Convex optimization on CAT(0) cubical complexes

TL;DR

This paper develops a convex optimization framework for CAT(0) cubical complexes, enabling solutions to problems like minimum enclosing ball, mean, median, and intersection feasibility using Euclidean cutting plane algorithms.

Contribution

It introduces a decomposition approach that leverages geodesic computations in CAT(0) complexes to solve convex optimization problems efficiently.

Findings

Effective algorithms for geodesic computation in CAT(0) complexes.

Application of Euclidean cutting plane methods to complex optimization problems.

Unified approach for various geometric optimization tasks in CAT(0) spaces.

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.