Applications of variational analysis to a generalized Heron problem

Abstract

This paper 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 completely solve optimal location problems of this type in some important settings.

Recent Developments on Primal–Dual Splitting Methods with Applications to Convex Minimization

“…Example 1 (Example 5.5 in [30]). Consider problem (52) with the constraint setb eing the closed ball centered at .5; 0/ having radius 2 and the sets˝i , i D 1; : : : ; 8, being pairwise disjoint squares in right position in R 2 (i. e. the edges are parallel to the x-and y-axes, respectively), with centers .…”

“…Baillon-Haddad Theorem, [2,Corollary 18.16]) in each of the infimal convolutions k k ı˝i , i D 1; : : : ; m, fact which is obviously here not the case. Notice that In the following we test our algorithms on some examples taken from [29,30].…”

“…The following numerical experiments address the generalized Heron problem which has been recently investigated in [29,30] and where for its solving subgradient-type methods have been used.…”

“…4a). Figure 4 gives an insight into the performance of the proposed primal-dual methods when compared with the subgradient algorithm used in [30]. After a few milliseconds both splitting algorithms reach machine precision with respect to the root-mean-square error where the following parameters were used:…”

Nonsmooth Algorithms and Nesterov's Smoothing Technique for Generalized Fermat--Torricelli Problems

“…This practical problem has been the inspiration for many new problems in the fields of computational geometry, logistics, and location science. Many generalized versions of the Fermat-Torricelli have been introduced and studied over the years; see [14,15,17,19,20,21,26] and the references therein. In particular, the generalized Fermat-Torricelli problems involving distances to sets were the topics of recent research; see [2,4,7,19,20].…”