Hoàng Xuân Phú

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ROUGH CONVERGENCE IN NORMED LINEAR SPACES

2001-03-31

Hoàng Xuân Phú

x * ∈ X is said to be an r-limit point of a sequence (xi ) in some normed linear space (X,∥ · ∥) if (r ≥ 0). The set … x * ∈ X is said to be an r-limit point of a sequence (xi ) in some normed linear space (X,∥ · ∥) if (r ≥ 0). The set of all r-limit points of (xi , denoted by LIM r x i , is bounded closed and convex. Further properties, in particular the relation between this rough convergence and other convergence notions, and the dependence of LIM r x i on the roughness degree r, are investigated. For instance, the set-valued mapping r ↦ LIM r x i is strictly increasing and continuous on (), where . For a so-called ρ-Cauchy sequence (xi ) satisfying it is shown in case X = R n that r = (n/(n + 1))ρ (or for Euclidean space) is the best convergence degree such that LIM r x i ≠ Ø.

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Rough Convergence in Infinite Dimensional Normed Spaces

2003-01-08

Hoàng Xuân Phú

Abstract For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, … Abstract For given r, ρ ≥ 0, a sequence (xi ) in some normed linear space X is said to be r-convergent if the r-limit set defined by is nonempty, and it is called a ρ-Cauchy sequence if This article investigates different aspects of this rough convergence, especially in infinite dimensional spaces, such as analytical properties of r-limit set, relation to other convergence notions, and the dependence of the r-limit set on the roughness degree r. Moreover, by using the Jung constant we find the minimal value of r such that an arbitrary ρ-Cauchy sequence in X is certainly r-convergent.

ROUGH CONTINUITY OF LINEAR OPERATORS

2002-01-05

Hoàng Xuân Phú

ABSTRACT Let and be two linear normed spaces and r ≥ 0. A mapping f:X → Y is called roughly continuous at x ε X w.r.t. the roughness degree r … ABSTRACT Let and be two linear normed spaces and r ≥ 0. A mapping f:X → Y is called roughly continuous at x ε X w.r.t. the roughness degree r (or shortly: r-continuous at x) if for all ϵ > 0 there exists a δ > 0 such that where and . Under the assumption and r > 0 we show that every linear operator f:X → Y is r-continuous at every point x ε X.

Adaptive RBF-FD method for elliptic problems with point singularities in 2D

2017-07-05

Đặng Thị Oanh , Oleg Davydov , Hoàng Xuân Phú

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Six Kinds of Roughly Convex Functions

1997-02-01

Hoàng Xuân Phú

On the stability of solutions to quadratic programming problems

2001-02-01

Hoàng Xuân Phú , N. D. Yen

Stable generalization of convex functions

1996-01-01

Hoàng Xuân Phú , Phan Thanh An

A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is … A kind of generalized convex functions is said to be stable with respect to some property (P) if this property is maintaincd during an arbitrary function from this class is disturbed by a linear functional with sufficiently small norm. Unfortunately. known generallzed convexities iike quasicunvexity, explicit quasiconvexity. and pseudoconvexity are not stable with respect to such optimization properties which are expected to be true by these generalizations, even if the domain ol the functions is compact. Therefore, we introduce the notion of s-quasiconvex functions. These functions are quasiconvex, explicitly quasicon vex. and pseudoconvex if they are continuously differentiable. Especially, the s-quasiconvexity is stable with respect to the following important properties: (Pl) all lower level sets are convex, (P2) each local minimum is a global minimum. and (P3) each stationary point is a global minimizer. In this paper, different aspects. of s–quasiconvexity and its stability are investigated.

γ-Subdifferential and γ-convexity of functions on a normed space

1995-06-01

Hoàng Xuân Phú

Essential supremum and supremum of summable functions

1996-01-01

Hoàng Xuân Phú , Armin Hoffmann

Let D ≌ RN , 0 < μ(D) < +∞ and f : D → R is an arbitary summable function. Then the function is continuous, non-negative, non-increasing, convex, and … Let D ≌ RN , 0 < μ(D) < +∞ and f : D → R is an arbitary summable function. Then the function is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative . Further on, it holds ess sup ƒ = sup{α ↦ R : F(α) > 0}, where ess sup ƒ denotes the essential supremum of ƒ. These properties can be used for computing ess sup ƒ. As example, two algorithms are stated. If the function ƒ is dense, or lower semicontinuous, or if −ƒ is robust, then sup ƒ = ess sup ƒ. In this case, the algorithms mentioned can be applied for determining the supremum of ƒ, i.e., also the global maximum of ƒ if it exists.

