Solving k-center problems involving sets based on optimization techniques

The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in $$\mathbb {R}^n$$. In this paper, we propose and study the following generalized version of the k-center problem: Given a finite number of nonempty closed convex sets in $$\mathbb {R}^n$$, find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms.