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LOCAL MONOTONICITY AND FULL STABILITY FOR PARAMETRIC VARIATIONAL SYSTEMS

SIAM JOURNAL ON OPTIMIZATION 2016年第2期26卷 1032-1059页

作者： Mordukhovich, B. S. Nghia, T. T. A. Wayne State Univ Dept Math Detroit MI 48202 USA Oakland Univ Dept Math & Stat Rochester MI 48309 USA

This paper introduces and characterizes new notions of Lipschitzian and Holderian full stability of solutions to general parametric variational systems defined via partial subdifferential of prox-regular functions acting in finite-dimensional and Hilbert spaces. These notions, which postulate certain quantitative properties of single-valued localizations of solution maps, are closely related to local strong maximal monotonicity of associated set-valued mappings. Based on advanced tools of variational analysis and generalized differentiation, we derive verifiable characterizations of the local strong maximal monotonicity and full stability notions under consideration via some positive-definiteness conditions involving second-order constructions of variational analysis.

关键词： variational analysis parametric variational systems variational inequalities local monotonicity strong maximal monotonocity full stability generalized differentiation subdifferentials coderivatives

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CHARACTERIZING GENERALIZED DERIVATIVES OF SET-VALUED MAPS: EXTENDING THE TANGENTIAL AND NORMAL APPROACHES

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2013年第1期51卷 145-171页

作者： Pang, C. H. Jeffrey Natl Univ Singapore NUS Dept Math Singapore 119076 Singapore

For a set-valued map, we characterize, in terms of its (unconvexified or convexified) graphical derivatives near the point of interest, positively homogeneous maps that are generalized derivatives in the sense of [C. H. J. Pang, Math. Oper. Res., 36 (2011), pp. 377-397]. This result generalizes the Aubin criterion in [A. L. Dontchev, M. Quincampoix, and N. Zlateva, J. Convex Anal., 3 (2006), pp. 45-63]. A second characterization of these generalized derivatives is easier to check in practice, especially in the finite dimensional case. Finally, the third characterization in terms of limiting normal cones and coderivatives generalizes the Mordukhovich criterion in the finite dimensional case. The convexified coderivative has a bijective relationship with the set of possible generalized derivatives. We conclude by illustrating a few applications of our result.

关键词： Aubin criterion Mordukhovich criterion tangent cones multifunctions Aubin property generalized derivatives normal cones coderivatives

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Primal-Dual Stability in Local Optimality

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第2期203卷 1325-1354页

作者： Benko, Matus Rockafellar, R. Tyrrell Johann Radon Inst Computat & Appl Math Linz Austria Univ Washington Dept Math Box 354350 Seattle WA 98195 USA

Much is known about when a locally optimal solution depends in a single-valued Lipschitz continuous way on the problem's parameters, including tilt perturbations. Much less is known, however, about when that solution and a uniquely determined multiplier vector associated with it exhibit that dependence as a primal-dual pair. In classical nonlinear programming, such advantageous behavior is tied to the combination of the standard strong second-order sufficient condition (SSOC) for local optimality and the linear independent gradient condition (LIGC) on the active constraint gradients. But although second-order sufficient conditons have successfully been extended far beyond nonlinear programming, insights into what should replace constraint gradient independence as the extended dual counterpart have been lacking. The exact answer is provided here for a wide range of optimization problems in finite dimensions. Behind it are advances in how coderivatives and strict graphical derivatives can be deployed. New results about strong metric regularity in solving variational inequalities and generalized equations are obtained from that as well.

关键词： Second-order variational analysis Local optimality Primal-dual stability Tilt stability, full stability, metric regularity Kummer's inverse theorem Implicit mapping theorems Graphically Lipschitzian mappings Crypto-continuity Strict graphical derivatives coderivatives Variational sufficiency

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LIPSCHITZIAN STABILITY OF CONSTRAINT SYSTEMS AND GENERALIZED EQUATIONS

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 1994年第2期22卷 173-206页

作者： MORDUKHOVICH, B Department of Mathematics Wayne State University Detroit MI 48202 U.S.A.

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Variational Geometric Approach to Generalized Differential and Conjugate Calculi in Convex Analysis

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SET-VALUED AND VARIATIONAL ANALYSIS 2017年第4期25卷 731-755页

作者： Mordukhovich, B. S. Nam, N. M. Rector, R. B. Tran, T. Wayne State Univ Dept Math Detroit MI 48202 USA RUDN Univ Moscow 117198 Russia Portland State Univ Fariborz Maseeh Dept Math & Stat Portland OR 97207 USA

This paper develops a geometric approach of variational analysis for the case of convex objects considered in locally convex topological spaces and also in Banach space settings. Besides deriving in this way new results of convex calculus, we present an overview of some known achievements with their unified and simplified proofs based on the developed geometric variational schemes.

