel branch-and-bound algorithm that makes use of the Relaxation Perspectification Technique (RPT), as well as binary, integer, and eigenvector cuts. We demonstrate the performance of this approach on four representative nonconvex problems, as well as one real-world nonconvex optimization problem, and we benchmark its performance on BARON and SCIP, two state-of-the-art optimization solvers for nonconvex mixed-integer problems. Our results show that our method stands well against BARON, and often outperforms BARON, in terms of computational time and optimal objective value. Moreover, our results show that, while SCIP continues to lead as the state-of-the-art solver, the proposed algorithm demonstrates strong performance on challenging instances, successfully solving problems to global optimality that SCIP and BARON are unable to solve within the time limit.