Nguyen Mau Nam

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Subgradients of marginal functions in parametric mathematical programming

2007-04-27

Boris S. Mordukhovich , Nguyen Mau Nam , N. D. Yen

Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming

2006-10-01

Boris S. Mordukhovich , Nguyen Mau Nam , N. D. Yen

We develop various (exact) calculus rules for Fréchet lower and upper subgradients of extended-real-valued functions in real Banach spaces. Then we apply this calculus to derive new necessary optimality conditions … We develop various (exact) calculus rules for Fréchet lower and upper subgradients of extended-real-valued functions in real Banach spaces. Then we apply this calculus to derive new necessary optimality conditions for some remarkable classes of problems in constrained optimization including minimization problems for difference-type functions under geometric and operator constraints as well as subdifferential optimality conditions for the so-called weak sharp minima.

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Second-Order Analysis of Polyhedral Systems in Finite and Infinite Dimensions with Applications to Robust Stability of Variational Inequalities

2010-01-01

René Henrion , Boris S. Mordukhovich , Nguyen Mau Nam

This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with … This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions.

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An Easy Path to Convex Analysis and Applications

2014-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of … Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of di

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Applications of Variational Analysis to a Generalized Fermat-Torricelli Problem

2010-11-17

Boris S. Mordukhovich , Nguyen Mau Nam

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Variational Stability and Marginal Functions via Generalized Differentiation

2005-11-01

Boris S. Mordukhovich , Nguyen Mau Nam

Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and … Robust Lipschitzian properties of set-valued mappings and marginal functions play a crucial role in many aspects of variational analysis and its applications, especially for issues related to variational stability and optimization. We develop an approach to variational stability based on generalized differentiation. The principal achievements of this paper include new results on coderivative calculus for set-valued mappings and singular subdifferentials of marginal functions in infinite dimensions with their extended applications to Lipschitzian stability. In this way we derive efficient conditions ensuring the preservation of Lipschitzian and related properties for set-valued mappings under various operations, with the exact bound/modulus estimates, as well as new sufficient conditions for the Lipschitz continuity of marginal functions.

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Variational Analysis of Marginal Functions with Applications to Bilevel Programming

2011-10-12

Boris S. Mordukhovich , Nguyen Mau Nam , Hung M. Phan

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Solving k-center problems involving sets based on optimization techniques

2019-10-23

Nguyen Thai An , Nguyen Mau Nam , Xiaolong Qin

Solving a Generalized Heron Problem by Means of Convex Analysis

2012-01-01

Boris S. Mordukhovich , Nguyen Mau Nam , Juan Salinas

The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points … The classical Heron problem states: on a given straight line in the plane, find a point C such that the sum of the distances from C to the given points A and B is minimal. This problem can be solved using standard geometry or differential calculus. In the light of modern convex analysis, we are able to investigate more general versions of this problem. In this paper we propose and solve the following problem: on a given nonempty closed convex subset of ℝs, find a point such that the sum of the distances from that point to n given nonempty closed convex subsets of ℝs is minimal.

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An Easy Path to Convex Analysis and Applications

2013-12-21

Boris S. Mordukhovich , Nguyen Mau Nam

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Applications of variational analysis to a generalized Heron problem

2011-09-14

Boris S. Mordukhovich , Nguyen Mau Nam , Juan Salinas

This article is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here … This article is a continuation of our ongoing efforts to solve a number of geometric problems and their extensions by using advanced tools of variational analysis and generalized differentiation. Here we propose and study, from both qualitative and numerical viewpoints, the following optimal location problem as well as its further extensions: on a given nonempty subset of a Banach space, find a point such that the sum of the distances from it to n given nonempty subsets of this space is minimal. This is a generalized version of the classical Heron problem: on a given straight line, find a point C such that the sum of the distances from C to the given points A and B is minimal. We show that the advanced variational techniques allow us to solve optimal location problems of this type completely in some important settings.

