Convex optimization on CAT(0) cubical complexes

Abstract

We consider geodesically convex optimization problems involving distances to a finite set of points A in a CAT(0) cubical complex. Examples include the minimum enclosing ball problem, the weighted mean and median problems, and the feasibility and projection problems for intersecting balls with centers in A. We propose a decomposition approach relying on standard Euclidean cutting plane algorithms. The cutting planes are readily derivable from efficient algorithms for computing geodesics in the complex.