Some analytical properties of γ-convex functions on the real line

1996-12-01

Hoàng Xuân Phú , Nguyễn Ngọc Hải

Some properties of globally δ-convex functions∗

1995-01-01

Hoàng Xuân Phú

Abstract The notions of δ-convex and midpoint δ-convex functions were introduced by Hu, Klee, and Larman (SIAM Journal on Control and Optimization, Vol. 27 1989). It is known that such … Abstract The notions of δ-convex and midpoint δ-convex functions were introduced by Hu, Klee, and Larman (SIAM Journal on Control and Optimization, Vol. 27 1989). It is known that such functions have some important optimization properties: each r-local minimum is a global minimum, and if they assume their global maximum on a bounded convex domain of a Hilbert space then they do so at least at some r-extreme points of this domain. In this paper some analytical properties of δ-convex and midpoint δ-convex functions are investigated. Concretely, it is shown when they are bounded (from above or from below). For instance, δ-convex functions defined on the entire real line is always locally bounded, and midpoint δ-convex function on the real line is either locally bounded or totally unbounded. Further on, it is proved that there are totally discontinuous (i.e., nowhere differentiable) δ-convex and midpoint δ-convex functions on the real line Keywords: Generalized ConvexityRoughly Convexδ-convexMidpoint δ-convexBoundednessContinuity ∗This research was supported by the Deutsche Forschungsgemeinschaft. The author thanks Prof Dr. E. Zeidler, Prof. Dr. H.G Bock, and Dr. J Schlöder for their hospitality and valuable support during his visit in Leipzig and Heidelberg ∗This research was supported by the Deutsche Forschungsgemeinschaft. The author thanks Prof Dr. E. Zeidler, Prof. Dr. H.G Bock, and Dr. J Schlöder for their hospitality and valuable support during his visit in Leipzig and Heidelberg Notes ∗This research was supported by the Deutsche Forschungsgemeinschaft. The author thanks Prof Dr. E. Zeidler, Prof. Dr. H.G Bock, and Dr. J Schlöder for their hospitality and valuable support during his visit in Leipzig and Heidelberg

Representation of Bounded Convex Sets By Rational Convex Hull Of Its Gamma-Extreme Points

1994-01-01

Hoàng Xuân Phú

Abstract H. Minkowski proved C= convext C whenever C is a compact convex subset of a finite-dimensional linear space. If C is bounded but not closed, this representation does not … Abstract H. Minkowski proved C= convext C whenever C is a compact convex subset of a finite-dimensional linear space. If C is bounded but not closed, this representation does not hold anymore. In this case, we introduce the set of so-called γ-extreme points extγC of C and show C = convext γ C = raco extγ C, where raco M denotes the rational convex hull of M. Keywords: Bounded convex setsextreme pointsγ-extreme pointsconvex hullrational convex hull52A01AMS(MOS) subject classifications

Symmetricallyγ-Convex Functions

1999-01-01

Nguyễn Ngọc Hải , Hoàng Xuân Phú

Given γ>0, a real-valued function f defined on a noempty convex set D of a normed space X is said to be symmetrically γ-convex iff for all x oand X … Given γ>0, a real-valued function f defined on a noempty convex set D of a normed space X is said to be symmetrically γ-convex iff for all x oand X 1∊D satisfying , the Jensen inequality is fulfilled for and for . Though the Jensen inequality is only required to hold true at two points and on the segment [x 0,x 1] satisfying , such a function has interesting properties similar to those of classical convex functions. For instance, a symmetrically γ-convex function from a finite-dimensional normed space X is locally Lipschitzian at each γ-interior point of D, i.e., at each point x∊D for which there is some ε>0 such that implies . This and other properties of this function class are established in our paper

Outer Γ-Convexity in Vector Spaces

2008-09-11

Hoàng Xuân Phú

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ⊂ X if for all x 0, x 1 ∊ … A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ⊂ X if for all x 0, x 1 ∊ S there exists a closed subset Λ ⊂ [0,1] such that {x λ | λ ∊ Λ} ⊂ S and [x 0, x 1] ⊂ {x λ | λ ∊ Λ} + 0.5 Γ, where x λ: = (1 − λ)x 0 + λ x 1. A real-valued function f:D → ℝ defined on some convex D ⊂ X is called outer Γ-convex if for all x 0, x 1 ∊ D there exists a closed subset Λ ⊂ [0,1] such that [x 0, x 1] ⊂ {x λ | λ ∊ Λ} + 0.5 Γ and f(x λ) ≤ (1 − λ)f(x 0) + λ f(x 1) holds for all λ ∊ Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.

Convergence Speed of an Integral Method for Computing the Essential Supremum

1997-01-01

Jens Hichert , Armin Hoffmann , Hoàng Xuân Phú

Strictly and Roughly Convexlike Functions

2003-04-01

Hoàng Xuân Phú

Some properties of boundedly perturbed strictly convex quadratic functions

2010-05-09

Hoàng Xuân Phú , Vo Minh Pho

We investigate the problem of minimizing subject to x ∈ D, where f(x) := x T Ax + b T x, A is a symmetric positive definite n-by-n matrix, b … We investigate the problem of minimizing subject to x ∈ D, where f(x) := x T Ax + b T x, A is a symmetric positive definite n-by-n matrix, b ∈ ℝ n , D ⊂ ℝ n is convex and p : ℝ n → ℝ satisfies sup x∈D |p(x)| ≤ s for some given s < +∞. Function p is called a perturbation, but it may also describe some correcting term, which arises when investigating a real inconvenient objective function by means of an idealized convex quadratic function f. We prove that is strictly outer Γ-convex for some specified balanced set Γ ⊂ ℝ n . As a consequence, a Γ-local optimal solution of is global optimal and the difference of two arbitrary global optimal solutions of is contained in Γ. By the property that holds if x* is the optimal solution of the problem of minimizing f on D and is an arbitrary global optimal solution of , we show that the set S s of global optimal solutions of is stable with respect to the Hausdorff metric d H (.,.). Moreover, the roughly generalized subdifferentiability of and a generalization of Kuhn–Tucker theorem for are presented.