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Limiting subgradients of minimal time functions in Banach spaces

2009-05-22

Boris S. Mordukhovich , Nguyen Mau Nam

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Variational Geometric Approach to Generalized Differential and Conjugate Calculi in Convex Analysis

2017-06-05

Boris S. Mordukhovich , Nguyen Mau Nam , R. Blake Rector +1 more

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Convergence analysis of a proximal point algorithm for minimizing differences of functions

2016-11-24

Nguyen Thai An , Nguyen Mau Nam

Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A significant progress to go beyond convexity was made … Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A significant progress to go beyond convexity was made by considering the class of functions representable as differences of convex functions. In this paper, we introduce a generalized proximal point algorithm to minimize the difference of a nonconvex function and a convex function. We also study convergence results of this algorithm under the main assumption that the objective function satisfies the Kurdyka–ᴌojasiewicz property.

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Coderivative calculus and metric regularity for constraint and variational systems

2007-12-28

W. Geremew , Boris S. Mordukhovich , Nguyen Mau Nam

Variational Analysis of Directional Minimal Time Functions and Applications to Location Problems

2013-03-05

Nguyen Mau Nam , Constantin Zălinescu

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Geometric approach to convex subdifferential calculus

2015-11-07

Boris S. Mordukhovich , Nguyen Mau Nam

AbstractIn this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural … AbstractIn this paper, we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic results of convex subdifferential calculus in full generality and also derive new results of convex analysis concerning optimal value/marginal functions, normals to inverse images of sets under set-valued mappings, calculus rules for coderivatives of single-valued and set-valued mappings, and calculating coderivatives of solution maps to parameterized generalized equations governed by set-valued mappings with convex graphs.Keywords: convex analysisgeneralized differentiationgeometric approachconvex separationnormal conesubdifferentialcoderivativecalculus rulesmaximum functionoptimal value functionAMS Subject Classifications: 49J5249J5390C31 AcknowledgementsThe authors are grateful to two anonymous referees and the handling Editor for their valuable remarks, which allowed us to improve the original presentation.NotesNo potential conflict of interest was reported by the authors.Dedicated to Franco Giannessi and Diethard Pallaschke with great respect.Additional informationFundingMordukhovich was partially supported by the National Science Foundation [grant number DMS-1007132], [grant number DMS-1512846]; the Air Force Office of Scientific Research [grant number 15RT0462]. Nam was partially supported by the NSF [grant number DMS-1411817]; the Simons Foundation [grant number 208785].

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Subgradient of distance functions with applications to Lipschitzian stability

2005-07-14

Boris S. Mordukhovich , Nguyen Mau Nam

Metric Regularity of Mappings and Generalized Normals to Set Images

2009-10-16

Boris S. Mordukhovich , Nguyen Mau Nam , Bingwu Wang

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Partial Second-Order Subdifferentials in Variational Analysis and Optimization

2014-07-03

Boris S. Mordukhovich , Nguyen Mau Nam , Nguyen Thi Yen Nhi

This article presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. … This article presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. The main results concern developing extended calculus rules for these second-order constructions in both finite-dimensional and infinite-dimensional frameworks. We also provide new applications of partial second-order subdifferentials to Lipschitzian stability of stationary point mappings in parametric constrained optimization and discuss some other applications.

Nonsmooth Algorithms and Nesterov's Smoothing Technique for Generalized Fermat--Torricelli Problems

2014-01-01

Nguyen Mau Nam , Nguyen Thai An , R. Blake Rector +1 more

We present algorithms for solving a number of new models of facility location which generalize the classical Fermat--Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization … We present algorithms for solving a number of new models of facility location which generalize the classical Fermat--Torricelli problem. Our first approach involves using Nesterov's smoothing technique and the minimization majorization principle to build smooth approximations that are convenient for applying smooth optimization schemes. Another approach uses subgradient-type algorithms to cope directly with the nondifferentiability of the cost functions. Convergence results of the algorithms are proved and numerical tests are presented to show the effectiveness of the proposed algorithms.

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The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

2012-04-19

Boris S. Mordukhovich , Nguyen Mau Nam , C.I Villalobos

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Applications of Convex Analysis to the Smallest Intersecting Ball Problem

2011-01-01

Nguyen Mau Nam , Nguyen Thai An , Juan Salinas

The smallest enclosing circle problem asks for the circle of smallest radius enclosing a given set of finite points on the plane. This problem was introduced in the 19th century … The smallest enclosing circle problem asks for the circle of smallest radius enclosing a given set of finite points on the plane. This problem was introduced in the 19th century by Sylvester [17]. After more than a century, the problem remains very active. This paper is the continuation of our effort in shedding new light to classical geometry problems using advanced tools of convex analysis and optimization. We propose and study the following generalized version of the smallest enclosing circle problem: given a finite number of nonempty closed convex sets in a reflexive Banach space, find a ball with the smallest radius that intersects all of the sets.

