Applications of Convex Optimization to Weighted Generalized Fermat-Torricelli and Heron Problems

In the Fermat-Torricelli problem, a point in the plane is to be found such that the sum of the distances from this point to the three given target points is smallest. The Heron problem asks for a point on a line constraint such that the sum of the distances to two given target points is minimal. These problems can be solved by differential calculus or geometry. In this research project, we study generalized versions of the classical Fermat-Torricelli and Heron problems. Our idea is to replace the given points and the line constraint in these problems with sets. Additionally, we modify the sum of the distances to points in each problem by a general weighted sum of distances to sets. Using convex analysis and optimization, we are able to study these problems from both theoretical and numerical viewpoints. The new problems are mathematically interesting and have promising applications to location science and optimal networks.