On Global Maximization of Bounded Hessian Functions over Strongly Convex Domains

In this paper we present two frameworks in which global maximization of a bounded hessian function over a strongly convex set can be reduced to convex optimization. The first presented framework is a continuation of one of our previous papers [11]. We improve the results and give an explicit algorithm for the computation in polynomial time of the farthest point in a finite intersection of n-disks to $C \in \mathbb{R}^{n \times 1}$ under the requirement that $C$ does not belong to the convex hull of the centers of the n-disks. Finally, in order to overcome this limitation we present a second framework which characterizes the furthest in the finite intersection of n-disks with $C$ in the convex hull. Unfortunately this second framework requires the ability to decide if a polytope that we define is included in the intersection, which is hard in general. However, as a particular application of our second framework we are able solve in P time some instances of the subset sum problem with real entries: given a set of real numbers decide if there is a subset which adds up to zero.