On the convex hull of convex quadratic optimization problems with indicators

TL;DR

This paper characterizes the convex hull of convex quadratic problems with indicator variables using extended formulations, semidefinite constraints, and finitely generated conic inequalities, advancing the understanding of mixed-integer nonlinear sets.

Contribution

It provides a unified convex hull description for quadratic problems with indicators, including a compact MILP formulation and conic inequalities, linking polyhedral and semidefinite approaches.

Findings

Convex hull described by a single semidefinite and linear constraints.

A compact MILP formulation matches the convex hull vertices.

Finitely generated conic inequalities characterize the convex hull in original space.

Abstract

We consider the convex quadratic optimization problem with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of a single positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are ``finitely generated." In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.