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Metric subregularity for composite-convex generalized equations in Banach spaces

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第10期74卷 3311-3323页

作者： Zheng, Xi Yin Ouyang, Wei Yunnan Univ Dept Math Kunming 650091 Peoples R China

In this paper we consider a convex-composite generalized constraint equation in Banach spaces. Using variational analysis technique, in terms of normal cones and coderivatives, we first establish sufficient conditions for such an equation to be metrically subregular. Under the Robinson qualification, we prove that these conditions are also necessary for the metric subregularity. In particular, some existing results on error bound and metric subregularity are extended to the composite-convexity case from the convexity case. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： Convex-composite multifunction Metric subregularity Normal cone coderivative Robinson qualification

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Estimates of Generalized Hessians for Optimal Value Functions in Mathematical Programming

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第3期30卷 847-871页

作者： Zemkoho, Alain B. Univ Southampton Sch Math Sci Highfield Campus Southampton SO17 1BJ Hants England

We consider the optimal value function of a parametric optimization problem. A large number of publications have been dedicated to the study of continuity and differentiability properties of the function. However, the differentiability aspect of works in the current literature has mostly been limited to first order analysis, with focus on estimates of its directional derivatives and subdifferentials, given that the function is typically nonsmooth. With the progress made in the last two to three decades in major subfields of optimization such as robust, minmax, semi-infinite and bilevel optimization, and their connection to the optimal value function, there is a need for a second order analysis of the generalized differentiability properties of this function. This could enable the development of robust solution algorithms, such as the Newton method. The main goal of this paper is to provide estimates of the generalized Hessian for the optimal value function. Our results are based on two handy tools from parametric optimization, namely the optimal solution and Lagrange multiplier mappings, for which completely detailed estimates of their generalized derivatives are either well-known or can easily be obtained.

关键词： Parametric optimization Optimal value function Generalized Hessian Second order subdifferential Sensitivity analysis coderivative

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Calmness of efficient solution maps in parametric vector optimization

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JOURNAL OF GLOBAL OPTIMIZATION 2011年第4期51卷 677-688页

作者： Chuong, T. D. Kruger, A. Y. Yao, J. -C. Kaohsiung Med Univ Ctr Gen Educ Kaohsiung 80708 Taiwan Dong Thap Univ Dept Math Cao Lanh City Dong Thap Provi Vietnam Univ Ballarat Grad Sch Informat Technol & Math Sci Ctr Informat & Appl Optimizat Ballarat Vic 3353 Australia

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Frechet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained.

关键词： Parametric vector optimization Efficient solution map Calmness coderivative Implicit multifunction

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Relationships Between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第2期199卷 766-786页

作者： Luan, Nguyen Ngoc Nam, Nguyen Mau Thieu, Nguyen Nang Yen, Nguyen Dong Hanoi Natl Univ Educ Dept Math & Informat 136 Xuan Thuy Hanoi Vietnam Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA Vietnam Acad Sci & Technol Inst Math 18 Hoang Quoc Viet Hanoi 10307 Vietnam

In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a polyhedral convex set obtained by Ng et al. (Nonlinear Anal. 55:845-858, 2003;Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets. We also introduce and study the notions of generalized polyhedral multifunctions and optimal value functions generated by generalized polyhedral convex multifunctions along with their generalized differentiation calculus rules.

关键词： Convex polyhedron Generalized convex polyhedron Separation Normal cone coderivative Subdifferential Optimal value function

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A Notion of Fenchel Conjugate for Set-Valued Mappings

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第2期203卷 1263-1292页

作者： Nam, Nguyen Mau Sandine, Gary Thieu, Nguyen Nang Yen, Nguyen Dong Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA Vietnam Acad Sci & Technol Inst Math 18 Hoang Quoc Viet Hanoi 10307 Vietnam

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 some 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.

