Constructing Tight Quadratic Relaxations for Global Optimization: II. Underestimating Difference-of-Convex (D.C.) Functions

TL;DR

This paper extends quadratic relaxation techniques to non-convex difference-of-convex functions, resulting in significantly tighter bounds for global optimization problems, especially in QCQP and D.C. functions.

Contribution

It introduces a hierarchy of quadratic underestimators for D.C. functions, improving the tightness of convex relaxations in global optimization.

Findings

Notable reduction in the hypervolume gap between underestimators and linear bounds.

Quadratic relaxations at the root node reduce the gap to the optimal solution by over 90%.

Demonstrated effectiveness on benchmark D.C. functions and optimization problems.

Abstract

Recent advances in the efficiency and robustness of algorithms solving convex quadratically constrained quadratic programming (QCQP) problems motivate developing techniques for creating convex quadratic relaxations that, although more expensive to compute, provide tighter bounds than their classical linear counterparts. In the first part of this two-paper series [Strahl et al., 2024], we developed a cutting plane algorithm to construct convex quadratic underestimators for twice-differentiable convex functions, which we extend here to address the case of non-convex difference-of-convex (d.c.) functions as well. Furthermore, we generalize our approach to consider a hierarchy of quadratic forms, thereby allowing the construction of even tighter underestimators. On a set of d.c. functions extracted from benchmark libraries, we demonstrate noteworthy reduction in the hypervolume between our quadratic underestimators and linear ones constructed at the same points. Additionally, we construct convex QCQP relaxations at the root node of a spatial branch-and-bound tree for a set of systematically created d.c. optimization problems in up to four dimensions, and we show that our relaxations reduce the gap between the lower bound computed by the state-of-the-art global optimization solver BARON and the optimal solution by an excess of 90%, on average.