arXiv subjects

Nguyen Duy Cuong

Publications and source records attributed to Nguyen Duy Cuong.

15 recordsLinked to original sources

arXiv2026

Sequential Extremal Principle: Refinements and Applications

Sequential extremality and stationarity properties are discussed with the emphasis on those corresponding to fixed sequences of translations. Exact quantitative characterisations of the properties are provided. We show, in particular, that the sequential (as well as conventional) extremality and approximate stationarity properties possess certain stability, while the (non-approximate) stationarity does not. Dual necessary conditions for the sequential extremality and stationarity properties with fixed and non-fixed sequences of translations are established. A version of the sequential extended extremal principle is formulated. In the statements, we employ certain generalised separation conditions $(GS)$ and $(GS_α)$ as well as a complementary primal-dual condition $(PD)$. To illustrate the model, we prove dual optimality/stationarity conditions for a constrained minimisation problem in which the minimal value is not necessarily attained.

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arXiv2026

Dual characterizations of norm minimization problems

The paper studies a general norm minimization problem on a product of normed vector spaces. We establish dual necessary and sufficient optimality conditions and derive explicit formulas for the corresponding solution sets. These formulas are obtained under the assumption that one optimal solution together with its associated dual vectors arising from the optimality conditions is known. Three important cases of product norms, namely the sum norm, maximum norm and $p$-norm, are also studied. Several examples in finite and infinite dimensional spaces equipped with various types of norms are presented to illustrate the established results.

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arXiv2026

Norm Minimisation Problems Involving Distances to Convex Sets

The paper studies product-space norm minimisation problems involving distances to convex sets. Using standard tools from convex and functional analysis, we establish complete dual necessary and sufficient optimality conditions and show that the entire solution set can be constructed from the dual vectors arising from the optimality conditions at a given solution. As a consequence, we study optimality conditions and solution set descriptions for generalised versions of the Fermat-Torricelli problem, the Chebyshev centre problem, and the p-Fermat-Torricelli problem. Comparisons with existing results are provided whenever applicable. Examples in finite and infinite dimensional spaces equipped with different norms are presented to illustrate the results.

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arXiv2026

A von Neumann-Jordan Constant of Non-Normable Metrics

The paper studies a generalized von Neumann-Jordan constant of non-normable metrics on vector spaces. To the best of our knowledge, all existing results of the von Neumann-Jordan constant and its generalizations have been established only in the normed setting. We identify reasonable conditions on non-normable metrics under which results known for norms remain valid. Several examples and counterexamples are provided to justify the established results. The computation for a class of non-normable metrics on product spaces is also investigated. In particular, we give precise formulas for the generalized von Neumann-Jordan constant of p-metrics under a metric-type Clarkson inequality. Comparisons with existing results are discussed throughout the paper whenever applicable.

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arXiv2026

Primal and dual characterizations of sign-symmetric norms

The paper studies primal and dual characterizations of a class of sign-symmetric norms on product vector spaces. Correspondences between these norms and a class of convex functions are established. Explicit formulas for the dual norm and the convex subdifferential of a given primal norm are derived. It is demonstrated that this class of norms is well-suited for studying properties and problems on product spaces. As an application, we study the von Neumann-Jordan constant of norms on product spaces and extend a classical result of Clarkson from Lebesgue spaces to general normed vector spaces.

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arXiv2026

Metric constructions and fixed point theorems in product spaces

The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are topologically equivalent to the conventional ones. As an application, we study fixed point and approximate fixed point properties for nonexpansive maps on a product space equipped with the constructed metric. We show that existing fixed point results of this type are consequences of our framework. Examples are provided to illustrate the established results. The construction machinery is also used to study products of length and geodesic spaces. The obtained results encompass existing ones and provide a background for potential studies of fixed point properties on these product spaces.

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arXiv2025

Generalized Separation of Collections of Sets

We show that the existing generalized separation statements including the conventional extremal principle and its extensions differ {in the ways norms on product spaces are defined}. We prove a general separation statement with arbitrary product norms covering the existing results of this kind. The proof is divided into a series of claims and exposes the key steps and arguments used when proving generalized separation statements. As an application, we prove dual necessary (sufficient) conditions for an abstract product norm extension of the approximate stationarity (transversality) property.

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arXiv2025

Sequential Extremal Principle and Necessary Conditions for Minimizing Sequences

The conventional definition of extremality of a finite collection of sets is extended by replacing a fixed point (extremal point) in the intersection of the sets by a collection of sequences of points in the individual sets with the distances between the corresponding points tending to zero. This allows one to consider collections of unbounded sets with empty intersection. Exploiting the ideas behind the conventional extremal principle, we derive an extended sequential version of the latter result in terms of Fréchet and Clarke normals. Sequential versions of the related concepts of stationarity, approximate stationarity and transversality of collections of sets are also studied. As an application, we establish sequential necessary conditions for minimizing (and more general firmly stationary, stationary and approximately stationary) sequences in a constrained optimization problem.

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arXiv2025

Extremality of families of sets and set-valued optimization

The paper explores a new extremality model involving collections of arbitrary families of sets. We demonstrate its applicability to set-valued optimization problems with general preferences, weakening the assumptions of the known results and streamlining their proofs.

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arXiv2024

Extremality of families of sets

The paper proposes another extension of the extremal principle. A new extremality model involving collections of arbitrary families of sets is studied. It generalizes the conventional model based on linear translations of given sets as well as its set-valued extensions. This approach leads to a more general and simpler version of fuzzy separation. The new model is capable of treating a wider range of optimization and variational problems.

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arXiv2023

Lyusternik-Graves Theorem for Holder Metric Regularity

The paper extends the well-known Lyusternik-Graves theorem for set-valued mappings to the Holder framework, offers an affirmative answer to an open problem proposed by Dontchev and improves recent results of He and Ng. Primal and dual necessary and sufficient conditions for Holder metric regularity are established. The results are applied to convergence analysis of a Newton-type method. Some open problems for future research are also discussed.

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arXiv2021

Error bounds revisited

We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. Employing special collections of slope operators, we introduce a succinct form of sufficient error bound conditions, which allows one to combine in a single statement several different assertions: nonlocal and local primal space conditions in complete metric spaces, and subdifferential conditions in Banach and Asplund spaces.

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arXiv2021

Uniform Regularity of Set-Valued Mappings and Stability of Implicit Multifunctions

We propose a unifying general (i.e. not assuming the mapping to have any particular structure) view on the theory of regularity and clarify the relationships between the existing primal and dual quantitative sufficient and necessary conditions including their hierarchy. We expose the typical sequence of regularity assertions, often hidden in the proofs, and the roles of the assumptions involved in the assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. As a consequence, we formulate primal and dual conditions for the stability properties of solution mappings to inclusions

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arXiv2020

Transversality Properties: Primal Sufficient Conditions

The paper studies 'good arrangements' (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Holder case is given a special attention. Our main objective is not formally extending our earlier results from the Holder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed.

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arXiv2018

Some new characterizations of intrinsic transversality in Hilbert spaces

Motivated by a number of research questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. Our dual space results clarify the picture of intrinsic transversality, its variants and the only existing sufficient dual condition for subtransversality, and actually unify them. New primal space characterizations of the intrinsic transversality which is originally a dual space condition lead to new understanding of the property in terms of primal space elements for the first time. As a consequence, the obtained analysis allows us to address a number of research questions asked by the two aforementioned researchers about the intrinsic transversality property in the Hilbert space setting.

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