Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

Mordukhovich, Boris S.; Nam, Nguyen Mau

doi:10.1007/s10957-010-9761-7

Cited by 51 publications

(39 citation statements)

References 15 publications

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“…Problems of this type have been recently studied from various perspectives [41,42,43]. Our approach, based on the proximal point algorithm, seems to be however novel even in linear spaces.…”

Section: Introductionmentioning

confidence: 99%

Computing Medians and Means in Hadamard Spaces

Bacak¹

2014

SIAM J. Optim.

154

164

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Abstract. The geometric median as well as the Fréchet mean of points in an Hadamard space are important in both theory and applications. Surprisingly, no algorithms for their computation are hitherto known. To address this issue, we use a split version of the proximal point algorithm for minimizing a sum of convex functions and prove that this algorithm produces a sequence converging to a minimizer of the objective function, which extends a recent result of D. Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only does it yield algorithms for the median and the mean, but it also applies to various other optimization problems. We moreover show that another algorithm for computing the Fréchet mean can be derived from the law of large numbers due to K.-T. Sturm (2002).In applications, computing medians and means is probably most needed in tree space, which is an instance of an Hadamard space, invented by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic trees. It turns out, however, that it can be also used to model numerous other tree-like structures. Since there now exists a polynomial-time algorithm for computing geodesics in tree space due to M. Owen and S. Provan (2011), we obtain efficient algorithms for computing medians and means, which can be directly used in practice.

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Section: Introductionmentioning

confidence: 99%

Computing Medians and Means in Hadamard Spaces

Bacak¹

2014

SIAM J. Optim.

154

164

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“…On the other hand, the next result shows that for the unconstrained version of (1.1), i.e., for the generalized Fermat-Torricelli problem [13] with disjoint sets Ω i , the projection nonemptiness at a local optimal solution automatically implies the projection uniqueness in arbitrary Hilbert spaces. Proposition 3.4 (projection uniqueness at optimal solutions).…”

Section: Proof To Justify Assertionmentioning

confidence: 85%

Applications of variational analysis to a generalized Heron problem

Mordukhovich

Nam²,

Salinas³

2011

Applicable Analysis

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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.

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“…Our next goal is to study optimality conditions and numerical algorithms for the smallest enclosing ball problem and the smallest intersecting ball problem. Based on the approach we have developed in [7][8][9][10], we foresee the potential of success of this future work.…”

Section: Discussionmentioning

confidence: 99%

The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

Mordukhovich

Nam²,

Villalobos³

2012

Optim Lett

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In this paper we study the following problems: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects all of the sets. These problems can be viewed as generalized versions of the smallest enclosing circle problem introduced in the 19th century by Sylvester [12] which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We will focus on the sufficient conditions for the existence and uniqueness of an optimal solution for each problem, while the study of optimality conditions and numerical implementation will be addressed in our next projects.

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Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

Cited by 51 publications

References 15 publications

Computing Medians and Means in Hadamard Spaces

Computing Medians and Means in Hadamard Spaces

Applications of variational analysis to a generalized Heron problem

The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

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