Minimizing Convex Functions with Bounded Perturbations

2010-01-01

Hoàng Xuân Phú

We investigate the problem of minimizing the perturbed convex function $\tilde{f}(x)=f(x)+p(x)$ over some convex subset D of a normed linear space X, where the function f is convex and the … We investigate the problem of minimizing the perturbed convex function $\tilde{f}(x)=f(x)+p(x)$ over some convex subset D of a normed linear space X, where the function f is convex and the perturbation p is bounded. The key tool for our investigation is a convexity modulus of f named $h_1$, whose generalized inverse function $h_1^{-1}$ is used to define the quantity $\gamma^*:=h_1^{-1}(2\sup_{x\in D}|p(x)|)$. Generally, by the irregular perturbation p, the perturbed function $\tilde{f}$ loses all usual analytical and optimization properties yielded by the convexity of f. But we show that some convexity trace remains in $\tilde{f}$, namely, $\tilde{f}$ is outer $\gamma$-convex for any $\gamma\geq\gamma^*$ and strictly outer $\gamma$-convex for any $\gamma>\gamma^*$. As a consequence, each $\gamma^*$-minimizer $x^*\in D$ defined by $\tilde{f}(x^*)=\inf_{x\in\bar{B}(x^*,\gamma^*)\cap D}\tilde{f}(x)$ is a global minimizer, i.e., $\tilde{f}(x^*)=\inf_{x\in D}\tilde{f}(x)$, and each $\gamma^*$-infimizer $x^*$ defined by ${\lim\inf}_{x\in D,\,x\to x^*}\tilde{f}(x)=\inf_{x\in\bar{B}(x^*,\gamma^*)\cap D}\tilde{f}(x)$ is a global infimizer, i.e., ${\lim\inf}_{x\in D,\,x\to x^*}\tilde{f}(x)=\inf_{x\in D}\tilde{f}(x)$. Moreover, the diameter of the set of global infimizers (including global minimizers) of $\tilde{f}$ is not greater than $\gamma^*$, and the distance between any global infimizer of $\tilde{f}$ and any global infimizer of f cannot exceed $\gamma^*$. The latter property is used for sensibility analysis.

Stability Of Generalized Convex Functions With Respect To Linear Disturbances*

1999-01-01

Hoàng Xuân Phú , Phan Thanh An

A function is said to be absolutely stable w.r.t. some property (P) if the sum of this function and an arbitrary continuous linear functional always fulfills (P). In this paper … A function is said to be absolutely stable w.r.t. some property (P) if the sum of this function and an arbitrary continuous linear functional always fulfills (P). In this paper we show that a lower semicontinuous function from a bounded closed interval into ℝ is convex if and only if it is absolutely stable w.r.t the property:(Po o) "each local minimum is a global minimum". This is a consequence of a more general theorem concerned with so-called γ-convexlike functions which are defined as follows:A functional ffrom som conve4ix subset Dof some linear normed space into ℝ is γ-convexlike[ ILM0002] if for all x 0and x 1in Dsatisfying , there exists a such that . Our theorem says that a lower semicontinuous function from a bounded closed interval into ℝ is γ-convexlike if and only if it is absolutely stable w.r.t. the property:(P γ) "each γ-local minimum is a global minimum" (i.e. for all x∈Dsatisfying [ ILM0007] imply for all X∈D. Hence, classical convexity and γ-convexlikeness can no more be generalized without loosing the absolute stability w.r.t. (P 0) or (P γ), respectively

Some Geometrical Properties of Outer γ-Convex Sets

2003-01-08

Hoàng Xuân Phú

Abstract A subset S of some normed linear space X is said to be outer γ-convex w.r.t. some given γ >0 provided that for all x 0, x 1∈ S … Abstract A subset S of some normed linear space X is said to be outer γ-convex w.r.t. some given γ >0 provided that for all x 0, x 1∈ S there exist which belongs to S and the segment [x 0,x 1] connecting x 0 and x 1 such that for i = 0,1,…,j, where and . This article is devoted to some geometrical properties of such a set. For instance, if S is closed and outer γ-convex then any lies in some simplex whose vertices belong to S and whose diameter does not exceed γ. This implies that for , where . As consequence, if S is outer γ-convex and if x∈ X satisfies for some then , and therefore, some non-zero continuous linear functional strictly separates x and S.

