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Generalized Metric Subregularity for Generalized Subsmooth Multifunctions in Asplund Spaces

SET-VALUED AND VARIATIONAL ANALYSIS 2025年第2期33卷 1-31页

作者： Gao, Ming Wei, Ouyang Zhang, Jin Zhu, Jiangxing Yunnan Univ Dept Math Kunming 650090 Yunnan Peoples R China Yunnan Normal Univ Sch Math Kunming 650500 Yunnan Peoples R China Southern Univ Sci & Technol Dept Math Shenzhen 518055 Guangdong Peoples R China

This paper introduces and considers the concept of generalized subsmoothness of a multifunction, which is a generalization of both the prox-regularity property and the subsmoothness property of multifunctions. Subsequently, it mainly deals with generalized metric subregularity (in particular, H & ouml;lder metric subregularity) for general set-valued mappings in Asplund spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive sufficient conditions for generalized metric subregularity, which extend even the known results for the conventional metric subregularity. In particular, our results improve/extend the main results established by Li and Mordukhovich (SIAM J. Optim. 22:1655-1684, 2012). Moreover, we also conduct local convergence analysis of an inexact quasi-Newton method for solving the generalized equation 0 is an element of f(x)+F(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\in f(x)+F(x)$\end{document} in Banach spaces, where the function f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f$\end{document} is continuous but not smooth and F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F$\end{document} is a set-valued mapping with closed graph.

关键词： Variational analysis H & ouml lder metric subregularity Normal cone coderivative

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coderivatives of implicit multifunctions and stability of variational systems

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JOURNAL OF GLOBAL OPTIMIZATION 2016年第3期65卷 615-635页

作者： Nguyen Thanh Qui Can Tho Univ Coll Informat & Commun Technol Campus 23-2 St Can Tho Vietnam

We establish formulas for computing/estimating the regular and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions ... 详细信息

关键词： coderivative Generalized equation Implicit multifunction Local Lipschitzlike property Normal cone operator Variational system

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coderivatives of a Karush-Kuhn-Tucker point set map and applications

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2014年 95卷 191-201页

作者： Lee, G. M. Yen, N. D. Pukyong Natl Univ Dept Appl Math Pusan 608737 South Korea Vietnam Acad Sci & Technol Inst Math Hanoi 10307 Vietnam

The problem of minimizing a linear-quadratic function over the Euclidean ball is encountered frequently in the theory of trust-region methods in nonlinear programming. By some tools from Variational Analysis, we investigate the stability of the Karush-Kuhn-Tucker point set map of that problem with respect to total perturbations of its data. Verifiable sufficient conditions for the local Lipschitz-like property of the map are obtained, and the connection of our results with the existing criteria for the lower semicontinuity of this Karush-Kuhn-Tucker point set map is shown. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Trust-region subproblem KKT point set map Normal cone coderivative Local Lipschitz-like property

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coderivatives of multivalued mappings, locally compact cones and metric regularity

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 1999年第7期35卷 925-945页

作者： Jourani, A Thibault, L Univ Bourgogne F-21011 Dijon France Univ Montpellier 2 Lab Anal Convexe F-34095 Montpellier 5 France

An abstract subdifferential was employed to obtain a verifiable criterion for metric regularity of infinite dimensional multivalued mappings, satisfying the given convergence condition, and to study some important properties of multivalued mappings that are separately partially compactly epi-Lipschitzian. Some subclasses of partially compactly epi-Lipschitzian multivalued mappings were used to give necessary and sufficient conditions for metric regularity of infinite-dimensional multivalued mappings in finite dimensions. The metric regularity at (x0,y0) of a multivalued mapping F with F-1 uniformly compactly epi-Lipschitzian implies that the kernel of the approximate coderivative of F at (x0,y0) is reduced to zero.

关键词： locally compact cone normal cone subdifferential coderivative metric regularity

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coderivativeS AND THE SOLUTION MAP OF A LINEAR CONSTRAINT SYSTEM

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SIAM JOURNAL ON OPTIMIZATION 2016年第2期26卷 986-1007页

作者： Duong Thi Kim Huyen Nguyen Dong Yen Vietnam Acad Sci & Technol Inst Math Grad Training Ctr 18 Hoang Quoc Viet Hanoi 10307 Vietnam Vietnam Acad Sci & Technol Inst Math 18 Hoang Quoc Viet Hanoi 10307 Vietnam

The Lipschitz-like property and the metric regularity in the sense of Robinson of the solution map of a parametric linear constraint system are investigated thoroughly by means of normal coderivative, the Mordukhovich criterion, and a related theorem due to Levy and Mordukhovich [Math. Program., 99 (2004), pp. 311-327]. Among other things, the obtained results yield uniform local error bounds and traditional local error bounds for the linear complementarity problem and the general affine variational inequality problem, as well as verifiable sufficient conditions for the Lipschitz-like property of the solution map of the linear complementarity problem and a class of affine variational inequalities, where all components of the problem data are subject to perturbations.

