Convex and Nonconvex Optimization Techniques for the Constrained Fermat-Torricelli Problem

Abstract

The Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to three given points in the plane. This problem was introduced by the French mathematician Fermat in the 17th century and was solved by the Italian mathematician and physicist Torricelli. In this thesis we introduce a constrained version of the Fermat-Torricelli problem in high dimensions that involves distances to a finite number of points with both positive and negative weights. Based on the distance penalty method, Nesterov’s smoothing technique, and optimization techniques for minimizing differences of convex functions, we provide effective algorithms to solve the problem.