Approximate Fixed-Point Theorems for Discontinuous Mappings

2004-05-18

Hoàng Xuân Phú

Abstract For a given r ≥ 0, a mapping T : M → M on some convex subset M of a normed linear space X is said to be around … Abstract For a given r ≥ 0, a mapping T : M → M on some convex subset M of a normed linear space X is said to be around r-continuous if for all x ∈ M and ϵ > 0 there exists δ > 0 such that ‖Ty − Tz‖ < r + ϵ holds whenever y, z ∈ M, ‖y − x‖ < δ, and ‖z − x‖ < δ. If δ does not depend on x then T is called uniformly r-continuous. By using the self-Jung constant J s (X) ∈ [1, 2], we state some theorems on approximate fixed points of such mappings. For instance, if M is compact and T is around r-continuous then, for all ρ > 0, there exists x * ∈ M satisfying , where ρ can be replaced by zero under some additional assumptions. This property remains true if M is only relatively compact but T is uniformly r-continuous, or if the relative compactness of M is replaced by the relative compactness of T(M).

Einige notwendige Optimalitätsbedingungen für einfache reguläre Aufgaben der optimalen Steuerung

1986-10-31

Hoàng Xuân Phú

Three necessary conditions for optimal solutions of special regular optimal control problems with state and control constraints are proved. These conditions are then applied to a class of problems. Three necessary conditions for optimal solutions of special regular optimal control problems with state and control constraints are proved. These conditions are then applied to a class of problems.

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Piecewise Constant Roughly Convex Functions

2003-05-01

Hoàng Xuân Phú , Nguyễn Ngọc Hải , Phan Thanh An

Supremizers of inner γ-convex functions

2007-10-23

Hoàng Xuân Phú

Fixed-Point Properties of Roughly Contractive Mappings

2003-09-30

Hoàng Xuân Phú

For given k \in (0,1) and r > 0 , a self-mapping T: \, M \to M is said to be r -roughly k -contractive provided \|Tx - Ty\| \le … For given k \in (0,1) and r > 0 , a self-mapping T: \, M \to M is said to be r -roughly k -contractive provided \|Tx - Ty\| \le k\,\|x - y\| + r \quad (x,y \in M). To state fixed-point properties of such a mapping, the self-Jung constant J_s(X) is used, which is defined as the supremum of the ratio 2\,r_{\mathrm{conv} S}(S)/\mathrm{diam} S over all non-empty, non-singleton and bounded subsets S of some normed linear space X , where r_{\mathrm{conv} S}(S) = \inf_{x\in\mathrm{conv} S} \sup_{y\in S} \|x - y\| is the self-radius of S and \mathrm{diam} S is its diameter. If M is a closed and convex subset of some finite-dimensional normed space X and if T: \, M \to M is r -roughly k -contractive, then for all \varepsilon > 0 there exists x^* \in M such that \|x^* - Tx^*\| < \frac12\,J_s(X)\, r + \varepsilon. If \mathrm{dim} X = 1 , or X is some two-dimensional strictly convex normed space, or X is some Euclidean space, then there is x^* \in M satisfying \|x^* - Tx^*\| \le {1 \over 2}\,J_s(X)\,r .

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Global infimum of strictly convex quadratic functions with bounded perturbations

2010-08-27

Hoàng Xuân Phú , Vo Minh Pho

Some properties of solution sets to nonconvex quadratic programming problems

2007-05-24

Hoàng Xuân Phú

This article deals with some properties of the global minimizer set GQ , the local minimizer set LQ , and the stationary point set SQ to the quadratic programming problem … This article deals with some properties of the global minimizer set GQ , the local minimizer set LQ , and the stationary point set SQ to the quadratic programming problem (Q) of minimizing the function f(x)=(1/2)xTAx+bTx on the polyhedron , where , i∈I={1,2,…, m}. In particular, we investigate the intersection of these solution sets with faces and pseudofaces , where J⊂I. Some selected results are the following. If GQ∩DJ≠ =∅ then GQ ∩ DJ and are relatively affine in the following sense: GQ∩DJ =aff(GQ∩DJ)∩DJ and . If LQ∩DJ≠ =∅ then LQ ∩ DJ is open relative to aff(LQ∩DJ)∩DJ , is open relative to , and LQ ∩ DJ and are convex. If GQ∩DJ≠ =∅ then each stationary point (in particular, each local minimizer) in is a global minimizer. If x0∈LQ∩DJ , , and x0≠ = x1 , then [x0,x1)⊂LQ∩DJ⊂LQ . Let and denote the maximal number of nonempty faces and the maximal cardinality of an antichain of nonempty faces of a polyhedron defined as intersection of m closed halfspaces in . Then GQ (or LQ , or SQ , respectively) contains a segment connecting two distinct points if it possesses more than (or , or , respectively) different points.

Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces

2005-08-09

Hoàng Xuân Phú , Nguyễn Ngọc Hải

Maximizing Strictly Convex Quadratic Functions with Bounded Perturbations

2011-01-04

Hoàng Xuân Phú , Vo Minh Pho , Phan Thanh An

Adaptive RBF-FD Method for Elliptic Problems with Point Singularities in 2D

2016-03-25

Đặng Thị Oanh , Oleg Davydov , Hoàng Xuân Phú

We describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with … We describe and test numerically an adaptive meshless generalized finite difference method based on radial basis functions that competes well with the finite element method on standard benchmark problems with reentrant corners of the boundary, sharp peaks and rapid oscillations in the neighborhood of an isolated point. This is achieved thanks to significant improvements introduced into the earlier algorithms of [Oleg Davydov and Dang~Thi Oanh, Adaptive meshless centers and RBF stencils for Poisson equation, Journal of Computational Physics, 230:287--304, 2011], including a new error indicator of Zienkiewicz-Zhu type.

Inner γ-Convex Functions in Normed Linear Spaces

2015-02-05

Hoàng Xuân Phú

Partial Differential Equations, Control, and Numerics

2021-08-04

Jean‐Michel Coron , Sorin Micu , Hoàng Xuân Phú

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Adaptive RBF-FD Method for Elliptic Problems with Point Singularities in 2D

2016-01-01

Đặng Thị Oanh , Oleg Davydov , Hoàng Xuân Phú

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Inner $$\delta $$-approximation of the convex hull of finite sets

2025-04-23

Nam-Dũng Hoàng , Nguyen Kieu Linh , Hoàng Xuân Phú

Fast Numerical Methods for Simulation of Chemically Reacting Flows in Catalytic Monoliths

2008-01-01

Hoang Duc Minh , Hans Georg Bock , Hoàng Xuân Phú +1 more

Inner $$\delta $$-approximation of the convex hull of finite sets

2025-04-23

Nam-Dũng Hoàng , Nguyen Kieu Linh , Hoàng Xuân Phú

Partial Differential Equations, Control, and Numerics

2021-08-04

Jean‐Michel Coron , Sorin Micu , Hoàng Xuân Phú

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Adaptive RBF-FD method for elliptic problems with point singularities in 2D

2017-07-05

Đặng Thị Oanh , Oleg Davydov , Hoàng Xuân Phú

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Adaptive RBF-FD Method for Elliptic Problems with Point Singularities in 2D

2016-03-25

Đặng Thị Oanh , Oleg Davydov , Hoàng Xuân Phú

Adaptive RBF-FD Method for Elliptic Problems with Point Singularities in 2D

2016-01-01

Đặng Thị Oanh , Oleg Davydov , Hoàng Xuân Phú

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Inner γ-Convex Functions in Normed Linear Spaces

2015-02-05

Hoàng Xuân Phú

Maximizing Strictly Convex Quadratic Functions with Bounded Perturbations

2011-01-04

Hoàng Xuân Phú , Vo Minh Pho , Phan Thanh An

Global infimum of strictly convex quadratic functions with bounded perturbations

2010-08-27

Hoàng Xuân Phú , Vo Minh Pho

Some properties of boundedly perturbed strictly convex quadratic functions

2010-05-09

Hoàng Xuân Phú , Vo Minh Pho

Minimizing Convex Functions with Bounded Perturbations

2010-01-01

Hoàng Xuân Phú

Outer Γ-Convexity in Vector Spaces

2008-09-11

Hoàng Xuân Phú

Fast Numerical Methods for Simulation of Chemically Reacting Flows in Catalytic Monoliths

2008-01-01

Hoang Duc Minh , Hans Georg Bock , Hoàng Xuân Phú +1 more

Supremizers of inner γ-convex functions

2007-10-23

Hoàng Xuân Phú

Some properties of solution sets to nonconvex quadratic programming problems

2007-05-24

Hoàng Xuân Phú

Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces

2005-08-09

Hoàng Xuân Phú , Nguyễn Ngọc Hải

Approximate Fixed-Point Theorems for Discontinuous Mappings

2004-05-18

Hoàng Xuân Phú

Fixed-Point Properties of Roughly Contractive Mappings

2003-09-30

Hoàng Xuân Phú

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Piecewise Constant Roughly Convex Functions

2003-05-01

Hoàng Xuân Phú , Nguyễn Ngọc Hải , Phan Thanh An

Strictly and Roughly Convexlike Functions

2003-04-01

Hoàng Xuân Phú

Some Geometrical Properties of Outer γ-Convex Sets

2003-01-08

Hoàng Xuân Phú

Rough Convergence in Infinite Dimensional Normed Spaces

2003-01-08

Hoàng Xuân Phú

ROUGH CONTINUITY OF LINEAR OPERATORS

2002-01-05

Hoàng Xuân Phú

ROUGH CONVERGENCE IN NORMED LINEAR SPACES

2001-03-31

Hoàng Xuân Phú

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On the stability of solutions to quadratic programming problems

