Relaxation via Separable Estimators: Arithmetic and Implementation

TL;DR

This paper introduces superposition relaxation, a method for tightly bounding multivariate functions with separable under- and overestimators, analyzing its convergence, implementation, and comparison to McCormick relaxations.

Contribution

It develops a new arithmetic for function relaxation that exploits monotonicity and convexity, with convergence analysis and practical parameterizations for implementation.

Findings

Superposition relaxations can be tighter than McCormick relaxations.

The method's local convergence properties are characterized, including quadratic convergence conditions.

Numerical case studies demonstrate the effectiveness and computational trade-offs of the approach.

Abstract

This article presents an arithmetic, called superposition relaxation, for bracketing the graph of a multivariate factorable function on a compact domain between a pair of underestimating and overestimating functions that are both separable. Propagation rules are established for affine and nonlinear composition operations, with a focus on exploiting global monotonicity and convexity properties in the composition. The local convergence properties of this arithmetic are also analyzed in both the pointwise and Hausdorff sense, including conditions under which quadratic pointwise convergence propagates through composition. Parameterizations of the univariate summands in a superposition relaxation either as piecewise-constant or continuous piecewise-linear functions are discussed for a practical implementation. It is shown through numerical case studies that superposition relaxations can be consistently tighter than McCormick relaxations, including for the relaxation of artificial neural networks. But superposition relaxations also incur a higher computational cost than McCormick relaxations. Further investigations are thus warranted as applications in global optimization seek to balance a relaxation's tightness with its computational cost.