Convex hull results on quadratic programs with non-intersecting constraints

Let F subset of R-n be a nonempty closed set. Understanding the structure of the closed convex hull (C) over bar (F) := <(conv{over bar>{( x, xx(T))|x is an element of F} in the lifted space is crucial for solving quadratic programs related to F. This paper discusses the relationship between C(F) and (C) over bar (G), where G results by adding non-intersecting quadratic constraints to F. We prove that C(G) can be represented as the intersection of C(F) and half spaces defined by the added constraints. The proof relies on a complete description of the asymptotic cones of sets defined by a single quadratic equality and a partial characterization of the recession cone of C(G). Our proof generalizes an existing result for bounded quadratically defined F with non-intersecting hollows and several results on C(G) for G defined by non-intersecting quadratic constraints. The result also implies a sufficient condition for when the lifted closed convex hull of an intersection equals the intersection of the lifted closed convex hulls.

Automated summary

This paper discusses the structure of the closed convex hull (C) over bar (F) and proves that C(G) can be represented as the intersection of C(F) and half spaces defined by added constraints. This result is important for solving quadratic programs related to F.