Tighter McCormick Relaxations through Subgradient Propagation

TL;DR

This paper introduces a heuristic method using subgradient propagation to tighten McCormick relaxations, leading to more accurate bounds and improved computational efficiency in global optimization tasks.

Contribution

It proposes a novel heuristic that enhances McCormick relaxations by using subgradient propagation to obtain tighter bounds, with demonstrated improvements on benchmark problems.

Findings

Significant tightening of relaxations observed in benchmark tests.

Reduction in computational time for many cases.

Heuristic provides notable improvements despite no guaranteed success.

Abstract

Tight convex and concave relaxations are of high importance in the field of deterministic global optimization. We present a heuristic to tighten relaxations obtained by the McCormick technique. We use the McCormick subgradient propagation (Mitsos et al., SIAM J. Optim., 2009) to construct simple affine under- and overestimators of each factor of the original factorable function. Then, we minimize and maximize these affine relaxations in order to obtain possibly improved range bounds for every factor resulting in possibly tighter final McCormick relaxations. We discuss the heuristic and its limitations, in particular the lack of guarantee for improvement. Subsequently, we provide numerical results for benchmark cases found in the COCONUT library and case studies presented in previous works and discuss computational efficiency. We see that the presented heuristic provides a significant improvement in tightness and decrease in computational time in many cases.