Explicit convex hull description of bivariate quadratic sets with indicator variables

TL;DR

This paper provides an explicit convex hull description for a specific bivariate quadratic set with indicator variables, aiding in solving related nonconvex optimization problems more effectively.

Contribution

It introduces an explicit convex hull description for the bivariate quadratic set with indicator variables and a separation algorithm to improve relaxations in optimization.

Findings

Explicit convex hull description for the set $\

A separation algorithm for supporting hyperplanes is developed.

Enhanced relaxation techniques for nonconvex problems are proposed.

Abstract

We consider the nonconvex set $\mathcal S_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \geq 0,\; z \in \{0,1\}^n\}$, which is closely related to the feasible region of several difficult nonconvex optimization problems such as the best subset selection and constrained portfolio optimization. Utilizing ideas from convex analysis and disjunctive programming, we obtain an explicit description for the closure of the convex hull of $\mathcal S_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of $\mathcal S_2$, we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.