Calculus of Variations and Geometric Measure Theory

A. De Rosa - A. Khajavirad

Explicit convex hull description of bivariate quadratic sets with indicator variables

created by derosa on 20 Aug 2022



Inserted: 20 aug 2022

Year: 2022

ArXiv: 2208.08703 PDF


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.