Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints

TL;DR

This paper investigates the semidefinite representation of convex hulls of semialgebraic sets parameterized by quadratic polynomials, providing conditions under which these convex hulls can be exactly characterized by spectrahedral projections.

Contribution

It establishes that the convex hull of such quadratic parameterized sets can be represented by first order moment semidefinite relaxations under specific quadratic constraints.

Findings

Convex hull equals the first order moment relaxation when T is defined by a single quadratic constraint.

Results extend to homogeneous quadratic polynomials with two constraints.

Applicable to sets with rational quadratic parameterizations.

Abstract

Let V be a semialgebraic set parameterized by quadratic polynomials over a quadratic set T. This paper studies semidefinite representation of its convex hull by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that its convex hull is equal to the first order moment type semidefinite relaxation of $V$, up to taking closures. Similar results hold when every quadratic polynomial is homogeneous and T is defined by two homogeneous quadratic constraints,or V is defined by rational quadratic parameterizations.