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Normal and mixed partial second-order subdifferentials in variational analysis and optimization

POSITIVITY 2018年第3期22卷 671-685页

作者： Sadeqi, Ildar Nadi, Somayeh Sahand Univ Technol Dept Math Tabriz Iran

A partial second-order subdifferential is defined here for extended real valued functions of two variables corresponding to its variables through coderivatives of first-order partial subdifferential mappings. In addition, some rules are presented to calculate these second-order structures along with defining some conditions to insure the equality and . Moreover, as an application, some conditions are stated which show the relation between local minimum of a function and positiveness of principal minors of its hessian matrix.

关键词： coderivatives Second-order subdifferentials Stability Strictly differentiable mappings Variational analysis

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Tilt stability of a local minimum

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SIAM JOURNAL ON OPTIMIZATION 1998年第2期8卷 287-299页

作者： Poliquin, RA Rockafellar, RT Univ Alberta Dept Math Sci Edmonton AB T6G 2G1 Canada Univ Washington Dept Math Seattle WA 98195 USA Univ Perpignan F-66025 Perpignan France

The behavior of a minimizing point when an objective function is tilted by adding a small linear term is studied from the perspective of second-order conditions for local optimality. The classical condition of a positive-definite Hessian in smooth problems without constraints is found to have an exact counterpart much more broadly in the positivity of a certain generalized Hessian mapping. This fully characterizes the case where tilt perturbations cause the minimizing point to shift in a Lipschitzian manner.

关键词： variational analysis second-order optimality tilt stability perturbations sensitivity prox-regularity amenable functions generalized Hessians coderivatives monotone mappings

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Algebraic core and convex calculus without topology

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OPTIMIZATION 2022年第2期71卷 309-335页

作者： Dang Van Cuong Mordukhovich, Boris S. Nguyen Mau Nam Cartmell, Addison Duy Tan Univ Fac Nat Sci Dept Math Da Nang Vietnam Duy Tan Univ Inst Res & Dev Da Nang Vietnam Wayne State Univ Dept Math Detroit MI 48202 USA Portland State Univ Farlborz Maseeh Dept Math & Stat Portland OR 97207 USA

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.

关键词： Algebraic core convex separation normals coderivatives subgradients

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

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2009年第2期350卷 663-679页

作者： Mordukhovich, Boris S. Nam, Nguyen Mau Wayne State Univ Dept Math Detroit MI 48202 USA Univ Texas Pan Amer Dept Math Edinburg TX 78539 USA

This paper is devoted to the development of variational analysis and generalized differentiation in the framework of Asplund spaces. We mainly concern the Study of a special class of set-valued mapping given in the form S(x) = {y is an element of Y vertical bar 0 is an element of F(x, y)}, x is an element of X, where both F and Q are set-valued mappings between Asplund spaces. Models of this type are associated With Solutions maps to the so-called (extended) generalized equations and play a significant role in many aspects of variational analysis and its applications to optimization, stability, control theory, etc. In this paper we conduct a local variational analysis of such extended solution [naps S and their remarkable specifications based on dual-space generalized differential constructions of the coderivative type. The major part of our analysis revolves around coderivative calculus largely developed and implemented in this paper and then applied to establishing verifiable conditions for robust Lipschitzian stability of extended generalized equations and related objects. (C) 2008 Elsevier Inc. All rights reserved.

关键词： Variational analysis Asplund spaces Generalized differentiation coderivatives Generalized equations Lipschitzian suability

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The radius of metric regularity

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 2003年第2期355卷 493-517页

作者： Dontchev, AL Lewis, AS Rockafellar, RT Amer Math Soc Math Reviews Ann Arbor MI 48107 USA Simon Fraser Univ Dept Math Burnaby BC V5A 1S6 Canada Univ Washington Dept Math Seattle WA 98195 USA

Metric regularity is a central concept in variational analysis for the study of solution mappings associated with "generalized equations", including variational inequalities and parameterized constraint systems. Here it is employed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in numerical analysis are obtained in particular.

关键词： metric regularity perturbations distance to irregularity distance to infeasibility Eckart-Young theorem Lusternik-Graves theorem Robinson-Ursescu theorem coderivatives

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Linearly perturbed generalized polyhedral normal cone mappings and applications

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OPTIMIZATION 2016年第1期65卷 9-34页

作者： Ban, Liqun Song, Wen Harbin Univ Sci & Technol Dept Math Harbin Peoples R China Harbin Normal Univ Dept Math Harbin Peoples R China

Under a mild regularity assumption, we derive an exact formula for the Frechet coderivative and some estimates for the Mordukhovich coderivative of the normal cone mappings of perturbed generalized polyhedra in reflexive Banach spaces. Assume in addition that the generating elements are linearly independent and some qualification condition holds, the Lipschitzian stability of the parameterized variational inequalities over the right-hand side perturbed generalized polyhedra is characterized using the initial data.