Subgradients of Minimal Time Functions Under Minimal Requirements

2010-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

This paper concerns the study of a broad class of minimal time functions corresponding to control problems with constant convex dynamics and closed target sets in arbitrary Banach spaces. In … This paper concerns the study of a broad class of minimal time functions corresponding to control problems with constant convex dynamics and closed target sets in arbitrary Banach spaces. In contrast to other publications, we do not impose any nonempty interior and/or calmness assumptions on the initial data and deal with generally non-Lipschitzian minimal time functions. The major results present refined formulas for computing various subgradients of minimal time functions under minimal requirements in both cases of convex and nonconvex targets. Our technique is based on advanced tools of variational analysis and generalized differentiation.

A Generalized Sylvester Problem and a Generalized Fermat-Torricelli Problem

2012-01-01

Nguyen Mau Nam , Nguyen Hoang

In this paper, we introduce and study the following problem and its further generalizations: given two finite collections of sets in a normed space, find a ball whose center lies … In this paper, we introduce and study the following problem and its further generalizations: given two finite collections of sets in a normed space, find a ball whose center lies in a given constraint set with the smallest radius that encloses all the sets in the first collection and intersects all the sets in the second one. This problem can be considered as a generalized version of the Sylvester smallest enclosing circle problem introduced in the 19th century by Sylvester which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We also consider a generalized version of the Fermat-Torricelli problem: given two finite collections of sets in a normed space, find a point in a given constraint set that minimizes the sum of the farthest distances to the sets in the first collection and shortest distances (distances) to the sets in the second collection.

Variational analysis of extended generalized equations via coderivative calculus in Asplund spaces

2008-05-29

Boris S. Mordukhovich , Nguyen Mau Nam

Nesterov’s smoothing technique and minimizing differences of convex functions for hierarchical clustering

2017-08-29

Nguyen Mau Nam , W. Geremew , Samuel E. Reynolds +1 more

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Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics

2012-04-02

Nguyen Mau Nam , M. C. Villalobos , Nguyen Thai An

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Minimizing Differences of Convex Functions with Applications to Facility Location and Clustering

2017-02-07

Nguyen Mau Nam , R. Blake Rector , Daniel Giles

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Quasi-relative interiors for graphs of convex set-valued mappings

2019-06-26

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam

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Extremality of convex sets with some applications

2017-01-06

Boris S. Mordukhovich , Nguyen Mau Nam

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Subdifferential formulas for a class of non-convex infimal convolutions

2014-10-22

Nguyen Mau Nam

In this paper, we provide a number of subdifferential formulas for a class of non-convex infimal convolutions in normed spaces. The formulas obtained unify several results on subdifferentials of the … In this paper, we provide a number of subdifferential formulas for a class of non-convex infimal convolutions in normed spaces. The formulas obtained unify several results on subdifferentials of the distance function and the minimal time function. In particular, we generalize the results obtained recently by Zhang et al.

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Metric regularity, tangential distances and generalized differentiation in Banach spaces

2011-02-20

Nguyen Mau Nam , Bingwu Wang

Introduction to Mathematical Analysis

2015-06-01

Beatriz Lafferriere , Gerardo Lafferriere , Nguyen Mau Nam

A DC programming approach for solving multicast network design problems via the Nesterov smoothing technique

2018-06-11

W. Geremew , Nguyen Mau Nam , Alexander Semenov +2 more

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Smoothing techniques and difference of convex functions algorithms for image reconstructions

2019-08-03

Nguyen Mau Nam , Hoai An Le Thi , Daniel Giles +1 more

In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov's smoothing techniques in infinite dimensions. As an … In this paper, we study characterizations of differentiability for real-valued functions based on generalized differentiation. These characterizations provide the mathematical foundation for Nesterov's smoothing techniques in infinite dimensions. As an application, we provide a simple approach to image reconstructions based on Nesterov's smoothing and algorithms for minimizing differences of convex (DC) functions that involve the ℓ1−ℓ2 regularization.

