arXiv subjects

R. Blake Rector

Publications and source records attributed to R. Blake Rector.

4 recordsLinked to original sources

arXiv2017

Variational Geometric Approach to Generalized Differential and Fenchel Conjugate Calculi in Convex Analysis

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes.

math.OC↗

arXiv2016

Minimizing Differences of Convex Functions and Applications to Facility Location and Clustering

In this paper we develop algorithms to solve generalized weighted Fermat-Torricelli problems with positive and negative weights and multifacility location problems involving distances generated by Minkowski gauges. We also introduce a new model of clustering based on squared distances to convex sets. Using the Nesterov smoothing technique and an algorithm for minimizing differences of convex functions called the DCA introduced by Tao and An, we develop effective algorithms for solving these problems.

math.OC↗

arXiv2015

The Log-Exponential Smoothing Technique and Nesterov's Accelerated Gradient Method for Generalized Sylvester Problems

The Sylvester smallest enclosing circle problem involves finding the smallest circle that encloses a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov's accelerated gradient method, we present an effective numerical algorithm for solving these problems.

math.OC↗

arXiv2013

A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings

In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the pioneer works by Clarke considering generalized differentiation theory of Lipschitz continuous functions. Although Clarke generalized differentiation theory is applicable for nonconvex functions, convexity still plays a crucial role in Clarke subdifferential calculus. In the mid 1970's, Mordukhovich developed another generalized differentiation theory for nonconvex functions and set-valued mappings in which the "umbilical cord with convexity" no longer exists. The primary goal of this paper is to present a unified approach and shed new light on convex and Clarke generalized differentiation theories using the concepts and techniques from Mordukhovich's developments.

math.OC↗