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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 thefeasible region of several difficult nonconvex optimization problems such asthe best subset selection and constrained portfolio optimization. Utilizingideas from convex analysis and disjunctive programming, we obtain an explicitdescription for the closure of the convex hull of $\mathcal S_2$ in the spaceof original variables. In order to generate valid inequalities corresponding tosupporting hyperplanes of the convex hull of $\mathcal S_2$, we present asimple separation algorithm that can be incorporated in branch-and-cut basedsolvers to enhance the quality of existing relaxations.

Explicit convex hull description of bivariate quadratic sets with indicator variables