arXiv subjects
Publications and source records attributed to M. D. Voisei.
This article employs techniques from convex analysis to present characterizations of (maximal) $n-$monotonicity, similar to the well-established characterizations of (maximal) monotonicity found in the existing literature. These characterizations are further illustrated through examples.
In this note new properties for the Gossez example are presented in regard to its representability, closedness, and maximal monotonicity with respect to the two dual systems it naturally inhabits.
We study monotone extension problems in the general framework of dual systems, without assuming separation. The paper develops a compact target-set formulation that includes multivalued operators as a special case and allows the initial set to be nonmonotone. Using tools from nonlinear and convex analysis, we establish existence results and necessary and sufficient conditions for monotone extensions. Applications to variational inequalities are also obtained.
Our goal is to present a new shorter proof for the maximal monotonicity of the Minkowski sum of two maximal monotone multi-valued operators defined in a reflexive Banach space under the classical interiority condition involving their domains.
This paper introduces the metric-dependent core subdifferential, a local robust affine-support construction for extended-real functions on metrizable topological vector spaces. A core subgradient is a continuous linear slope for which the affine lower support holds on metric balls with an error negligible relative to the supporting radius, along arbitrarily fine admissible scales. The main minimality results show that these slopes are unavoidable: on complete metrizable spaces they belong to the graph closure of any subdifferential satisfying a local-minimum principle together with a mild stability condition under metric-distance perturbations, and on Banach spaces they belong to the nearby graph closure of any abstract subdifferential satisfying the usual fuzzy minimum principle. For the norm metric, the construction contains the Fr\'echet subdifferential, coincides with the Fenchel subdifferential on convex functions, is contained in the limiting subdifferential whenever the relevant Fr\'echet fuzzy calculus is available, and is contained in the Clarke--Rockafellar subdifferential for lower semicontinuous functions on Banach spaces. The paper also records metric-dependence phenomena, strict comparison examples, constrained optimality and variational-inequality conditions, a scale-slope/error-bound characterization, and the relation with Goldstein-type stationarity used in finite-dimensional nonsmooth optimization.
Abstract Equivalent conditions that make the convex subdifferential maximal monotone are investigated in the general settings of locally convex spaces.
Equivalent conditions that make the normal cone maximal monotone are investigated in the general settings of locally convex spaces. Some consequences such as Bishop Phelps and sum representability results are presented in the last part.
New perspectives, proofs, and some extensions of known results are presented concerning the behavior of the Fitzpatrick function of a monotone type operator in the general context of a locally convex space.
In this paper we study, in the relaxed context of locally convex spaces, intrinsic properties of monotone operators needed for the sum conjecture for maximal monotone operators to hold under classical interiority-type domain constraints.
The local equicontinuity of an operator $T:X\rightrightarrows X^{*}$ with proper Fitzpatrick function $φ_{T}$ and defined in a barreled locally convex space $X$ has been shown to hold on the algebraic interior of $\operatorname*{Pr}\,_{X}(\operatorname*{dom}φ_{T})$). The current note presents direct consequences of the aforementioned result with regard to the local equicontinuity of a maximal monotone operator defined in a barreled locally convex space.
The local boundedness of classes of operators is analyzed on different subsets directly related to their Fitzpatrick functions and characterizations of the topological vector spaces for which that local boundedness holds is given in terms of the uniform boundedness principle. For example the local boundedness of a maximal monotone operator on the algebraic interior of its domain convex hull is a characteristic of barreled locally convex spaces.
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.
In the context of general Banach spaces characterizations for the maximal monotonicity of operators with non-empty domain interior as well as stronger continuity properties for such operators are provided.
In this note we prove that a recent result stated by D.Y. Gao and R.W. Ogden on global minimizers and local extrema in a phase transition problem is false. Our goal is achieved by providing a thorough analysis of the context and result in question and counter-examples.
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].
On [3, p. 199] one says "We mention parenthetically that the proof of [99, Lemma 41.3] is incorrect, and we do not know whether it, [99, Theorem 41.5] and [99, Theorem 41.6] are true". The previously cited reference [99] is our reference [2]. The aim of this short note is to provide a result that improves upon [2, Lemma 41. 3].
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.
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.