78 research outputs found
Subdifferential Formulas for a Class of Nonconvex Infimal Convolutions
In this paper, we provide a number of subdifferential formulas for a class of
nonconvex infimal convolutions in normed spaces. The formulas obtained unify
several results on subdifferentials of the distance function and the minimal
time function. In particular, we generalize and validate the results obtained
recently by Zhang, He, and Jian
A Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem
In this paper, we introduce and study the following problem and its further
generalizations: given two finite collections of sets in a normed space, find a
ball whose center lies in a given constraint set with the smallest radius that
encloses all the sets in the first collection and intersects all the sets in
the second one. This problem can be considered as a generalized version of the
Sylvester smallest enclosing circle problem introduced in the 19th century by
Sylvester which asks for the circle of smallest radius enclosing a given set of
finite points in the plane. We also consider a generalized version of the
Fermat-Torricelli problem: given two finite collections of sets in a normed
space, find a point in a given constraint set that minimizes the sum of the
farthest distances to the sets in the first collection and shortest distances
(distances) to the sets in the second collection
Convergence Analysis of a Proximal Point Algorithm for Minimizing Differences of Functions
Several optimization schemes have been known for convex optimization
problems. However, numerical algorithms for solving nonconvex optimization
problems are still underdeveloped. A progress to go beyond convexity was made
by considering the class of functions representable as differences of convex
functions. In this paper, we introduce a generalized proximal point algorithm
to minimize the difference of a nonconvex function and a convex function. We
also study convergence results of this algorithm under the main assumption that
the objective function satisfies the Kurdyka - \L ojasiewicz property
Geometric Approach to Subdifferential Calculus
In this paper we develop a geometric approach to convex subdifferential
calculus in finite dimensions with employing some ideas of modern variational
analysis. This approach allows us to obtain natural and rather easy proofs of
basic results of convex subdifferential calculus in full generality and also
derive new results of convex analysis concerning marginal/value functions,
normal of inverse images of sets under set-valued mappings, calculus rules for
coderivatives of single-valued and set-valued mappings, and calculating
coderivatives of solution maps to convex generalized equations
Variational Analysis of Directional Minimal Time Functions and Applications to Location Problems
This paper is devoted to the study of directional minimal time functions that
specify the minimal time for a vector to reach an object following its given
direction. We provide a careful analysis of general and generalized
differentiation properties of this class of functions. The analysis allows us
to study a new model of facility location that involves sets. This is a
continuation of our effort in applying variational analysis to facility
location problems
Subgradients of Minimal Time Functions Under Minimal Requirements
This paper concerns the study of a broad class of minimal time functions
corresponding to control problems with constant convex dynamics and closed
target sets in arbitrary Banach spaces. In contrast to other publications, we
do not impose any nonempty interior and/or calmness assumptions on the initial
data and deal with generally non-Lipschitzian minimal time functions. The major
results present refined formulas for computing various subgradients of minimal
time functions under minimal requirements in both cases of convex and nonconvex
targets. Our technique is based on advanced tools of variational analysis and
generalized differentiation
Applications of variational analysis to a generalized Fermat-Torricelli problem
In this paper we develop new applications of variational analysis and
generalized differentiation to the following optimization problem and its
specifications: given n closed subsets of a Banach space, find such a point for
which the sum of its distances to these sets is minimal. This problem can be
viewed as an extension of the celebrated Fermat-Torricelli problem: given three
points on the plane, find another point such that the sum of its distances to
the designated points is minimal. The generalized Fermat-Torricelli problem
formulated and studied in this paper is of undoubted mathematical interest and
is promising for various applications including those frequently arising in
location science, optimal networks, etc. Based on advanced tools and recent
results of variational analysis and generalized differentiation, we derive
necessary as well as necessary and sufficient optimality conditions for the
extended version of the Fermat-Torricelli problem under consideration, which
allow us to completely solve it in some important settings. Furthermore, we
develop and justify a numerical algorithm of the subgradient type to find
optimal solutions in convex settings and provide its numerical implementations
The Fermat-Torricelli Problem and Weiszfeld's Algorithm in the Light of Convex Analysis
In the early 17th century, Pierre de Fermat proposed the following problem:
given three points in the plane, find a point such that the sum of its
Euclidean distances to the three given points is minimal. This problem was
solved by Evangelista Torricelli and was named the {\em Fermat-Torricelli
problem}. A more general version of the Fermat-Torricelli problem asks for a
point that minimizes the sum of the distances to a finite number of given
points in . This is one of the main problems in location science. In
this paper we revisit the Fermat-Torricelli problem from both theoretical and
numerical viewpoints using some ingredients of convex analysis and
optimization
Lipschitz Properties of Nonsmooth Functions and Set-Valued Mappings via Generalized Differentiation and Applications
In this paper, we revisit the Mordukhovich's subdifferential criterion for
Lipschitz continuity of nonsmooth functions and coderivative criterion for the
Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The
criteria are useful and beautiful results in modern variational analysis
showing the state of the art of the field. As an application, we establish
necessary and sufficient conditions for Lipschitz continuity of the minimal
time function and the scalarization function, that play an important role in
many aspects of nonsmooth analysis and optimization
Applications of Convex Analysis to the Smallest Intersecting Ball Problem
The smallest enclosing circle problem asks for the circle of smallest radius
enclosing a given set of finite points on the plane. This problem was
introduced in the 19th century by Sylvester [17]. After more than a century,
the problem remains very active. This paper is the continuation of our effort
in shedding new light to classical geometry problems using advanced tools of
convex analysis and optimization. We propose and study the following
generalized version of the smallest enclosing circle problem: given a finite
number of nonempty closed convex sets in a reflexive Banach space, find a ball
with the smallest radius that intersects all of the sets
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