Dan Butnariu – Optimization Online

On the behavior of subgradient projections methods for convex feasibility problems in Euclidean spaces

Published: 2008/04/17, Updated: 2008/07/07

Convex Optimization convex feasibility, steering parameters, strategical relaxation, subgradient projections

We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user-flexibility but gives a mathematical guarantee for the algorithm’s behavior in … Read more

Mosco stability of proximal mappings in reflexive Banach spaces

Published: 2006/04/25

Dan Butnariu

Elena Resmerita

Convex Optimization bregman distance, legendre function, modulus of total convexity, mosco convergence of a sequence of functions, proximal mapping relative to a convex function, relative projection onto a convex set, uniformly convex function

In this paper we establish criteria for the stability of the proximal mapping \textrm{Prox} $_{\varphi }^{f}=(\partial \varphi +\partial f)^{-1}$ associated to the proper lower semicontinuous convex functions $\varphi $ and $f$ on a reflexive Banach space $X.$ We prove that, under certain conditions, if the convex functions $\varphi _{n}$ converge in the sense of Mosco … Read more

Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces

Published: 2005/01/16, Updated: 2005/01/18

Dan Butnariu

Elena Resmerita

Convex and Nonsmooth Optimization bregman distance, bregman projection, e-space, essential strict convexity, inverse monotone operator, inverse strongly-monotone operator, mosco convergence, relative projection, total convexity, uniform convexity

The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity … Read more

Reduntant axioms in the definitionof Bregman functions

Published: 2002/03/17, Updated: 2004/10/04

Dan Butnariu

Charles Byrne

Yair Censor

Constrained Nonlinear Optimization bregman distance, bregman function, kullback-leibler distance, modulus of total convexity, sequential consistency, totally convex function

The definition of a Bregman function, given by Censor and Lent in 1981 on the basis of Bregman’s seminal 1967 paper, was subsequently used in a plethora of research works as a tool for building sequential and inherently parallel feasibility and optimization algorithms. Solodov and Svaiter have recently shown that it is not CitationJournal of … Read more