57 research outputs found
A D.C. Algorithm via Convex Analysis Approach for Solving a Location Problem Involving Sets
We study a location problem that involves a weighted sum of distances to
closed convex sets. As several of the weights might be negative, traditional
solution methods of convex optimization are not applicable. After obtaining
some existence theorems, we introduce a simple, but effective, algorithm for
solving the problem. Our method is based on the Pham Dinh - Le Thi algorithm
for d.c. programming and a generalized version of the Weiszfeld algorithm,
which works well for convex location problems
Applications of Variational Analysis to a Generalized Heron Problem
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 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
A Look at the Generalized Heron Problem through the Lens of Majorization-Minimization
In a recent issue of this journal, Mordukhovich et al.\ pose and solve an
interesting non-differentiable generalization of the Heron problem in the
framework of modern convex analysis. In the generalized Heron problem one is
given closed convex sets in \Real^d equipped with its Euclidean norm
and asked to find the point in the last set such that the sum of the distances
to the first sets is minimal. In later work the authors generalize the
Heron problem even further, relax its convexity assumptions, study its
theoretical properties, and pursue subgradient algorithms for solving the
convex case. Here, we revisit the original problem solely from the numerical
perspective. By exploiting the majorization-minimization (MM) principle of
computational statistics and rudimentary techniques from differential calculus,
we are able to construct a very fast algorithm for solving the Euclidean
version of the generalized Heron problem.Comment: 21 pages, 3 figure
Nonsmooth Algorithms and Nesterov\u27s Smoothing Technique for Generalized Fermat-Torricelli Problems
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\u27s smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradient-type 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
- …