Subgradients of Minimal Time Functions Under Minimal Requirements

2010-09-08

Boris S. Mordukhovich , Nguyen Mau Nam

The Log-Exponential Smoothing Technique and Nesterov’s Accelerated Gradient Method for Generalized Sylvester Problems

2015-09-29

Nguyen Thai An , Daniel Giles , Nguyen Mau Nam +1 more

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SUBGRADIENTS OF DISTANCE FUNCTIONS AT OUT-OF-SET POINTS

2006-02-01

Boris S. Mordukhovich , Nguyen Mau Nam

This paper deals with the classical distance function to closed sets and its extension to the case of set-valued mappings. It has been well recognized that the distance functions play … This paper deals with the classical distance function to closed sets and its extension to the case of set-valued mappings. It has been well recognized that the distance functions play a crucial role in many aspects of variational analysis, optimization, and their applications. One of the most remarkable properties of even the classical distance function is its intrinsic nonsmoothness, which requires the usage of generalized differential constructions for its study and applications. In this paper we present new results in theser directions using mostly the generalized differential constructions introduced earlier by the first author, as well as their recent modifications. We pay the main attention to studying subgradients of the distance functions in out-of-set points, which is essentially more involved in comparison with the in-set case. Most of the results obtained are new in both finite-dimensional and infinite-dimensional settings; some of them of provide essential improvements of known results even for convex sets.

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On a convex set with nondifferentiable metric projection

2015-01-12

Shyan Akmal , Nguyen Mau Nam , J. J. P. Veerman

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Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions

2014-12-02

Nguyen Mau Nam , Dang Van Cuong

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The Fermat-Torricelli Problem in the Light of Convex Analysis

2013-02-21

Nguyen Mau Nam

In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to … In the early 17th century, Pierre de Fermat proposed the following problem: given three points in the plane, find a point such that the sum of its Euclidean distances to the three given points is minimal. This problem was solved by Evangelista Torricelli and was named the Fermat-Torricelli problem. A more general version of the Fermat-Torricelli problem asks for a point that minimizes the sum of the distances to a finite number of given points in $\Bbb R^n$. This is one of the main problems in location science. In this paper, we revisit the Fermat-Torricelli problem from both theoretical and numerical viewpoints using some ingredients of of convex analysis and optimization.

Constructions of Solutions to Generalized Sylvester and Fermat–Torricelli Problems for Euclidean Balls

2013-07-15

Nguyen Mau Nam , Nguyen Hoang , Nguyen Thai An

Exact calculus for proximal subgradients with applications to optimization

2007-04-01

Boris S. Mordukhovich , Nguyen Mau Nam

The paper contains new exact calculus rules for proximal subgradients of extended-real-valued functions defined on arbitrary real Banach spaces. We also develop efficient formulas for evaluating proximal subgradients of marginal/value … The paper contains new exact calculus rules for proximal subgradients of extended-real-valued functions defined on arbitrary real Banach spaces. We also develop efficient formulas for evaluating proximal subgradients of marginal/value functions in various problems of parametric optimization. The results obtained are employed to derive new necessary optimality conditions in unconstrained nondifferentiable programming.

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Variational Geometric Approach to Generalized Differential and Fenchel Conjugate Calculi in Convex Analysis

2016-06-28

Boris S. Mordukhovich , Nguyen Mau Nam , R. Blake Rector +1 more

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in … This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes.

Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors

2022-03-12

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

In this paper we provide further studies of Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong … In this paper we provide further studies of Fenchel duality theory in the general framework 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.

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Lipschitz Properties of Nonsmooth Functions and Set-Valued Mappings via Generalized Differentiation and Applications

2013-01-01

Nguyen Mau Nam , Gerardo Lafferriere

In this paper, we revisit the Mordukhovich's subdifferential criterion for Lipschitz continuity of nonsmooth functions and coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria … In this paper, we revisit the Mordukhovich's subdifferential criterion for Lipschitz continuity of nonsmooth functions and coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, that play an important role in many aspects of nonsmooth analysis and optimization.

Minimal time functions and the smallest intersecting ball problem generated by unbounded dynamics

2011-01-01

Nguyen Mau Nam , C.I Villalobos , Nguyen Thai An

The smallest enclosing circle problem introduced in the 19th century by J. J. Sylvester [20] aks for the circle of smallest radius enclosing a given set of finite points in … The smallest enclosing circle problem introduced in the 19th century by J. J. Sylvester [20] aks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of the smallest enclosing circle problem called the smallest intersecting ball problem was considered in [17,18]: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to extensions of the smallest intersecting ball problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location.