关键词： reflexive Banach spaces perturbed generalized polyhedral sets parametric variational inequalities robust Lipschitz stability coderivatives

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VARIATIONAL ANALYSIS IN SEMI-INFINITE AND INFINITE PROGRAMMING, I: STABILITY OF LINEAR INEQUALITY SYSTEMS OF FEASIBLE SOLUTIONS

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SIAM JOURNAL ON OPTIMIZATION 2009年第3期20卷 1504-1526页

作者： Canovas, M. J. Lopez, M. A. Mordukhovich, B. S. Parra, J. Miguel Hernandez Univ Elche Ctr Operat Res Alicante 03202 Spain Univ Alicante Dept Stat & Operat Res E-03080 Alicante Spain Wayne State Univ Dept Math Detroit MI 48202 USA

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to parametric problems of semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Part I is primarily devoted to the study of robust Lipschitzian stability of feasible solutions maps for such problems described by parameterized systems of infinitely many linear inequalities in Banach spaces of decision variables indexed by an arbitrary set T. The parameter space of admissible perturbations under consideration is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set T is finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l(infinity) type. By using advanced tools of variational analysis and exploiting specific features of linear infinite systems, we establish complete characterizations of robust Lipschitzian stability entirely via their initial data with computing the exact bound of Lipschitzian moduli. A crucial part of our analysis addresses the precise computation of the coderivative of the feasible set mapping and its norm. The results obtained are new in both semi-infinite and infinite frameworks.

关键词： semi-infinite and infinite programming variational analysis linear infinite inequality systems robust stability generalized differentiation coderivatives

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Coderivative analysis of variational systems

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JOURNAL OF GLOBAL OPTIMIZATION 2004年第3-4期28卷 347-362页

作者： Mordukhovich, BS Wayne State Univ Dept Math Detroit MI 48202 USA

The paper mostly concerns applications of the generalized differentiation theory in variational analysis to Lipschitzian stability and metric regularity of variational systems in infinite dimensional spaces. The main tools of our analysis involve coderivatives of set-valued mappings that turn out to be proper extensions of the adjoint derivative operator to nonsmooth and set-valued mappings. The involved coderivatives allow us to give complete dual characterizations of certain fundamental properties in variational analysis and optimization related to Lipschitzian stability and metric regularity. Based on these characterizations and extended coderivative calculus, we obtain efficient conditions for Lipschitzian stability of variational systems governed by parametric generalized equations and their specifications.

关键词： coderivatives generalized differentiation infinite-dimensional spaces Lipschitzian stability and metric regularity variational analysis

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PARTIAL SECOND-ORDER SUBDIFFERENTIALS IN VARIATIONAL ANALYSIS AND OPTIMIZATION

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2014年第7-9期35卷 1113-1151页

作者： Mordukhovich, Boris S. Nguyen Mau Nam Nguyen Thi Yen Nhi Wayne State Univ Dept Math Detroit MI 48202 USA King Fahd Univ Petr & Minerals Dhahran 31261 Saudi Arabia Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA

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.

关键词： coderivatives Generalized differentiation Parametric constrained optimization Second-order subdifferentials Stability Stationary point mappings Variational analysis

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Metric Regularity of Mappings and Generalized Normals to Set Images

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SET-VALUED AND VARIATIONAL ANALYSIS 2009年第4期17卷 359-387页

作者： Mordukhovich, Boris S. Nam, Nguyen Mau Wang, Bingwu Wayne State Univ Dept Math Detroit MI 48202 USA Univ Texas Pan Amer Dept Math Edinburg TX 78539 USA Eastern Michigan Univ Dept Math Ypsilanti MI 48197 USA

The primary goal of this paper is to study some notions of normals to nonconvex sets in finite-dimensional and infinite-dimensional spaces and their images under single-valued and set-valued mappings. The main motivation for our study comes from variational analysis and optimization, where the problems under consideration play a crucial role in many important aspects of generalized differential calculus and applications. Our major results provide precise equality formulas (sometimes just efficient upper estimates) allowing us to compute generalized normals in various senses to direct and inverse images of nonconvex sets under single-valued and set-valued mappings between Banach spaces. The main tools of our analysis revolve around variational principles and the fundamental concept of metric regularity properly modified in this paper.

关键词： Variational analysis Metric regularity Generalized differentiation Normal cones coderivatives Calculus rules Set images

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