关键词： parametric linear constraint system linear complementarity problem affine variational inequality solution map coderivative Lipschitz-like property metric regularity in the sense of Robinson uniform local error bound full rank condition

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coderivatives of gap function for Minty vector variational inequality

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2015年第1期2015卷 1页

作者： Xue, Xiaowei Zhang, Yu Nanyang Normal Univ Coll Math & Stat Nanyang 473061 Henan Peoples R China Yunnan Univ Finance & Econ Coll Stat & Math Kunming 650221 Yunnan Peoples R China

The purpose of this paper is to investigate coderivatives of the gap function involving the Minty vector variational inequality. First, we discuss the regular coderivative, the normal coderivative, and the mixed coderivative of a class of set-valued maps. Then, by using the relationships between the coderivatives of a set-valued map and its efficient points set-valued map, we obtain the coderivatives of the gap function for the Minty vector variational inequality.

关键词： Minty vector variational inequality gap function normal cone coderivative

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coderivatives of the generalized perturbation maps

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POSITIVITY 2011年第2期15卷 309-329页

作者： Xue, X. W. Li, S. J. Chongqing Univ Coll Math & Sci Chongqing 400044 Peoples R China

This paper is devoted to considering the coderivatives of the generalized perturbation maps in general Banach spaces. Under some mild conditions, the upper estimate of coderivatives of the generalized perturbation maps are obtained. Their exact calculus rules are obtained under some additional conditions. Furthermore, the generalized perturbation maps are shown to be differentiably regular under some strong conditions.

关键词： Set-valued maps Normal cone coderivative Strictly differentiable Generalized perturbation map

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Subdifferentials and coderivatives of Efficient Point Multifunctions in Parametric Convex Vector Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第2期202卷 745-770页

作者： An, Duong Thi Viet Hung, Nguyen Huy Tuyen, Nguyen Van Thai Nguyen Univ Sci Dept Math & Informat Thai Nguyen 250000 Vietnam Hanoi Pedag Univ Dept Math 2 Xuan Hoa Phuc Yen Vinh Phuc Vietnam

In this paper, by revisiting coderivative calculus rules for convex multifunctions in finite-dimensional spaces, we derive formulae for estimating/computing the basic subdifferential and the coderivative of the efficient point multifunction of parametric convex vector optimization problems. These results are then applied to a broad class of conventional convex vector optimization problems with the presence of operator constraints and equilibrium ones. Examples are also designed to analyze and illustrate the obtained results.

关键词： Parametric convex vector optimization Efficient point multifunction Subdifferential coderivative Sensitivity analysis

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On coderivatives and Lipschitzian properties of the dual pair in optimization

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第3期75卷 1461-1482页

作者： Lopez, Marco A. Ridolfi, Andrea B. Vera de Serio, Virginia N. Univ Alicante Dept Stat & Operat Res Alicante Spain Univ Ballarat Grad Sch Informat Technol & Math Sci Ballarat Vic 3353 Australia Natl Univ Cuyo Fac Sci Appl Ind Mendoza Argentina Consejo Nacl Invest Cient & Tecn Mendoza Argentina Natl Univ Cuyo Fac Econ Mendoza Argentina Natl Univ Cuyo ICB Mendoza Argentina

In this paper, we apply the concept of coderivative and other tools from the generalized differentiation theory for set-valued mappings to study the stability of the feasible sets of both the primal and the dual problem in infinite-dimensional linear optimization with infinitely many explicit constraints and an additional conic constraint. After providing some specific duality results for our dual pair, we study the Lipschitz-like property of both mappings and also give bounds for the associated Lipschitz moduli. The situation for the dual shows much more involved than the case of the primal problem. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Semi-infinite and infinite-dimensional programming Dual pair and duality theory Stability Lipschitz-like property and Lipschitz moduli coderivative

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Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2012年第3期75卷 1305-1323页

作者： Truong Xuan Duc Ha Inst Math Hanoi 10307 Vietnam

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality. We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Set-valued optimization problem Set-valued equilibrium problem Vector equilibrium problem Vector variational inequality Normal cone coderivative Necessary optimality conditions Sufficient optimality conditions Pareto efficient solutions Weakly efficient solutions Quasi weakly efficient solutions Strongly efficient solutions Properly efficient solutions Relative efficient solutions

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