A Unified Approach to Convex and Convexified Generalized Differentiation of Nonsmooth Functions and Set-Valued Mappings

2014-06-02

Nguyen Mau Nam , Nguyễn Đình Hoàng , R. Blake Rector

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Clustering and multifacility location with constraints via distance function penalty methods and dc programming

2018-09-02

Nguyen Mau Nam , Nguyen Thai An , Samuel E. Reynolds +1 more

This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and … This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and apply it particularly to a number of problems in clustering and multifacility location in which the centers to be found must lie in some given set constraints. We also provide numerical examples to test our method.

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Lagrange Multipliers and Duality with Applications to Constrained Support Vector Machine

2025-01-02

Nguyen Mau Nam , Gary Sandine , Quoc Tran-Dinh

In this paper, we employ the concept of quasi-relative interior to analyze the method of Lagrange multipliers and establish strong Lagrangian duality for nonsmooth convex optimization problems in Hilbert spaces. … In this paper, we employ the concept of quasi-relative interior to analyze the method of Lagrange multipliers and establish strong Lagrangian duality for nonsmooth convex optimization problems in Hilbert spaces. Then, we generalize the classical support vector machine (SVM) model by incorporating a new geometric constraint or a regularizer on the separating hyperplane, serving as a regularization mechanism for the SVM. This new SVM model is examined using Lagrangian duality and other convex optimization techniques in both theoretical and numerical aspects via a new subgradient algorithm as well as a primal-dual method.

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Nash equilibrium and minimax theorems via variational tools of convex analysis

2024-12-26

Nguyen Xuan Duy Bao , Boris S. Mordukhovich , Nguyen Mau Nam

In this paper, we first provide a simple variational proof of the existence of Nash equilibrium in Hilbert spaces by using optimality conditions in convex minimization and Schauder's fixed-point theorem. … In this paper, we first provide a simple variational proof of the existence of Nash equilibrium in Hilbert spaces by using optimality conditions in convex minimization and Schauder's fixed-point theorem. Then applications of convex analysis and generalized differentiation are given to the existence of Nash equilibrium and extended versions of von Neumann's minimax theorem in locally convex topological vector spaces. Our analysis in this part combines generalized differential tools of convex analysis with elements of fixed point theory revolving around Kakutani's fixed-point theorem and related issues.

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Properties of generalized polyhedral convex multifunctions

2024-10-28

Nguyen Ngoc Luan , Nguyen Mau Nam , N. D. Yen

This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains … This paper presents a study of generalized polyhedral convexity under basic operations on multifunctions. We address the preservation of generalized polyhedral convexity under sums and compositions of multifunctions, the domains and ranges of generalized polyhedral convex multifunctions, and the direct and inverse images of sets under such mappings. Then we explore the class of optimal value functions defined by a generalized polyhedral convex objective function and a generalized polyhedral convex constrained mapping. The new results provide a framework for representing the relative interior of the graph of a generalized polyhedral convex multifunction in terms of the relative interiors of its domain and mapping values in locally convex topological vector spaces. Among the new results in this paper is a significant extension of a result by Bonnans and Shapiro on the domain of generalized polyhedral convex multifunctions from Banach spaces to locally convex topological vector spaces.

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On Strong Quasiconvexity of Functions in Infinite Dimensions

2024-09-25

Nguyen Mau Nam , Jacob Sharkansky

In this paper, we explore the concept of $\sigma$-quasiconvexity for functions defined on normed vector spaces. This notion encompasses two important and well-established concepts: quasiconvexity and strong quasiconvexity. We start … In this paper, we explore the concept of $\sigma$-quasiconvexity for functions defined on normed vector spaces. This notion encompasses two important and well-established concepts: quasiconvexity and strong quasiconvexity. We start by analyzing certain operations on functions that preserve $\sigma$-quasiconvexity. Next, we present new results concerning the strong quasiconvexity of norm and Minkowski functions in infinite dimensions. Furthermore, we extend a recent result by F. Lara [16] on the supercoercive properties of strongly quasiconvex functions, with applications to the existence and uniqueness of minima, from finite dimensions to infinite dimensions. Finally, we address counterexamples related to strong quasiconvexity.