关键词： Fenchel conjugate coderivative Subdifferential Convex set-valued mapping Relative interior Quasi-relative interior Strong quasi-relative interior

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Generalized Differentiation and Duality in Infinite Dimensions under Polyhedral Convexity

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第4期30卷 1503-1526页

作者： Cuong, D. V. Mordukhovich, B. S. Nam, N. M. Sandine, G. Duy Tan Univ Dept Math Fac Nat Sci Da Nang Vietnam Duy Tan Univ Amer Degree Program Da Nang Vietnam Wayne State Univ Dept Math Detroit MI 48202 USA Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA

This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar's proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension 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 (Nonlinear Anal. 55, 845-858, 12).

关键词： Convex analysis Generalized differentiation Geometric approach Normal cone Relative interior coderivative Calculus rules Solution maps

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Characterization of metric regularity for σ-subsmooth multifunctions

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2014年第0期100卷 111-121页

作者： Zheng, Xi Yin He, Qing Hai Yunnan Univ Dept Math Kunming 650091 Peoples R China

In terms of the coderivative, we provide characterizations of metric subregularity for sigma-subsmooth multifunctions and extend some existing results on metric subregularity. We also establish Robinson-Ursescu type theorems for sigma-subsmooth multifunctions and approximately convex multifunctions. As applications, we study error bounds for nonconvex inequalities and Lipschitz property for multifunctions. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Metric regularity Linear openness Subsmoothness coderivative

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KKT REFORMULATION AND NECESSARY CONDITIONS FOR OPTIMALITY IN NONSMOOTH BILEVEL OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2014年第4期24卷 1639-1669页

作者： Dempe, Stephan Zemkoho, Alain B. Tech Univ Bergakad Freiberg Dept Math & Comp Sci D-09596 Freiberg Germany Univ Birmingham Sch Math Birmingham B15 2TT W Midlands England

For a long time, the bilevel programming problem has essentially been considered as a special case of mathematical programs with equilibrium constraints, in particular when the so-called KKT reformulation is in question. Recently though, this widespread believe was shown to be false in general. In this paper, other aspects of the difference between both problems are revealed as we consider the KKT approach for the nonsmooth bilevel program. It turns out that the new inclusion (constraint) which appears as a consequence of the partial subdifferential of the lower-level Lagrangian (PSLLL) places the KKT reformulation of the nonsmooth bilevel program in a new class of mathematical program with both set-valued and complementarity constraints. While highlighting some new features of this problem, we attempt here to establish close links with the standard optimistic bilevel program. Moreover, we discuss possible natural extensions for C-, M-, and S-stationarity concepts. Most of the results rely on a coderivative estimate for the PSLLL that we also provide in this paper.

关键词： nonsmooth bilevel optimization parametric optimization coderivative variational analysis constraint qualifications stationarity conditions

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Stability of generalized equations under nonlinear perturbations

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OPTIMIZATION LETTERS 2018年第4期12卷 799-815页

作者： Nguyen Thanh Qui Hoang Ngoc Tuan Can Tho Univ Coll Informat & Commun Technol Campus 23-2 St Can Tho Vietnam Hanoi Pedag Univ 2 Dept Math Phuc Yen Vinh Phuc Vietnam

This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.

关键词： Generalized equation Nonlinear perturbation Local Lipschitz-like property Normal cone mapping coderivative Partial second-order subdifferential

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Directional Metric Pseudo Subregularity of Set-valued Mappings: a General Model

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SET-VALUED AND VARIATIONAL ANALYSIS 2020年第1期28卷 61-87页

作者： Ngai Huynh Van Tron Nguyen Huu Vu Nguyen Van Thera, Michel Quy Nhon Univ Dept Math 170 An Duong Vuong Quy Nhon Binh Dinh Vietnam Univ Limoges CNRS UMR 7252 XLIM Limoges France Federat Univ Australia CIAO Mt Helen Australia

This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations... 详细信息

关键词： Abstract subdifferential Metric regularity Directional metric regularity Metric subregularity Directional Holder metric subregularity Directional metric pseudo-subregularity coderivative Slope

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