Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints

Abstract

Let V be a semialgebraic set parameterized as

$$\{(f_1(x),\ldots,f_m(x)):x\in T)\}$$

for quadratic polynomials f ₀,…,f _m and a subset T of Rⁿ. This paper studies semidefinite representation of the convex hull conv(V) or its closure, i.e., describing conv(V) by projections of spectrahedra (defined by linear matrix inequalities). When T is defined by a single quadratic constraint, we prove that conv(V) is equal to the first-order moment type semidefinite relaxation of V, up to taking closures. Similar results hold when every f _i is a quadratic form and T is defined by two homogeneous (modulo constants) quadratic constraints, or when all f _i are quadratic rational functions with a common denominator and T is defined by a single quadratic constraint, under some proper conditions.

Mathematics Subject Classification. 14P10, 90C22, 90C25.