Applications of variational analysis to a generalized Heron problem

Boris S. Mordukhovich; Nguyen Mau Nam; Juan Salinas

doi:10.1080/00036811.2011.604849

Applications of variational analysis to a generalized Heron problem

Boris S. Mordukhovich
, Nguyen Mau Nam
, Juan Salinas

King Fahd University of Petroleum & Minerals

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

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.

Original language	English
Pages (from-to)	1915-1942
Number of pages	28
Journal	Applicable Analysis
Volume	91
Issue number	10
DOIs	https://doi.org/10.1080/00036811.2011.604849
State	Published - Oct 2012

Bibliographical note

Funding Information:
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).

Keywords

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

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1080/00036811.2011.604849

Cite this

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title = "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, convex and nonconvex sets, generalized differentiation, minimal time function, variational analysis and optimization",

author = "Mordukhovich, \{Boris S.\} and Nam, \{Nguyen Mau\} and Juan Salinas",

note = "Funding Information: The authors are grateful to Jon Borwein, Mari{\'a}n 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).",

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AU - Nam, Nguyen Mau

AU - Salinas, Juan

N1 - Funding Information: 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).

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N2 - 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.

AB - 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.

KW - Heron problem and its extensions

KW - convex and nonconvex sets

KW - generalized differentiation

KW - minimal time function

KW - variational analysis and optimization

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DO - 10.1080/00036811.2011.604849

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EP - 1942

JO - Applicable Analysis

JF - Applicable Analysis

IS - 10

ER -

Applications of variational analysis to a generalized Heron problem

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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