2001-02-01

Hoàng Xuân Phú , N. D. Yen

Symmetricallyγ-Convex Functions

1999-01-01

Nguyễn Ngọc Hải , Hoàng Xuân Phú

Stability Of Generalized Convex Functions With Respect To Linear Disturbances*

1999-01-01

Hoàng Xuân Phú , Phan Thanh An

Six Kinds of Roughly Convex Functions

1997-02-01

Hoàng Xuân Phú

Convergence Speed of an Integral Method for Computing the Essential Supremum

1997-01-01

Jens Hichert , Armin Hoffmann , Hoàng Xuân Phú

Some analytical properties of γ-convex functions on the real line

1996-12-01

Hoàng Xuân Phú , Nguyễn Ngọc Hải

Essential supremum and supremum of summable functions

1996-01-01

Hoàng Xuân Phú , Armin Hoffmann

Stable generalization of convex functions

1996-01-01

Hoàng Xuân Phú , Phan Thanh An

γ-Subdifferential and γ-convexity of functions on a normed space

1995-06-01

Hoàng Xuân Phú

Some properties of globally δ-convex functions∗

1995-01-01

Hoàng Xuân Phú

Representation of Bounded Convex Sets By Rational Convex Hull Of Its Gamma-Extreme Points

1994-01-01

Hoàng Xuân Phú

Einige notwendige Optimalitätsbedingungen für einfache reguläre Aufgaben der optimalen Steuerung

1986-10-31

Hoàng Xuân Phú

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Coauthor	Papers Together
Phan Thanh An	4
Nguyễn Ngọc Hải	4
Oleg Davydov	3
Đặng Thị Oanh	3
Vo Minh Pho	3
Armin Hoffmann	2
Jean‐Michel Coron	1
Hans Georg Bock	1
Johannes P. Schiöder	1
Hoang Duc Minh	1
Sorin Micu	1
Nam-Dũng Hoàng	1
Jens Hichert	1
Nguyen Kieu Linh	1
N. D. Yen	1

γ-Subdifferential and γ-convexity of functions on a normed space

1995-06-01

Hoàng Xuân Phú

?-Subdifferential and ?-convexity of functions on the real line

1993-03-01

Ho�ng Xu�n Ph�

Optimization of Globally Convex Functions

1989-09-01

T. C. Hu , Victor Klee , David Larman

Convex functions have nice properties with respect to both minimization and maximization. Similar properties are established here for functions that are permitted to have bad local behavior but are globally … Convex functions have nice properties with respect to both minimization and maximization. Similar properties are established here for functions that are permitted to have bad local behavior but are globally convex in the sense that they behave “convexly” on triples of collinear points that are widely dispersed. The results illustrate a development that seems desirable in the interest of more realistic mathematical modeling: the “globalization” of important function properties. In connection with the maximization of globally convex functions over convex bodies in a given finite-dimensional normed space E, there is interest in estimating the maximum, for points c of bodies $C \subset E$, of the ratio between two measures of how close c comes to being an extreme point of C. Good estimates are obtained for the cases in which E is Euclidean or has the “max” norm.

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Local boundedness and continuity of generalized convex functions

1992-01-01

Helga Hartwig

This article deals with generalizations of the usual convexity of real-valued functions in such a manner that “convex” is extended to “-convex” and -convexity is required only on straight lines … This article deals with generalizations of the usual convexity of real-valued functions in such a manner that “convex” is extended to “-convex” and -convexity is required only on straight lines with directions from a given cone K. Under certain assumptions on the generating family and on K, for functions of such kind (called -convex on K-lines) local boundedness and continuity properties are obtained. The main results are applied to a number of examples. In particular, Morrey’s rank 1 convexity and a special type of “rough convexity” are considered

Nonlinear Functional Analysis and its Applications

1990-01-01

Eberhard Zeidler

Six Kinds of Roughly Convex Functions

1997-02-01

Hoàng Xuân Phú

On generalized convex functions

1983-01-01

Helga Hartwig

This paper deals with an extension of Beckenbach's classical generalized convexity notion, called F-convexity, in such a manner that much of, “modern”convexities of functions on the Euclidean R n(as log-, … This paper deals with an extension of Beckenbach's classical generalized convexity notion, called F-convexity, in such a manner that much of, “modern”convexities of functions on the Euclidean R n(as log-, ω-, [explicit] quasi- or K- convexicity can be included. General theorems concerning connections between several kinds of convexity, convexity properties of composite functions, first and second order characterizations of differentiable F-convex functions are presented.

Some analytical properties of γ-convex functions on the real line

1996-12-01

Hoàng Xuân Phú , Nguyễn Ngọc Hải

Representation of Bounded Convex Sets By Rational Convex Hull Of Its Gamma-Extreme Points

