Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone

Abstract Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212–221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.

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On continuity properties of some classes of vector-valued functions

Mathematica Slovaca ◽

10.2478/s12175-011-0056-8 ◽

2011 ◽

Vol 61 (6) ◽

Cited By ~ 2

Author(s):

K. Naralenkov

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Continuity Properties ◽

Weakly Continuous ◽

Necessary And Sufficient ◽

Vector Valued Functions ◽

Vector Valued

AbstractWe extend the V BG* property to the context of vector-valued functions and give some characterizations of this property. Necessary and sufficient conditions for vector-valued VBG* functions to be continuous or weakly continuous, except at most on a countable set, are obtained.

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The order completeness of some spaces of vector-valued functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270004363x ◽

1974 ◽

Vol 11 (1) ◽

pp. 57-61 ◽

Cited By ~ 17

Author(s):

Donald I. Cartwright

Keyword(s):

Banach Lattice ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Banach Lattices ◽

Nuclear Operators ◽

Order Completeness ◽

Necessary And Sufficient ◽

Vector Valued Functions ◽

Order Intervals ◽

Vector Valued

Let E be a Banach lattice. Necessary and sufficient conditions are given for the order completeness of the Banach lattices C(X, E) and L1(μ, E) in terms of the compactness of the order intervals in E. The results have interpretations in terms of spaces of compact and nuclear operators.

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Error bounds for vector-valued functions: Necessary and sufficient conditions

Nonlinear Analysis ◽

10.1016/j.na.2011.05.098 ◽

2012 ◽

Vol 75 (3) ◽

pp. 1124-1140 ◽

Cited By ~ 9

Author(s):

Ewa M. Bednarczuk ◽

Alexander Y. Kruger

Keyword(s):

Error Bounds ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient ◽

Vector Valued Functions ◽

Vector Valued

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Conditions of unicity of the best non-symmetric $L_1$-approximant for continuous vector-valued functions by one-dimensional subspace

Researches in Mathematics ◽

10.15421/241218 ◽

2012 ◽

Vol 20 ◽

pp. 129

Author(s):

M.Ye. Tkachenko ◽

V.M. Traktyns'ka

Keyword(s):

Sufficient Conditions ◽

Dimensional Subspace ◽

Necessary And Sufficient Conditions ◽

Continuous Vector ◽

One Dimensional ◽

Necessary And Sufficient ◽

Vector Valued Functions ◽

Vector Valued

The necessary and sufficient conditions of the unicity of the best $(\overline{\alpha};\overline{\beta})$-approximant for the continuous vector-valued functions on the metric compact by one-dimensional subspace are obtained.

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Analogs of Fricke's theorems for analytic vector-valued functions in the unit ball having bounded L-index in joint variables.

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2019-33-1 ◽

2019 ◽

Vol 33 ◽

pp. 16-26 ◽

Cited By ~ 1

Author(s):

Vitalina Baksa ◽

Andriy Bandura ◽

Oleg Skaskiv

Keyword(s):

Unit Ball ◽

Vector Function ◽

Sufficient Conditions ◽

Maximum Modulus ◽

Function Class ◽

Analytic Vector ◽

Continuous Vector ◽

Necessary And Sufficient ◽

Vector Valued Functions ◽

Vector Valued

In this paper, we present necessary and sufficient conditions of boundedness of $\mathbb{L}$-index in joint variables for vector-functions analytic in the unit ball, where $\mathbf{L}=(l_1,l_2): \mathbb{B}^2\to\mathbb{R}^2_+$ is a positive continuous vector-function, $\mathbb{B}^2=\{z\in\mathbb{C}^2: |z|=\sqrt{|z_1|^2+|z_2|^2}\le 1\}.$ Particularly, we deduce analog of Fricke's theorems for this function class, give estimate of maximum modulus on the skeleton of bidisc. The first theorem concerns sufficient conditions. In this theorem we assume existence of some radii, for which the maximum of norm of vector-function on the skeleton of bidisc with larger radius does not exceed maximum of norm of vector-function on the skeleton of bidisc with lesser radius multiplied by some costant depending only on these radii. In the second theorem we show that boundedness of $\mathbf{L}$-index in joint variables implies validity of the mentioned estimate for all radii.