关键词： Convex and variational analysis Fenchel conjugates Normals and subgradients coderivatives Convex calculus Optimal value functions

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VARIATIONAL ANALYSIS IN SEMI-INFINITE AND INFINITE PROGRAMMING, II: NECESSARY OPTIMALITY CONDITIONS

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SIAM JOURNAL ON OPTIMIZATION 2010年第6期20卷 2788-2806页

作者： Canovas, M. J. Lopez, M. A. Mordukhovich, B. S. Parra, J. Miguel Hernandez Univ 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

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [Canovas et al., SIAM J. Optim., 20 (2009), pp. 1504-1526] from the viewpoint of robust Lipschitzian stability. The main results establish necessary optimality conditions for broad classes of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. The results obtained are new in both smooth and nonsmooth settings of semi-infinite and infinite programming. We illustrate our model and results by considering a practically meaningful model of water resource optimization via systems of reservoirs.

关键词： semi-infinite and infinite programming parametric optimization variational analysis necessary optimality conditions linear infinite inequality systems generalized differentiation coderivatives lower and upper subdifferentials

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Sensitivity Analysis of Multiobjective Optimization Problems with Parameterized Quasi-Variational Inequalities

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2018年第2期39卷 155-178页

作者： Pang, Li-Ping Meng, Fan-Yun Lv, Jian Tian, Qi Dalian Univ Technol Sch Math Sci Dalian 116024 Peoples R China Qingdao Technol Univ Comp Engn Inst Qingdao Peoples R China Zhejiang Univ Finance & Econ Dept Finance Hangzhou Zhejiang Peoples R China

This paper mainly establishes the sensitivity analysis of a multiobjective optimization problem with parameterized quasi-variational inequalities (QVIs). Using the (regular) coderivative of the associated epigraphical multifunction, the (regular) subdifferentials of the efficient frontier maps are estimated, which involve the (regular) coderivatives of the solution mapping to the parameterized QVIs. Under the linear independent constraint qualification, the defined auxiliary set-valued mappings in the parameterized QVIs are clam. The detailed formulae of subdifferentials of the efficient frontier maps are obtained and examples are simultaneously provided for analyzing and illustrating the obtained results.

关键词： coderivatives efficient frontier maps multiobjective optimization parameterized quasi-variational inequalities

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Generalized differentiation of piecewise linear functions in second-order variational analysis

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2016年 132卷 240-273页

作者： Mordukhovich, Boris S. Sarabi, M. Ebrahim Wayne State Univ Dept Math Detroit MI 48202 USA

The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to optimization and stability. This class consists of lower semicontinuous functions with possibly infinite values on finite-dimensional spaces, which are labeled as "piecewise linear" ones and can be equivalently described via the convexity of their epigraphs. In this paper we calculate the second-order subdifferentials (generalized Hessians) of arbitrary convex piecewise linear functions, together with the corresponding geometric objects, entirely in terms of their initial data. The obtained formulas allow us, in particular, to justify a new exact (equality-type) second-order sum rule for such functions in the general nonsmooth setting. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Nonlinear and variational analysis Piecewise linear extended-real-valued functions Normal cones coderivatives First-order and second-order subdifferentials

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Lipschitz-Like Property for Linear Constraint Systems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第3期199卷 1281-1296页

作者： Yao, Wenfang Yang, Xiaoqi Hong Kong Polytech Univ Dept Appl Math Hong Kong Peoples R China

In this paper, we consider a linear constraint system with a set constraint. We investigate the Lipschitz-like property of such systems with an explicit set constraint under full perturbations (including the matrix perturbation) and derive some sufficient and necessary conditions for this property. We also make use of some other approaches like outer-subdifferentials and error bounds to characterize such a property. We later apply the obtained results to linear portfolio selection problems with different settings and obtain some sufficient conditions for the parametric feasible set mapping to enjoy the Lipschitz-like property with various stock selection constraints.

关键词： Linear systems coderivatives Lipschitz-like property Linear portfolio selection

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Full Stability of General Parametric Variational Systems

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SET-VALUED AND VARIATIONAL ANALYSIS 2018年第4期26卷 911-946页

作者： Mordukhovich, B. S. Nghia, T. T. A. Pham, D. T. Wayne State Univ Dept Math Detroit MI 48202 USA RUDN Univ Moscow 117198 Russia Oakland Univ Dept Math & Stat Rochester MI 48309 USA

The paper introduces and studies the notions of Lipschitzian and Holderian full stability of solutions to three-parametric variational systems described in the generalized equation formalism involving nonsmooth base mappings and partial subgradients of prox-regular functions acting in Hilbert spaces. Employing advanced tools and techniques of second-order variational analysis allows us to establish complete characterizations of, as well as directly verifiable sufficient conditions for, such full stability properties under mild assumptions. Furthermore, we derive exact formulas and effective quantitative estimates for the corresponding moduli. The obtained results are specified for important classes of variational inequalities and variational conditions in both finite and infinite dimensions.

关键词： Variational analysis Parametric variational systems Variational inequalities and variational conditions Lipschitzian and Holderian full stability Prox-regularity Legendre forms Polyhedricity Generalized differentiation Subgradients coderivatives Primary 49J53 Secondary 49J52 90C31

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