1994-01-01

Hoàng Xuân Phú

Uniformly normal structure and related coefficients

1984-04-01

Elisabetta Maluta

It is shown that uniformly normal structure implies reflexivity.In spaces with uniformly normal structure some estimates are given for the uniformity constant and for a related coefficient. Introduction.Our aim is … It is shown that uniformly normal structure implies reflexivity.In spaces with uniformly normal structure some estimates are given for the uniformity constant and for a related coefficient. Introduction.Our aim is to study two constants of a Banach space X connected with normal structure.We recall that a normed space (or a convex subset) X is said to have normal structure if for every convex bounded non-empty non-singleton subset C of X, the Chebyshev radius of C relative to C, r(C, C), is strictly smaller than the diameter of C.This concept was introduced by Brodskii and MiΓman (1948), who also gave the following characterization in terms of sequences.A space X has normal structure if and only if there exists in X no bounded non-constant sequence {x n } such that d(x n ,co{x J } n ι ' 1 ), i.e. the distance from the nth element of the sequence to the convex hull of the preceding elements, approaches the diameter of the sequence as n approaches infinity.(For normal type structures and their applications to fixed point theory, we refer to the exhaustive survey of Kirk [9].)The first constant we consider, N( X), is the already known constant of uniformity of normal structure.It has a clear geometrical meaning, for it is the supremum, taken with respect to the convex bounded subsets C of X, of the ratio between r(C,C) and the diameter of C. Hence N(X) < 1 characterizes uniformly normal structure.The second constant, D( X), is a sequence coefficient which controls the behavior, as n approaches infinity, of d(x n ,co{XjY λ ~x)\ more precisely, D(X) describes how closely this distance can approach the diameter of the sequence.For the two constants we give some evaluations and estimates.We prove they both must be one in nonreflexive spaces, thus answering in the affirmative the following question raised in [9].Does uniformly normal structure imply reflexivity?This question follows naturally from the fact that in [4] it was proved, without requiring any hypothesis of reflexivity, that a space X with N{ X) < 1 has the fixed point property for nonexpansive mappings.We prove also that, in infinite-dimensional spaces, N(X) >: 2~1 /2 , thus showing that the best value of N(X) is achieved by the space / 2 , and

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On Jung’s constant and related constants in normed linear spaces

1985-05-01

Dan Amir

In this paper several results on certain constants related to the notion of Chebyshev radius are obtained.It is shown in the first part that the Jung constant of a finite-codimensional … In this paper several results on certain constants related to the notion of Chebyshev radius are obtained.It is shown in the first part that the Jung constant of a finite-codimensional subspace of a space C(T) is 2, where T is a compact Hausdorff space which is not extremally disconnected.Several consequences are stated, e.g. the fact that every linear projection from a space C(Γ), T a perfect compact Hausdorff space, onto a finite-codimensional proper subspace has norm at least 2.The second discusses mainly the "self-Jung constant" which measures "uniform normal structure."It is shown that this constant, unlike Jung's constant, is essentially determined by the finite subsets of the space.

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Convex Regions and Projections in Minkowski Spaces

1938-04-01

F. Bohnenblust

By a well known theorem of Hahn-Banach every linear functional of norm M defined over a closed linear subspace of a Banach space' can be extended linearly to the entire … By a well known theorem of Hahn-Banach every linear functional of norm M defined over a closed linear subspace of a Banach space' can be extended linearly to the entire space without increasing its norm. For operations the corresponding problem takes the following form: Given a linear operation u(x) whose domain of definition is a closed linear subspace of a Banach space B, and whose range lies in a Banach space B2, does there exist an operation U(x) defined over B1, with range in B2 and which coincide with u(x) over the subspace? How small can the norm of the extended operation be made? In the particular case where u(x) = x, B1 = domain of definition of u; the existence of an extension U(x) is equivalent to the existence of a complementary subspace or, as F. J. Murray2 has shown, to the existence of a projection of B, on the subspace. Conversely, if A(X) is such a projection and u(x) any linear operation, the linear operation U(X) = u(A(X)) is an extension of u(x) and its norm is < 11 u 11.11 A 11. The question of extension is thus equivalent to the discussion of best projections, i.e. of projections of least norm. It is relatively easy to exhibit examples of Banach spaces for which certain closed subspaces have no projections; even more, F. J. Murray3 has shown that this occurs within the function spaces L, and the sequence spaces lp! For such subspaces an extension of an operation is not always possible. If the Banach space is finite dimensional, i.e. if we are dealing with a Minkowski space, of n dimensions say, the existence of projections is trivial but the discussion of the best possible projections (i.e. those with a minimal norm) is interesting and may help to explain why, in certain infinitely dimensional spaces, projections fail to exist. We give in this paper a complete answer to the question of best possible projections in the case where the dimensionality of the subspace on which we project is by 1 less than the dimensionality of the entire space. The results obtained lead to some interesting theorems on convex regions; they are considered in section 6.

Stability of the solution of definite quadratic programs

1973-12-01

James W. Daniel

On the stability of solutions to quadratic programming problems

2001-02-01

Hoàng Xuân Phú , N. D. Yen

Ueber die kleinste Kugel, die eine räumliche Figur einschliesst.

