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Subdifferential analysis of differential inclusions via discretization

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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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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coderivatives OF FRONTIER AND SOLUTION MAPS IN PARAMETRIC MULTIOBJECTIVE OPTIMIZATION

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TAIWANESE JOURNAL OF MATHEMATICS 2008年第8期12卷 2083-2111页

作者： Huy, N. Q. Mordukhovich, B. S. Yao, J. C. Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan Hanoi Pedag Univ 2 Dept Math Vinh Phuc Vietnam Wayne State Univ Dept Math Detroit MI 48202 USA

This paper concerns sensitivity analysis for general parametric constrained problems of multiobjective optimization in infinite-dimensional spaces by using advanced tools of modem variational analysis and generalized differentiation. We pay the main attention to computing and estimating coderivatives of frontier and efficient solution maps in parametric multiobjective problems with respect to generalized order optimality that include a vast majority of conventional multiobjective problems in the presence of geometric, operator, functional, and equilibrium constraints. The obtained results are new in both finite-dimensional and infinite-dimensional spaces.

关键词： Variational analysis Parametric multiobjective optimization Generalized order and generalized Pareto optimality Frontier and efficient solution maps Set-valued mappings coderivatives Lipschitzian properties

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Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach spaces

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2011年第2期74卷 441-461页

作者： Ban, Liqun Mordukhovich, Boris S. Song, Wen NE Normal Univ Dept Math Changchun 130024 Peoples R China Harbin Univ Sci & Technol Dept Math Harbin 150080 Peoples R China Wayne State Univ Dept Math Detroit MI 48202 USA Harbin Normal Univ Sch Math Sci Harbin 150080 Peoples R China

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis. (C) 2010 Elsevier Ltd. All rights reserved.

关键词： Variational analysis Reflexive Banach spaces Generalized polyhedral sets Parametric variational inequalities Robust Lipschitzian stability Generalized differentiation coderivatives

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Generalized Differentiation with Positively Homogeneous Maps: Applications in Set-Valued Analysis and Metric Regularity

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MATHEMATICS OF OPERATIONS RESEARCH 2011年第3期36卷 377-397页

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

We propose a new concept of generalized differentiation of set-valued maps that captures first-order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a directional property similar to our concept of generalized differentiation. Finally, we discuss the relationship between the robust form of generalized differentiation and its one-sided counterpart.

关键词： multifunction differentiability metric regularity coderivatives calmness Lipschitz continuity

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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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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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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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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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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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