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Qualitative Analysis and Adaptive Boosted DCA for Generalized Multi-Source Weber Problems

2024-09-20

Vo Trong Long , Nguyen Mau Nam , Tuyen Tran +1 more

This paper has two primary objectives. First, we investigate fundamental qualitative properties of the generalized multi-source Weber problem formulated using the Minkowski gauge function. This includes proving the existence of … This paper has two primary objectives. First, we investigate fundamental qualitative properties of the generalized multi-source Weber problem formulated using the Minkowski gauge function. This includes proving the existence of global optimal solutions, demonstrating the compactness of the solution set, and establishing optimality conditions for these solutions. Second, we apply Nesterov's smoothing and the adaptive Boosted Difference of Convex functions Algorithm (BDCA) to solve both the unconstrained and constrained versions of the generalized multi-source Weber problems. These algorithms build upon the work presented in [6,19]. We conduct a comprehensive evaluation of the adaptive BDCA, comparing its performance to the method proposed in [19], and provide insights into its efficiency.

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Qualitative Properties of $k-$Center Problems

2024-09-18

Vo Trong Long , Nguyen Mau Nam , Jacob Sharkansky +1 more

In this paper, we study generalized versions of the k-center problem, which involves finding k circles of the smallest possible equal radius that cover a finite set of points in … In this paper, we study generalized versions of the k-center problem, which involves finding k circles of the smallest possible equal radius that cover a finite set of points in the plane. By utilizing the Minkowski gauge function, we extend this problem to generalized balls induced by various convex sets in finite dimensions, rather than limiting it to circles in the plane. First, we establish several fundamental properties of the global optimal solutions to this problem. We then introduce the notion of local optimal solutions and provide a sufficient condition for their existence. We also provide several illustrative examples to clarify the proposed problems.

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Nash Equilibrium and Minimax Theorems via Variational Tools of Convex Analysis

2024-08-26

Nguyen Xuan Duy Bao , Boris S. Mordukhovich , Nguyen Mau Nam

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A Notion of Fenchel Conjugate for Set-Valued Mappings

2024-05-28

Nguyen Mau Nam , Gary Sandine , Nguyen Nang Thieu +1 more

Generalized relative interiors and generalized convexity in infinite-dimensional spaces

2024-05-24

Vo Si Trong Long , Boris S. Mordukhovich , Nguyen Mau Nam

This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. … This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings.

Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

2023-07-18

Nguyen Ngoc Luan , Nguyen Mau Nam , Nguyen Nang Thieu +1 more

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Near Convexity and Generalized Differentiation

2023-01-01

Nguyen Mau Nam , Nguyen Nang Thieu , Nguyen Dong Yen

In this paper, we introduce the concept of nearly convex set-valued mappings and investigate fundamental properties of these mappings. Additionally, we establish a geometric approach for generalized differentiation of nearly … In this paper, we introduce the concept of nearly convex set-valued mappings and investigate fundamental properties of these mappings. Additionally, we establish a geometric approach for generalized differentiation of nearly convex set-valued mappings and nearly convex functions. Our contributions expand the current knowledge of nearly convex sets and functions, while providing several new results pertaining to nearly convex set-valued mappings.

Nearly Convex Optimal Value Functions and Some Related Topics

2023-01-01

Nguyễn Quang Huy , Nguyen Mau Nam , Nguyen Dong Yen

In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory … In this paper, we introduce new properties of the relative interior calculus for nearly convex sets, functions, and set-valued mappings. These properties are important for the development of duality theory in optimization. Then we investigate optimal value functions defined by nearly convex functions and nearly convex set-valued mappings, and derive the near convexity of the optimal value function under a qualification condition. We also develop formulas for subgradients and Fenchel conjugates of this class of functions, and explore their applications to duality theory.

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Properties of Generalized Polyhedral Convex Multifunctions

2023-01-01

Nguyen Ngoc Luan , Nguyen Mau Nam , Nguyen Dong Yen

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Minimal Time Functions and Related Issues

2023-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

Our main attention in this chapter is drawn to the study of minimal time functions, which belong to a rather new class of nondifferentiable functions that are challenging theoretically and … Our main attention in this chapter is drawn to the study of minimal time functions, which belong to a rather new class of nondifferentiable functions that are challenging theoretically and important for various applications.