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Conditions of unicity of the best non-symmetric $L_1$-approximant

Researches in Mathematics ◽

10.15421/241312 ◽

2013 ◽

Vol 21 ◽

pp. 81

Author(s):

Yu.S. Zagorul'ko ◽

M.Ye. Tkachenko ◽

V.M. Traktyns'ka

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Continuous Vector ◽

Necessary And Sufficient ◽

Vector Valued Functions ◽

Vector Valued

The necessary and sufficient conditions of the unicity of the best $(\overline{\alpha},\overline{\beta})$-approximant with coefficient constraints for the continuous vector-valued functions on a metric compact are obtained.

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Regular semigroups, fundamental semigroups and groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700021649 ◽

1980 ◽

Vol 29 (4) ◽

pp. 475-503 ◽

Cited By ~ 13

Author(s):

D. B. McAlister

Keyword(s):

Regular Semigroup ◽

Partial Order ◽

Direct Product ◽

Homomorphic Image ◽

Sufficient Conditions ◽

Main Tool ◽

Natural Partial Order ◽

Regular Semigroups ◽

Necessary And Sufficient ◽

Fundamental Semigroups

AbstractIn this paper we obtain necessary and sufficient conditions on a regular semigroup in order that it should be an idempotent separating homomorphic image of a full subsemigroup of the direct product of a group and a fundamental or combinatorial regular semigroup. The main tool used is the concept of a prehomomrphism θ: S → T between regular semigroups. This is a mapping such that (ab) θ ≦ aθ bθ in the natural partial order on T.

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s-Goodness for Low-Rank Matrix Recovery

Abstract and Applied Analysis ◽

10.1155/2013/101974 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Lingchen Kong ◽

Levent Tunçel ◽

Naihua Xiu

Keyword(s):

Linear Transformation ◽

Optimization Problems ◽

Sufficient Conditions ◽

Nuclear Norm ◽

Low Rank ◽

Matrix Recovery ◽

Rank Matrix ◽

Necessary And Sufficient ◽

Low Rank Matrix Recovery ◽

Low Rank Matrix

Low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, and system identification and control. This class of optimization problems is generally𝒩𝒫hard. A popular approach replaces the rank function with the nuclear norm of the matrix variable. In this paper, we extend and characterize the concept ofs-goodness for a sensing matrix in sparse signal recovery (proposed by Juditsky and Nemirovski (Math Program, 2011)) to linear transformations in LMR. Using the two characteristics-goodness constants,γsandγ^s, of a linear transformation, we derive necessary and sufficient conditions for a linear transformation to bes-good. Moreover, we establish the equivalence ofs-goodness and the null space properties. Therefore,s-goodness is a necessary and sufficient condition for exacts-rank matrix recovery via the nuclear norm minimization.

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Invexity of supremum and infimum functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020335 ◽

2002 ◽

Vol 65 (2) ◽

pp. 289-306 ◽

Cited By ~ 2

Author(s):

Nguyen Xuan Ha ◽

Do Van Luu

Keyword(s):

Sufficient Conditions ◽

Infinite Family ◽

Lipschitz Functions ◽

Necessary And Sufficient Conditions ◽

Directional Derivatives ◽

Mathematical Programs ◽

Sufficient Conditions For Optimality ◽

Invex Function ◽

Necessary And Sufficient

Under suitable assumptions we establish the formulas for calculating generalised gradients and generalised directional derivatives in the Clarke sense of the supremum and the infimum of an infinite family of Lipschitz functions. From these results we derive the results ensuring such a supremum or infimum are an invex function when all functions of the invex. Applying these results to a class of mathematical programs, we obtain necessary and sufficient conditions for optimality.

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