A Computational Study of Perspective Cuts

TL;DR

This paper conducts a comprehensive computational analysis of perspective cuts in mixed-integer nonlinear programming, extending their application from convex to nonconvex cases and evaluating their impact on solver performance.

Contribution

It introduces a method to apply perspective cuts to nonconvex nonlinearities and assesses their effectiveness in general-purpose solvers through extensive experiments.

Findings

Perspective cuts improve solver performance for convex constraints.

Applying perspective cuts to nonconvex constraints reduces branch-and-bound tree sizes.

Nonconvex perspective cuts strengthen the root relaxation without significantly affecting overall time.

Abstract

The benefits of cutting planes based on the perspective function are well known for many specific classes of mixed-integer nonlinear programs with on/off structures. However, we are not aware of any empirical studies that evaluate their applicability and computational impact over large, heterogeneous test sets in general-purpose solvers. This paper provides a detailed computational study of perspective cuts within a linear programming based branch-and-cut solver for general mixed-integer nonlinear programs. Within this study, we extend the applicability of perspective cuts from convex to nonconvex nonlinearities. This generalization is achieved by applying a perspective strengthening to valid linear inequalities which separate solutions of linear relaxations. The resulting method can be applied to any constraint where all variables appearing in nonlinear terms are semi-continuous and depend on at least one common indicator variable. Our computational experiments show that adding perspective cuts for convex constraints yields a consistent improvement of performance, and adding perspective cuts for nonconvex constraints reduces branch-and-bound tree sizes and strengthens the root node relaxation, but has no significant impact on the overall mean time.