A Computationally Viable Numerical Gradient-based Technique for Optimal Covering Problems

The problem of optimally covering a given compact subset of $\mathbb{R}^N$ with a preassigned number $n$ of Euclidean metric balls has a long-standing history and it is well-recognized to be computationally hard. This article establishes a numerically viable algorithm for obtaining optimal covers of compact sets via two key contributions. The first is a foundational result establishing Lipschitz continuity of the marginal function of a certain parametric non-convex maximization problem in the optimal covering problem, and it provides the substrate for numerical gradient algorithms to be employed in this context. The second is an adaptation of a stochastically smoothed numerical gradient-based (zeroth-order) algorithm for a non-convex minimization problem, that, equipped with randomized restarts, spurs global convergence to an optimal cover. Several numerical experiments with complicated nonconvex compact sets demonstrate the excellent performance of our techniques.