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Marius Durea

Publications and source records attributed to Marius Durea.

13 recordsLinked to original sources
arXiv2026

Nonlinear and orbital directional regularities for set-valued mappings and applications to fixed points

Motivated by developments in metric regularity theory for set-valued mappings, this paper introduces several directional regularity notions that are weaker than the corresponding classical regularity properties. We investigate the relationships among these concepts and establish a classification framework based on directional minimal time functions. Particular attention is devoted to directional orbital regularity and directional orbital Aubin continuity, which are shown to provide a natural setting for the analysis of fixed point phenomena. Using these notions, we derive approximate and exact fixed point theorems for set-valued mappings under directional assumptions. The obtained results are expressed in terms of minimal time functions and orbital approximation procedures, allowing for anisotropic and asymmetric behaviors that are not captured by the classical nondirectional framework. We further apply the developed theory to directional coincidence point results, coupled fixed point theorems, and Milyutin-type perturbation stability properties. The proposed approach provides a unified directional perspective on fixed point theory and variational analysis. In particular, it recovers several known results from the literature as special cases and yields new extensions under substantially weaker assumptions than classical contraction or global regularity conditions.

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arXiv2026

Dually cone-boundedness of a set and applications

We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the cone. We show that this is a weaker notion when compared to other similar ones and we explore several links with the existing literature. We subsequently demonstrate that this concept furnishes the properties required to obtain various generalizations of important results and techniques, including conic cancellation rules and the R{\aa}dstr\"om embedding procedure.

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arXiv2025

Necessary conditions for approximate solutions of vector and set optimization problems with variable domination structure

We consider vector and set optimization problems with respect to variable domination structures given by set-valued mappings acting between the preimage space and the image space of the objective mapping, as well as by set-valued mappings with the same input and output space, that coincides with the image space of the objective mapping. The aim of this paper is to derive necessary conditions for approximately nondominated points of problems with a single-valued objective function, employing an extension of Ekeland's Variational Principle for problems with respect to variable domination structures in terms of generalized differentiation in the sense of Mordukhovich. For set-valued objective mappings, we derive necessary conditions for approximately nondominated points of problems with variable domination structure taking into account the incompatibility between openness and optimality and a directional openness result for the sum of set-valued maps. We describe the necessary conditions for approximately nondominated points of set optimization problems with variable domination structure in terms of the limiting (Mordukhovich) generalized differentiation objects.

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arXiv2023

Conic cancellation laws and some applications

We discuss, on finite and infinite dimensional normed vector spaces, some versions of Radstr\"{o}m cancellation law (or lemma) that are suited for applications to set optimization problems. In this sense, we call our results "conic" variants of the celebrated result of Radstr\"{o}m, since they involve the presence of an ordering cone on the underlying space. Several adaptations to this context of some topological properties of sets are studied and some applications to subdifferential calculus associated to set-valued maps and to necessary optimality conditions for constrained set optimization problems are given. Finally, a stability problem is considered.

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arXiv2018

Directional Pareto efficiency: concepts and optimality conditions

We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations concerning the objective mapping and the constraints are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation and the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation.

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arXiv2012

Metric regularity of composition set-valued mappings: metric setting and coderivative conditions

The paper concerns a new method to obtain a direct proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions) and allow us to obtain two new proofs of a recent result having deep roots in the topic of regularity of mappings. Moreover, we make clear the idea that it is possible to use (co)derivative conditions as tools of proof for openness results in very general situations.

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arXiv2012

On Subregularity Properties of Set-Valued Mappings. Applications to Solid Vector Optimization

In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties to deduce implicit theorems for set-valued maps. Finally, we present some applications to the study of multicriteria optimization problems.

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arXiv2011

Chain Rules for Linear Openness in Metric Spaces. Applications to Parametric Variational Systems

In this work we present a general theorem concerning chain rules for linear openness of set-valued mappings acting between metric spaces. As particular cases, we obtain classical and also some new results in this field of research, including the celebrated Lyusternik-Graves Theorem. The applications deal with the study of the well-posedness of the solution mappings associated to parametric variational systems. Sharp estimates for the involved regularity moduli are given.

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arXiv2011

Openness Stability and Implicit Multifunction Theorems. Applications to Variational Systems

In this paper we aim to present two general results regarding, on one hand, the openness stability of set-valued maps and, on the other hand, the metric regularity behavior of the implicit multifunction related to a generalized variational system. Then, these results are applied in order to obtain, in a natural way, and in a widely studied case, several relations between the metric regularity moduli of the field maps defining the variational system and the solution map. Our approach allows us to complete and extend several very recent results in literature.

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arXiv2011

On Parametric Vector Optimization via Metric Regularity of Constraint Systems

Some metric and graphical regularity properties of generalized constraint systems are investigated. Then, these properties are applied in order to penalize (in the sense of Clarke) various scalar and vector optimization problems. This method allows us to present several necessary optimality conditions in solid constrained vector optimization.

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arXiv2011

Filling the Gap between Metric Regularity and Fixed Points: The Linear Openness of Compositions

This paper is devoted to the investigation of an important issue recently brought into attention by a recent paper of Arutyunov: the relation between openness of composition of set-valued maps and fixed point results. More precisely, we prove a general result concerning the openness of compositions and then we show that this result covers and implies most of the known openness results. In particular, we reobtain several recent results in this field, including a fixed point theorem of Dontchev and Frankowska.

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