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FULL STABILITY OF LOCALLY OPTIMAL SOLUTIONS IN SECOND-ORDER CONE PROGRAMS

SIAM JOURNAL ON OPTIMIZATION 2014年第4期24卷 1581-1613页

作者： Mordukhovich, Boris S. Outrata, Jiri V. Sarabi, M. Ebrahim Wayne State Univ Dept Math Detroit MI 48202 USA King Fahd Univ Petr & Minerals Dhahran 31261 Saudi Arabia Acad Sci Czech Republ Inst Informat Theory & Automat CR-18208 Prague Czech Republic Federat Univ Australia Ctr Informat & Appl Optimizat Ballarat Vic 3353 Australia

The paper presents complete characterizations of Lipschitzian full stability of locally optimal solutions to second-order cone programs (SOCPs) expressed entirely in terms of their initial data. These characterizations are obtained via appropriate versions of the quadratic growth and strong second-order sufficient conditions under the corresponding constraint qualifications. We also establish close relationships between full stability of local minimizers for SOCPs and strong regularity of the associated generalized equations at nondegenerate points. Our approach is mainly based on advanced tools of second-order variational analysis and generalized differentiation.

关键词： variational analysis second-order cone programming full stability of local minimizers nondegeneracy strong regularity quadratic growth second-order subdifferentials coderivatives

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On the Application of the SCD Semismooth* Newton Method to Variational Inequalities of the Second Kind

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SET-VALUED AND VARIATIONAL ANALYSIS 2022年第4期30卷 1453-1484页

作者： Gfrerer, Helmut Outrata, Jiri V. Valdman, Jan Johannes Kepler Univ Linz Inst Computat Math A-4040 Linz Austria Czech Acad Sci Inst Informat Theory & Automat Prague 18208 Czech Republic Federat Univ Australia Ctr Informat & Appl Optimizat POB 663 Ballarat Vic 3350 Australia Czech Tech Univ Fac Informat Technol Dept Appl Math Thakurova 9 Prague 160006 6 Czech Republic

The paper starts with a description of SCD (subspace containing derivative) mappings and the SCD semismooth* Newton method for the solution of general inclusions. This method is then applied to a class of variational inequalities of the second kind. As a result, one obtains an implementable algorithm which exhibits locally superlinear convergence. Thereafter we suggest several globally convergent hybrid algorithms in which one combines the SCD semismooth* Newton method with selected splitting algorithms for the solution of monotone variational inequalities. Finally, we demonstrate the efficiency of one of these methods via a Cournot-Nash equilibrium, modeled as a variational inequality of the second kind, where one admits really large numbers of players (firms) and produced commodities.

关键词： Newton method Semismoothness Superlinear convergence Global convergence Generalized equation coderivatives

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Coderivative calculus and metric regularity for constraint and variational systems

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2009年第1期70卷 529-552页

作者： Geremew, W. Mordukhovich, B. S. Nam, N. M. Wayne State Univ Dept Math Detroit MI 48202 USA Richard Stockton Coll New Jersey Dept Math Pomona NJ 08240 USA Univ Texas Pan Amer Dept Math Edinburg TX 78539 USA

This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： Variational analysis and optimization Generalized differentiation coderivatives Calculus rules Metric regularity Exact bounds Conditioning

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Frechet subdifferential calculus and optimality conditions in nondifferentiable programming

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OPTIMIZATION 2006年第5-6期55卷 685-708页

作者： Mordukhovich, B. S. Nam, N. M. Yen, N. D. Wayne State Univ Dept Math Detroit MI 48202 USA Inst Math Hanoi 10307 Vietnam

We develop various (exact) calculus rules for Frechet lower and upper subgradients of extended-real-valued functions in real Banach spaces. Then we apply this calculus to derive new necessary optimality conditions for some remarkable classes of problems in constrained optimization including minimization problems for difference-type functions under geometric and operator constraints as well as subdifferential optimality conditions for the so-called weak sharp minima.

关键词： variational analysis and constrained optimization Frechet normals subgradients coderivatives subdifferential calculus necessary optimality conditions difference-type functions geometric and operator constraints weak sharp minima

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Subdifferential analysis of differential inclusions via discretization

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JOURNAL OF DIFFERENTIAL EQUATIONS 2012年第1期253卷 203-224页

作者： Pang, C. H. Jeffrey MIT Dept Math Cambridge MA 02139 USA

The framework of differential inclusions encompasses modern optimal control and the calculus of variations. Necessary optimality conditions in the literature identify potentially optimal paths, but do not show how to perturb paths to optimality. We first look at the corresponding discretized inclusions, estimating the subdifferential dependence of the optimal value in terms of the endpoints of the feasible paths. Our approach is to first estimate the coderivative of the reachable map. The discretized (nonsmooth) Euler-Lagrange and Transversality Conditions follow as a corollary. We obtain corresponding results for differential inclusions by passing discretized inclusions to the limit. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Differential inclusions Optimality conditions Subdifferential Reachable map coderivatives

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SUBOPTIMALITY CONDITIONS FOR MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS

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TAIWANESE JOURNAL OF MATHEMATICS 2008年第9期12卷 2569-2592页

作者： Bao, Truong Q. Gupta, Pankaj Mordulchovich, Boris S. Wayne State Univ Dept Math Detroit MI 48202 USA Univ Delhi Dept Operat Res Delhi 110007 India

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techniques are mainly based on modem tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle.

