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Dang Van Cuong

Publications and source records attributed to Dang Van Cuong.

14 recordsLinked to original sources
arXiv2025

Duality Theory on Vector Spaces

In this paper, we study the Fenchel-Rockafellar duality and the Lagrange duality in the general frame work of vector spaces without topological structures. We utilize the geometric approach, inspired from its successful application by B. S. Mordukhovich and his coauthors in variational and convex analysis. After revisiting coderivative calculus rules and providing the subdifferential maximum rule in vector spaces, we establish conjugate calculus rules under qualifying conditions through the algebraic interior of the function's domains. Then we develop sufficient conditions which guarantee the Fenchel-Rockafellar strong duality. Finally, after deriving some necessary and sufficient conditions for optimal solutions to convex minimization problems, under a Slater condition via the algebraic interior, we then obtain a sufficient condition for the Lagrange strong duality.

math.OC↗
arXiv2023

Revisiting Rockafellar's Theorem on Relative Interiors of Convex Graphs with Applications to Convex Generalized Differentiation

In this paper we revisit a theorem by Rockafellar on representing the relative interior of the graph of a convex set-valued mapping in terms of the relative interior of its domain and function values. Then we apply this theorem to provide a simple way to prove many calculus rules of generalized differentiation of set-valued mappings and nonsmooth functions in finite dimensions. These results improve upon those in [14] by replacing the relative interior qualifications on graphs with qualifications on domains and/or ranges.

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arXiv2023

Fenchel-Rockafellar Theorem in Infinite Dimensions via Generalized Relative Interiors

In this paper we provide further studies of the Fenchel duality theory in the general frame work of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel-Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.

math.FA↗
arXiv2023

Quasi-Relative Interiors for Graphs of Convex Set-Valued Mappings

This paper aims at providing further studies of the notion of quasi-relative interior for convex sets introduced by Borwein and Lewis. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.

math.OC↗
arXiv2021

Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

This paper addresses the study and applications of polyhedral duality of locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar's proper separation theorem for two convex sets one which is polyhedral and then present its LCTV extension with replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by developing a geometric approach. We also establish in this way new results on conjugate calculus and duality in convex optimization with relaxed qualification conditions in polyhedral settings. Our developments contain significant improvements to a number of existing results obtained by Ng and Song in [31].

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arXiv2020

Convex Analysis of Minimal Time and Signed Minimal Time Functions

In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.

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arXiv2020

Extremal Systems of Convex Sets with Applications to Convex Calculus in Vector Spaces

In this paper we introduce and study the concept of set extremality for systems of convex sets in vector spaces without topological structures. Characterizations of the extremal systems of sets are obtained in the form of the convex extremal principle, which is shown to be equivalent to convex separation under certain qualification conditions expressed via algebraic cores. The obtained results are applied via a variational geometric approach to deriving enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued convex functions including the optimal value ones. These rules of the equality type are established under refined qualification conditions in terms of algebraic cores in arbitrary vector spaces. Our new developments partially answer the question on how far we can go with set-valued and convex analysis without any topological structure on the underlying spaces.

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arXiv2020

Algebraic Core and Convex Calculus without Topology

In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.

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arXiv2017

Subgradients of Minimal Time Functions without Calmness

In recent years there has been great interest in variational analysis of a class of nonsmooth functions called the minimal time function. In this paper we continue this line of research by providing new results on generalized differentiation of this class of functions, relaxing assumptions imposed on the functions and sets involved for the results. In particular, we focus on the singular subdifferential and the limiting subdifferential of this class of functions.

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arXiv2014

Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions

This paper is devoted to the study of generalized differentiation properties of the infimal convolution. This class of functions covers a large spectrum of nonsmooth functions well known in the literature. The subdifferential formulas obtained unify several known results and allow us to characterize the differentiability of the infimal convolution which plays an important role in variational analysis and optimization.

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arXiv2013

The Bi-normal fields on spacelike surfaces in $\mathbb R_1^4$

A normal field on a spacelike surface in $R^4_1$ is called bi-normal if $K^ν$, the determinant of Weingarten map associated with $ν$, is zero. In this paper we give a relationship between the spacelike pseudo-planar surfaces and spacelike pseudo-umbilical surfaces, then study the bi-normal fields on spacelike ruled surfaces and spacelike surfaces of revolution.

math.DG↗
arXiv2012

Surfaces of Revolution with Constant Gaussian Curvature in Four-Space

In this paper, we show that the constant property of the Gaussian curvature of surfaces of revolution in both $\mathbb R^4$ and $\mathbb R_1^4$ depend only on the radius of rotation. We then give necessary and sufficient conditions for the Gaussian curvature of the general rotational surfaces whose meridians lie in two dimensional planes in $\mathbb R^4$ to be constant, and define the parametrization of the meridians when both the Gaussian curvature is constant and the rates of rotation are equal.

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arXiv2011

$HS_{r}$-valued Gauss maps and umbilic spacelike surfaces of codimension two

To study spacelike surfaces of codimension two in the Lorentz-Minkowski space $\Bbb R^{n+1}_1,$ we construct a pair of maps whose values are in $HS_r:=H_+^n(\textbf v,1)\cap \{x_{n+1}=r\},$ called $\textbf n_r^{\pm}$-Gauss maps. It is showed that they are well-defined and useful to study practically flat as well as umbilic spacelike surfaces of codimension two in $\Bbb R^{n+1}_1.$

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