Fenchel Conjugate and Further Topics in Subdifferentiation

2023-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

Duality is one of the central themes of convex analysis and its applications. A large part of this chapter is devoted to Fenchel conjugates, their calculus, and duality relationships. We … Duality is one of the central themes of convex analysis and its applications. A large part of this chapter is devoted to Fenchel conjugates, their calculus, and duality relationships. We also consider here some other related topics of generalized differentiation addressing directional derivatives and convex subdifferentiation for a broad class of supremum functions.

Remarkable Consequences of Convexity

2023-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

This chapter is devoted to remarkable topics of convex analysis that, together with subdifferential calculus, Fenchel conjugates and related results of previous chapters, constitute major achievements of this discipline highly … This chapter is devoted to remarkable topics of convex analysis that, together with subdifferential calculus, Fenchel conjugates and related results of previous chapters, constitute major achievements of this discipline highly important for many applications—first of all, to convex optimization.

Convex Separation and Some Consequences

2023-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

This chapter mainly concerns convex separation theorems that play a fundamental role in convex analysis and its numerous applications. We also study here the notion of the relative interior of … This chapter mainly concerns convex separation theorems that play a fundamental role in convex analysis and its numerous applications. We also study here the notion of the relative interior of convex sets and its connection to proper convex separation. Among the major applications of convex separation and relative interior results, we establish the normal cone intersection rule for convex sets, which is crucial for developing calculus rules of generalized differentiation for convex functions and set-valued mappings via the geometric approach of convex analysis.

Fenchel Conjugate of Set-Valued Mappings

2023-01-01

Nguyen Mau Nam , Gary Sandine , Nguyen Nang Thieu +1 more

In this paper, we present a novel concept of the Fenchel conjugate for set-valued mappings and investigate its properties in finite and infinite dimensions. After establishing the fundamental properties of … In this paper, we present a novel concept of the Fenchel conjugate for set-valued mappings and investigate its properties in finite and infinite dimensions. After establishing the fundamental properties of the Fenchel conjugate for set-valued mappings, we derive its main calculus rules in various settings. Our approach is geometric and draws inspiration from the successful application of this method by B. S. Mordukhovich and coauthors in variational and convex analysis. Subsequently, we demonstrate that our new findings for the Fenchel conjugate of set-valued mappings can be utilized to obtain many old and new calculus rules of convex generalized differentiation in both finite and infinite dimensions.

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Generalized Relative Interiors and Generalized Convexity in Infinite Dimensions

2023-01-01

Vo Trong Long , Boris S. Mordukhovich , Nguyen Mau Nam

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Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

2022-08-08

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors

2022-03-12

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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

2022-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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 … 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.

CONVEX GENERALIZED DIFFERENTIATION

2022-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

MISCELLANEOUS TOPICS ON CONVEXITY

2022-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

VARIATIONAL TECHNIQUES AND FURTHER SUBGRADIENT STUDY

2022-01-01

Boris S. Mordukhovich , Nguyen Mau Nam

Relationships between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

2022-01-01

Nguyen Ngoc Luan , Nguyen Mau Nam , Nguyen Nang Thieu +1 more

In this paper we study some relationships between polyhedral convex sets (PCS) and generalized polyhedral convex sets (GPCS). In particular, we clarify by a counterexample that the necessary and sufficient … In this paper we study some relationships between polyhedral convex sets (PCS) and generalized polyhedral convex sets (GPCS). In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a PCS obtained by Kung Fu Ng and Wen Song in [Fenchel duality in finite-dimensional setting and its applications, Nonlinear Anal. 55(2003), 845--858; Theorem~3.1] are no longer valid when considering GPCS instead of PCS. We also introduce and study the notions of generalized polyhedral set-valued mappings and optimal value functions generated by generalized polyhedral convex set-valued mappings along with their generalized differentiation calculus rules.

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Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

2021-06-30

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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 … 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].

Convex analysis of minimal time and signed minimal time functions

2021-04-12

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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 … 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 the 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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Fenchel-Rockafellar Theorem in Infinite Dimensions via Generalized Relative Interiors

2021-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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 … 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.