关键词： Mathematical programs with equilibrium constraints Variational analysis Nonsmooth optimization Extremal principle Subdifferential variational principle Generalized differentiation coderivatives

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Implicit multifunction theorems

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SET-VALUED ANALYSIS 1999年第3期7卷 209-238页

作者： Ledyaev, YS Zhu, QJJ Western Michigan Univ Dept Math & Stat Kalamazoo MI 49008 USA

We prove a general implicit function theorem for multifunctions with a metric estimate on the implicit multifunction and a characterization of its coderivative. Traditional open covering theorems, stability results, and sufficient conditions for a multifunction to be metrically regular or pseudo-Lipschitzian can be deduced from this implicit function theorem. We prove this implicit multifunction theorem by reducing it to an implicit function/solvability theorem for functions. This approach can also be used to prove the Robinson-Ursescu open mapping theorem. As a tool for this alternative proof of the Robinson-Ursescu theorem, we also establish a refined version of the multidirectional mean value inequality which is of independent interest.

关键词： nonsmooth analysis subdifferentials coderivatives implicit function theorem solvability stability open mapping theorem metric regularity multidirectional mean value inequality

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Variational Analysis of Paraconvex Multifunctions

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2022年第1-3期193卷 180-218页

作者： Huynh Van Ngai Nguyen Huu Tron Nguyen Van Vu Thera, Michel Quy Nhon Univ Dept Math 170 An Duong Vuong Quy Nhon Binh Dinh Vietnam Univ Limoges XLIM UMR CNRS 7252 123 Ave Albert Thomas F-87060 Limoges France Federation Univ Ctr Informat & Appl Optimizat Sch Engn Informat Technol & Phys Sci Ballarat Vic Australia

Our aim in this article is to study the class of so-called rho-paraconv ex multifunctions from a Banach space X into the subsets of another Banach space Y. These multifunctions are defined in relation with a modulus function rho : X -> [0, +infinity) satisfying some suitable conditions. This class of multifunctions generalizes the class of gamma-paraconvex multifunctions with gamma > 1 introduced and studied by Rolewicz, in the eighties and subsequently studied by A. Jourani and some others authors. We establish some regular properties of graphical tangent and normal cones to paraconvex multifunctions between Banach spaces as well as a sum rule for coderivatives for such class of multifunctions. The use of subdifferential properties of the lower semicontinuous envelope function of the distance function associated to a multifunction established in the present paper plays a key role in this study.

关键词： Weak convexity Lower C-2 functions Paraconvexity Paramonotonicity Approximate convex function Frechet subdifferential Frechet normal cone coderivatives Fuzzy mean value theorem

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Quantitative stability of linear infinite inequality systems under block perturbations with applications to convex systems

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TOP 2012年第2期20卷 310-327页

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

The original motivation for this paper was to provide an efficient quantitative analysis of convex infinite (or semi-infinite) inequality systems whose decision variables run over general infinite-dimensional (resp. finite-dimensional) Banach spaces and that are indexed by an arbitrary fixed set J. Parameter perturbations on the right-hand side of the inequalities are required to be merely bounded, and thus the natural parameter space is l (a)(J). Our basic strategy consists of linearizing the parameterized convex system via splitting convex inequalities into linear ones by using the Fenchel-Legendre conjugate. This approach yields that arbitrary bounded right-hand side perturbations of the convex system turn on constant-by-blocks perturbations in the linearized system. Based on advanced variational analysis, we derive a precise formula for computing the exact Lipschitzian bound of the feasible solution map of block-perturbed linear systems, which involves only the system's data, and then show that this exact bound agrees with the coderivative norm of the aforementioned mapping. In this way we extend to the convex setting the results of Canovas et al. (SIAM J. Optim. 20, 1504-1526, 2009) developed for arbitrary perturbations with no block structure in the linear framework under the boundedness assumption on the system's coefficients. The latter boundedness assumption is removed in this paper when the decision space is reflexive. The last section provides the aimed application to the convex case.

关键词： Semi-infinite and infinite programming Parametric optimization Variational analysis Convex infinite inequality systems Quantitative stability Lipschitzian bounds Generalized differentiation coderivatives Block perturbations

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A derivative-coderivative inclusion in second-order nonsmooth analysis

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SET-VALUED ANALYSIS 1997年第1期5卷 89-105页

作者： Rockafellar, RT Zagrodny, D UNIV WASHINGTON DEPT MATH SEATTLE WA 98195 USA TECH UNIV LODZ MATH INST PL-90924 LODZ POLAND

For twice smooth functions, the symmetry of the matrix of second partial derivatives is automatic and can be seen as the symmetry of the Jacobian matrix of the gradient mapping. For nonsmooth functions, possibly even extended-real-valued, the gradient mapping can be replaced by a subgradient mapping, and generalized second derivative objects can then be introduced through graphical differentiation of this mapping, but the question of what analog of symmetry might persist has remained open. An answer is provided here in terms of a derivative-coderivative inclusion.

关键词： graphical derivatives coderivatives subgradient mappings generalized Hessians

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