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

Gokul Rajaraman, Debasish Chatterjee

Keywords: Optimal cover, Optimal n-net, Sensitivity of nonlinear programs, Numerical gradient algorithm

TL;DR: We present a viable algorithm for the problem of optimally covering a compact set with a fixed number of balls using results from sensitivity analysis and nonsmooth and nonconvex optimization.

Abstract: 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.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 13746