# Explicit convex hull description of bivariate quadratic sets with indicator variables

arXiv: https://arxiv.org/abs/2208.08703 · HTML: https://mlsanity.com/paper/2208.08703/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Aug 2022.

## Built on by (most cited first, 2 of 2)

- [On the Exactness of Dantzig-Wolfe Relaxation for Rank Constrained Optimization Problems](https://mlsanity.com/paper/2210.16191/) (2022)
- [Learning efficient representations of complex constraints for scalable optimization](https://mlsanity.com/paper/2603.08283/) (2026)

## Abstract (arXiv)

We consider the nonconvex set mathcal Sn = (x,X,z): X = x xT, ; x (1-z) =0,; x ≥ 0,; z ∈ 0,1n, 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 S2 in the space of original variables. In order to generate valid inequalities corresponding to supporting hyperplanes of the convex hull of mathcal S2, we present a simple separation algorithm that can be incorporated in branch-and-cut based solvers to enhance the quality of existing relaxations.