1901-01-01

Heinrich W. E. Jung

Zwei Extremalprobleme der Minkowski-Geometrie

1955-12-01

Kurt Leichtweiß

A RESULT IN HILBERT SPACE

1952-01-01

Norman Routledge

A RESULT IN HILBERT SPACE N. A. ROUTLEDGE N. A. ROUTLEDGE Cambridge Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume … A RESULT IN HILBERT SPACE N. A. ROUTLEDGE N. A. ROUTLEDGE Cambridge Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 3, Issue 1, 1952, Pages 12–18, https://doi.org/10.1093/qmath/3.1.12 Published: 01 January 1952 Article history Received: 20 July 1950 Revision received: 01 April 1951 Published: 01 January 1952

Outer Γ-Convexity in Vector Spaces

2008-09-11

Hoàng Xuân Phú

Characterizations of generalized monotone maps

1993-03-01

S. Karamardian , Siegfried Schaible , Jean-Pierre Crouzeix

Lower Semicontinuity of the Minimum in Parametric Convex Programs

1997-08-01

Diethard Klatte

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

2005-05-01

G. M. Lee , Nguyen Nang Tam , N. D. Yen

Characterizing an optimal input in perturbed convex programming

1983-01-01

S. Zlobec

Nonlinear Functional Analysis and Its Applications

1990-01-01

Eberhard Zeidler

On some covering and intersection properties in Minkowski spaces

1959-06-01

Branko Grünbaum

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Lipschitz behavior of convex semi-infinite optimization problems: a variational approach

2007-07-30

M. J. Cánovas , Abderrahim Hantoute , Marco A. López +1 more

Continuity of the Solution Map in Quadratic Programs under Linear Perturbations

2006-11-28

G. M. Lee , Nguyen Nang Tam , N. D. Yen

Estimates for Kuhn-Tucker points of perturbed convex programs

1988-01-01

Rüdiger Schultz

Abstract For saddle points of perturbed convex-concave functions in finite dimensional space we give a quantitative characterization of upper semicontinuity. Specifying these results to the standard Lagrangian we obtain bounds … Abstract For saddle points of perturbed convex-concave functions in finite dimensional space we give a quantitative characterization of upper semicontinuity. Specifying these results to the standard Lagrangian we obtain bounds for Kuhn-Tucker points of perturbed .convex programs without imposing differentiability of the problem functions. As a further application of the underlying techniques we present an estimate which relates the Hausdorff-distance of the graphs of the ∊-subdifferentials of convex, functions to the function distance with respect to the maximum norm. Keywords: Nondifferentiable convex programmingstability of Kuhn-Tucker pointssaddle pointsubdifferentialAMS 1980 Subject Classification: Primary:49 B 50Secondary:26 B 25

Perturbed convex programming in reflexive Banach spaces

1985-01-01

L. I. Trudzik

Some properties of globally δ-convex functions∗

1995-01-01

Hoàng Xuân Phú

Symmetricallyγ-Convex Functions

1999-01-01

Nguyễn Ngọc Hải , Hoàng Xuân Phú

Support set expansion sensitivity analysis in convex quadratic optimization

2007-05-24

Kamal Mirnia , ‎Alireza Ghaffari-Hadigheh

In support set expansion sensitivity analysis, one concerns to find the range of parameter variation where the perturbed problem has an optimal solution with the support set that includes the … In support set expansion sensitivity analysis, one concerns to find the range of parameter variation where the perturbed problem has an optimal solution with the support set that includes the support set of the given optimal solution of the unperturbed problem. In this article, we consider the perturbed convex quadratic optimization problem and present a method to identify the support set expansion sets for this problem.

Non-Linear Programming -- Theory and Methods

1977-01-01

A. G. Munford , Béla Martos

Duality theorems for perturbed convex optimization

1981-06-01

Ivan Singer

Strictly and Roughly Convexlike Functions

2003-04-01

Hoàng Xuân Phú

Seven kinds of monotone maps

1990-07-01

S. Karamardian , Siegfried Schaible

Stable generalization of convex functions

1996-01-01

Hoàng Xuân Phú , Phan Thanh An

Konvexe Mengen

1980-01-01

Kurt Leichtweiß

Stability of generalized equations and Kuhn-Tucker points of perturbed convex programs

2005-10-27

Bernd Kummer

Nonlinear Functional Analysis and its Applications

1986-01-01

Eberhard Zeidler

Generalized convexities of lower semicontinuous functions

1985-01-01

Helga Hartwig

In this paper a general theorem on the replacement of the condition “for all λ” in the definition of generalized convexity properties of lower semicontinuous functions by the condition “there … In this paper a general theorem on the replacement of the condition “for all λ” in the definition of generalized convexity properties of lower semicontinuous functions by the condition “there exists a λ” is shown. This result can be applied to a number of special kinds of convexity and completes, for instance, studies of Behbikgeb concerning (explicitly) quasiconvex functions.

On the lower semicontinuity of optimal sets in convex parametric optimization

1979-01-01

Diethard Klatte

Generalized Convexity

1994-01-01

Sandor Koml si , Tamás Rapcsák , Siegfried Schaible

Generalized convex functions

1937-01-01

E. F. Beckenbach

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Invex functions and constrained local minima

1981-12-01

B. D. Craven

If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for … If a certain weakening of convexity holds for the objective and all constraint functions in a nonconvex constrained minimization problem, Hanson showed that the Kuhn-Tucker necessary conditions are sufficient for a minimum. This property is now generalized to a property, called K -invex, of a vector function in relation to a convex cone K . Necessary conditions and sufficient conditions are obtained for a function f to be K -invex. This leads to a new second order sufficient condition for a constrained minimum.

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