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C. Zalinescu

Publications and source records attributed to C. Zalinescu.

18 recordsLinked to original sources
arXiv2025

On Berinde's method for comparing iterative processes

In the literature there are several methods for comparing two convergent iterative processes for the same problem. In this note we have in view mostly the one introduced by Berinde in [Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2004, no. 2, 97--105] because it seems to be very successful. In fact, if IP1 and IP2 are two iterative processes converging to the same element, then IP1 is faster than IP2 in the sense of Berinde. The aim of this note is to prove this almost obvious assertion and to discuss briefly several papers that cite the mentioned Berinde's paper and use his method for comparing iterative processes.

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arXiv2024

On a variant of Schu's lemma

In this note, we demonstrate that an incorrect statement has been propagated in multiple papers, stemming from the substitution of ``lim'' with ``limsup'' for a sequence in Lemma 1.3 of the paper [J. Schu: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull.\ Austral.\ Math.\ Soc.\ 43 (1991), 153--159]. This occurred over a span of more than 20 years, with the earliest paper we identified using this incorrect statement dating back to 2002.

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arXiv2022

On the lower semicontinuity and subdifferentiability of the value function for conic linear programming problems

Lemma 1 from the paper [N.E. Gretsky, J.M. Ostroy, W.R. Zame, Subdifferentiability and the duality gap, Positivity 6: 261--274, 2002] asserts that the value function $v$ of an infinite dimensional linear programming problem in standard form is lower semicontinuous whenever $v$ is proper and the involved spaces are normed vector spaces. In this note one shows that this statement is false even in finite-dimensional spaces, one provides an example of linear programming problem in Hilbert spaces whose (proper) value function is not lower semicontinuous (hence it is not subdifferentiable) at any point in its domain, one shows that the restriction of the value function to its domain in Kretschmer's gap example is not bounded on any neighborhood of any point of the domain, and discuss other assertions done in the same paper.

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arXiv2019

On constrained optimization problems solved using CDT

DY Gao together with some of his collaborators applied his Canonical duality theory (CDT) for solving a class of constrained optimization problems. Unfortunately, in several papers on this subject there are unclear statements, not convincing proofs, or even false results. It is our aim in this work to study rigorously these class of constrained optimization problems in finite dimensional spaces and to discuss several results published in the last ten years.

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arXiv2018

On unconstrained optimization problems solved using CDT and triality theory

DY Gao solely or together with some of his collaborators applied his Canonical duality theory (CDT) for solving a class of unconstrained optimization problems, getting the so-called "triality theorems". Unfortunately, the "double-min duality" from these results published before 2010 revealed to be false, even if in 2003 DY Gao announced that "certain additional conditions" are needed for getting it. After 2010 DY Gao together with some of his collaborators published several papers in which they added additional conditions for getting "double-min" and "double-max" dualities in the triality theorems. The aim of this paper is to treat rigorously this kind of problems and to discuss several results concerning the "triality theory" obtained up to now.

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arXiv2018

On quadratic optimization problems and canonical duality theory

DY Gao solely or together with some of his collaborators applied his Canonical duality theory (CDT) for solving some quadratic optimization problems with quadratic constraints. Unfortunately, in almost all papers we read on CDT there are unclear definitions, non convincing arguments in the proofs, and even false results. The aim of this paper is to treat rigorously quadratic optimization problems by the method suggested by CDT and to compare what we get with the results obtained by DY Gao and his collaborators on this topic in several papers.

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arXiv2018

On "two important theorems" in canonical duality theory

In this short note we show, providing counterexamples, that the "two important theorems" in the recent paper [Y, Yuan, Global optimization solutions to a class of non-convex quadratic minimization problems with quadratic constraints, in Canonical Duality Theory, D.Y. Gao et al.\ (eds), (AMMA, volume 37), Springer, 2017] are false.

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arXiv2015

Series of convex functions: subdifferential, conjugate and applications to entropy minimization

A formula for the sub\-differential of the sum of a series of convex functions defined on a Banach space was provided by X. Y. Zheng in 1998. In this paper, besides a slight extension to locally convex spaces of Zheng's results, we provide a formula for the conjugate of a countable sum of convex functions. Then we use these results for calculating the sub\-differentials and the conjugates in two situations related to entropy minimization, and we study a concrete example met in Statistical Physics.

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arXiv2013

Relations between the convexity of a set and the differentiability of its support function

It is known that, in finite dimensions, the support function of a compact convex set with non empty interior is differentiable excepting the origin if and only if the set is strictly convex. In this paper we realize a thorough study of the relations between the differentiability of the support function on the interior of its domain and the convexity of the set, mainly for unbounded sets. Then we revisit some results related to the differentiability of the cost function associated to a production function.

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arXiv2012

A non-convex variational problem appearing in a large deformation elasticity problem

A result concerning global extrema in a nonsmooth nonconvex variational problem that appears in applications (e.g. in a large deformation elasticity problem) is investigated in comparison with a result of D.Y. Gao and R.W. Ogden. The tools used are elementary and the results derived improve upon and correct a recent similar result, more precisely, Theorem 4 of the paper "Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem" by the fore-mentioned authors.

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arXiv2010

On three duality results

The aim of this short note is to give counterexamples to two results by D. Y. Gao [5, Th. 16], [4, Th. 2] and to improve a related result by S.-C. Fang, D. Y. Gao, R.-L. Sheu and S.-Y. Wu [1, Th. 3].

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arXiv2008

Linear Monotone Subspaces of Locally Convex Spaces

The main focus of this paper is to study multi-valued linear monotone operators in the contexts of locally convex spaces via the use of their Fitzpatrick and Penot functions. Notions such as maximal monotonicity, uniqueness, negative-infimum, and (dual-) representability are studied and criteria are provided.

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arXiv2008

Strongly-Representable Operators

Recently in [1] a new class of maximal monotone operators has been introduced. In this note we study domain range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly representable operator is locally maximal monotone, maximal monotone locally, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representative function.

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