Quadratic Optimization with Switching Variables: The Convex Hull for $n = 2$

TL;DR

This paper derives an exact convex hull representation for quadratic optimization problems with switching variables when n=2, extending known results from the case n=1 and proposing a simplified conjecture.

Contribution

It provides the first exact convex hull characterization for n=2 with switching variables, building on disjunctive representations and eliminating auxiliary variables.

Findings

Exact convex hull for n=2 derived

Disjunctive representation used for derivation

Conjecture proposed for simplified hull when ignoring y1y2

Abstract

We consider quadratic optimization in variables $(x,y)$ where $0\le x\le y$, and $y\in\{0,1\}^n$. Such binary $y$ are commonly refered to as "indicator" or "switching" variables and occur commonly in applications. One approach to such problems is based on representing or approximating the convex hull of the set $\{ (x,xx^T, yy^T) : 0\le x\le y\in\{0,1\}^n\}$. A representation for the case $n=1$ is known and has been widely used. We give an exact representation for the case $n=2$ by starting with a disjunctive representation for the convex hull and then eliminating auxilliary variables and constraints that do not change the projection onto the original variables. An alternative derivation for this representation leads to an appealing conjecture for a simplified representation of the convex hull for $n=2$ when the product term $y_1y_2$ is ignored.