An Easy Path to Convex Analysis and Applications

Mordukhovich, Boris S.; Nam, Nguyen Mau

doi:10.2200/s00554ed1v01y201312mas014

Cited by 66 publications

(96 citation statements)

References 20 publications

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“…Some of the reference books that contain surveys on the Fermat-Torricelli problem are Boltyanski et al [3], Drezner et al [10], Rȃdulescu et al [16] and Mordukhovich and Nam [21].…”

Section: Introductionmentioning

confidence: 99%

“…In this sense, there are presented structural results and also algorithms for this problem of non-convex optimization, which depends on the distance function and the number and the location of the passages on the barrier. Mordukhovich and Nam [20,21] developed new applications of the variational analysis to the following optimization problem: "given n closed subsets in a Banach space, find a point such that the sum of distances to these subsets is minimum". Mordukhovich and Nam considered that the study of this generalized problem is of great interest and has many applications in location theory and in optimal networks.…”

Section: Introductionmentioning

confidence: 99%

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Facility location in normed linear spaces

2015

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We study the Fermat-Torricelli problem in the framework of normed linear spaces by using some ingredients of convex analysis and optimization. Several general formulations of the Fermat-Torricelli problem are presented. Sufficient conditions for the existence and uniqueness of the minimum point are formulated. Existence conditions for the minimum point are related to reflexivity assumptions on the normed space. Uniqueness conditions are related to strict convexity assumptions on the normed space. In the second part of the paper we study the Fermat problem subject to constraints in the plane and on the sphere.

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“…Some of the reference books that contain surveys on the Fermat-Torricelli problem are Boltyanski et al [3], Drezner et al [10], Rȃdulescu et al [16] and Mordukhovich and Nam [21].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Facility location in normed linear spaces

2015

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“…Figures and examples are given to guide the reader's conceptual intuition. For a more comprehensive study of convex analysis the reader is referred to [5,10,16].…”

Section: Chapter 1 Preliminariesmentioning

confidence: 99%

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

Lawrence

2016

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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.

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“…To continue, we recall some basic concepts and results of convex analysis. A systematic development of convex analysis can be found, for instance, in [11,18,27]. Let f : R n → R be a convex function.…”

Section: This Implies Thatxmentioning

confidence: 99%

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

Nam

Rector

et al. 2014

SIAM J. Optim.

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Abstract. We present algorithms for solving a number of new models of facility location which generalize the classical Fermat-Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradienttype algorithms to cope directly with the nondifferentiability of the cost functions. Convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms.

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An Easy Path to Convex Analysis and Applications

Cited by 66 publications

References 20 publications

Facility location in normed linear spaces

Facility location in normed linear spaces

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

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

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