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Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

2021-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

Convex Analysis of Minimal Time and Signed Minimal Time Functions

2020-10-21

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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 … 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.

Algebraic core and convex calculus without topology

2020-08-04

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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 … 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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Solving a continuous multifacility location problem by DC algorithms

2020-05-29

Anuj Bajaj , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

The paper presents a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges … The paper presents a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as those of continuous optimization and then develop a new DC type algorithm for their solutions involving Nesterov's smoothing. The proposed algorithm is computationally implemented via MATLAB numerical tests on both artificial and real data sets.

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On the Equivalence of the Hahn-Banach Theorem and Convex Separation Theorems with Applications to Convexity in Vector Spaces

2020-01-01

Boris S. Mordukhovich , Nguyen Mau Nam , Addison Cartmell +1 more

Algebraic Core and Convex Calculus without Topology

2020-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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

2020-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam

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 … 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.

Convex Analysis of Minimal Time and Signed Minimal Time Functions

2020-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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Algebraic Core and Convex Calculus without Topology

2020-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

Solving k-center problems involving sets based on optimization techniques

2019-10-23

Nguyen Thai An , Nguyen Mau Nam , Xiaolong Qin

Smoothing techniques and difference of convex functions algorithms for image reconstructions

2019-08-03

Nguyen Mau Nam , Hoai An Le Thi , Daniel Giles +1 more

Quasi-relative interiors for graphs of convex set-valued mappings

2019-06-26

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam

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Solving Multifacility Location Problems by DC Algorithms

2019-06-25

Anuj Bajaj , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

Solving a Continuous Multifacility Location Problem by DC Algorithms

2019-06-25

Anuj Bajaj , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

Bornological Coderivative and Subdifferential Calculus in Smooth Banach Spaces

2019-01-02

Nguyen Mau Nam , Hung M. Phan , Bingwu Wang

Solving a Continuous Multifacility Location Problem by DC Algorithms

2019-01-01

Anuj Bajaj , Boris S. Mordukhovich , Nguyen Mau Nam +1 more

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Generalized Differential Calculus in Infinite-Dimensional Convex Analysis via Quasi-Relative Interiors

2018-12-03

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam

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 … 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.

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

2018-12-03

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam

Clustering and multifacility location with constraints via distance function penalty methods and dc programming

2018-09-02

Nguyen Mau Nam , Nguyen Thai An , Samuel E. Reynolds +1 more

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A DC programming approach for solving multicast network design problems via the Nesterov smoothing technique

2018-06-11

W. Geremew , Nguyen Mau Nam , Alexander Semenov +2 more

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

2018-01-01

Dang Van Cuong , Boris S. Mordukhovich , Nguyen Mau Nam

Coauthor	Papers Together
Boris S. Mordukhovich	69
Nguyen Thai An	23
Dang Van Cuong	22
R. Blake Rector	13
Nguyen Dong Yen	10
Gary Sandine	9
Tuyen Tran	7
W. Geremew	6
Nguyen Nang Thieu	5
Daniel Giles	5
Nguyen Hoang	5
Juan Salinas	5
Addison Cartmell	4
Anuj Bajaj	4
Gerardo Lafferriere	4
Nguyen Ngoc Luan	4
Bingwu Wang	4
Tuyen Quang Tran	3
Constantin Zălinescu	3
C.I Villalobos	3
J. J. P. Veerman	3
Nguyễn Đình Hoàng	3
Shyan Akmal	3
N. D. Yen	3
Eduardo L. Pasiliao	2
Vladimir Boginski	2
Alexander Semenov	2
Beatriz Lafferriere	2
Sam Raynolds	2
Nguyen Xuan Duy Bao	2
Han Le	2
Mike Wells	2
Hung M. Phan	2
Vo Trong Long	2
Jie Sun	2
Samuel E. Reynolds	2
Nguyen Thi Yen Nhi	1
Cristina Villalobos	1
Jacob Sharkansky	1
Nguyen Hoang	1
Nguyen Dong Yen	1
Nguyễn Huy Hoàng	1
Hoai An Le Thi	1
Jacob Sharkansky	1
Nam Tran Van	1
T.H. Tran	1
Tuyen Tran	1
Daniel Giles	1
Quoc Tran-Dinh	1
Nguyễn Quang Huy	1