Explicit convex hull description of bivariate quadratic sets with indicator variables

We consider the nonconvex set \({{\mathcal {S}}}_n = \{(x,X,z): X = x x^T, \; x (1-z) =0,\; x \ge 0,\; z \in \{0,1\}^n\}\) ${\mathcal{S}}_n=\{(x,X,z):X=x{x}^T,x(1-z)=0,x\ge 0,z\in {\{0,1\}}^n\}$ , 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 \({{\mathcal {S}}}_2\) ${\mathcal{S}}_2$ in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of \({{\mathcal {S}}}_2\) ${\mathcal{S}}_2$ , we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.