A note on quadratic constraints with indicator variables: Convex hull description and perspective relaxation

TL;DR

This paper analyzes the convex hull of a mixed-integer nonlinear set with quadratic constraints and indicator variables, revealing its structure and the effectiveness of perspective relaxation as an approximation.

Contribution

It characterizes the convex hull of the set and evaluates the approximation quality of perspective relaxation, providing insights into its structure and computational complexity.

Findings

Optimization over the set is NP-hard.

The convex hull can be characterized using polyhedral theory.

Perspective relaxation provides a close approximation to the convex hull.

Abstract

In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we characterize the structure of the convex hull, and show that it can be formally studied using polyhedral theory. Moreover, we show that although perspective relaxation in the literature for this set fails to match the structure of its convex hull, it is guaranteed to be a close approximation.