Applications of variational analysis to a generalized Heron problem

Abstract

This article is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point C such that the sum of the distances from C to the given points A and B is minimal. We show that the advanced variational techniques allow us to solve optimal location problems of this type completely in some important settings.

Keywords:

• Heron problem and its extensions
• variational analysis and optimization
• generalized differentiation
• minimal time function
• convex and nonconvex sets

AMS Subject Classifications::

• 49J52
• 49J53
• 90C31

Acknowledgments

The authors are grateful to Jon Borwein, Marián Fabian, and Doan The Hieu for valuable discussions on the material of this article. The first author acknowledges partial supports by the USA National Science Foundation under grant DMS-1007132, by the European Regional Development Fund (FEDER), and by the following Portuguese agencies: Foundation for Science and Technologies (FCT), Operational Program for Competitiveness Factors (COMPETE), and Strategic Reference Framework (QREN). The second author was partially supported by a grant from the Simons Foundation (#208785 to Mau Nam Nguyen).