Constructing Tight Quadratic Relaxations for Global Optimization: I. Outer-Approximating Twice-Differentiable Convex Functions

TL;DR

This paper introduces a novel cutting-plane algorithm for constructing the tightest quadratic underestimators of twice-differentiable convex functions, improving bounds in global optimization by leveraging recent advances in quadratic programming.

Contribution

It develops a new method to generate the tightest quadratic outer approximations for convex functions, enhancing the accuracy of relaxations in global optimization.

Findings

The algorithm produces significantly tighter quadratic underestimators.

The approach improves computational efficiency in solving global optimization problems.

Allowing overestimation in infeasible regions further tightens the relaxations.

Abstract

When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering that linear outer approximations sacrifice accuracy when approximating highly nonlinear functions and recognizing the recent advancements in the efficiency and robustness of available methods to solve optimization problems with quadratic objectives and constraints, we contemplate here the construction of quadratic outer approximations of twice-differentiable convex functions for use in deterministic global optimization. To this end, we present a novel cutting-plane algorithm that determines the tightest scaling parameter, $\alpha$, in the second-order Taylor series approximation quadratic underestimator proposed by Su et al. We use a representative set of convex functions extracted from optimization benchmark libraries to showcase--qualitatively and quantitatively--the tightness of the constructed quadratic underestimators and to demonstrate the overall computational efficiency of our algorithm. Furthermore, we extend our construction procedure to generate even tighter quadratic underestimators by allowing overestimation in infeasible polyhedral regions of optimization problems, as informed by the latter's linear constraints.