Minkowski Centers via Robust Optimization: Computation and Applications

Abstract

Properly defining the center of a set has been a longstanding question in applied mathematics, with implications in numerical geometry, physics, and optimization algorithms. Minkowski centers are one such definition, whose theoretical benefits are numerous and well documented. In this paper, we revisit the advantages of Minkowski centers from a computational, rather than theoretical, perspective. First, we show that Minkowski centers are solutions to a robust optimization problem. Under this lens, we then provide computationally tractable reformulations or approximations for a series of sets, including polyhedra, polyhedral projections, and intersections of ellipsoids. Computationally, we illustrate that Minkowski centers are viable alternatives to other centers, such as Chebyshev or analytic centers, and can speed up the convergence of numerical algorithms like hit-and-run and cutting-plane methods. We hope our work sheds new and practical light on Minkowski centers and exposes their potential benefits as a computational tool.