Results 1  10
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12
LAGRANGE MULTIPLIERS AND OPTIMALITY
, 1993
"... Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions ..."
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Cited by 89 (7 self)
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Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write firstorder optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of onesided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the gametheoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows blackandwhite constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.
Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity
, 1996
"... In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These r ..."
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Cited by 15 (9 self)
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In Gateaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Fr'echet subderivative: In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results for viscosity solutions of HamiltonJacobi equations in fismooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustrates the flexibility of viscosity subderivatives as a tool for analysis.
A Survey of Subdifferential Calculus with Applications
 TMA
, 1998
"... This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools. ..."
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Cited by 14 (6 self)
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This survey is an account of the current status of subdifferential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might profitably use subdifferentials as tools.
Proximal Analysis in Smooth Spaces
 CECM Research Report 9304 (1993), Setvalued Analysis
, 1996
"... . We provide a highlyrefined sequential description of the generalized gradients of Clarke and approximate Gsubdifferential of a lower semicontinuous extendedrealvalued function defined on a Banach space with a fismooth equivalent renorm. In the case of a Frech`et differentiable renorm we ..."
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Cited by 10 (8 self)
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. We provide a highlyrefined sequential description of the generalized gradients of Clarke and approximate Gsubdifferential of a lower semicontinuous extendedrealvalued function defined on a Banach space with a fismooth equivalent renorm. In the case of a Frech`et differentiable renorm we give a corresponding result for the corresponding singular objects. Keywords: Lipschitz functions, lower semicontinuous functions, subderivatives, variational principles, distance functions, tangent cones, normals, smooth renorms, Clarkesubdifferentials, Gsubdifferentials. AMS (1991) subject classification: Primary 49J52, Secondary 49J50, 58C20. 2 1 Introduction. There are two natural ways to define a (nonconvex) generalized derivative @f of a nondifferentiable realvalued function. As discussed in [1, 3] one can do so topologically as for example in [16] or sequentially as in [17, 18, 19, 21, 22]. Topological constructions have an intrinsically complicated structure usually ...
Partially Smooth Variational Principles and Applications
 CECM Research Report
"... . We discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis. Keywords: Smooth variational principle, fuzzy sum rules, mean value inequalities and partially smooth spaces. Short Title: Partially smooth variational prin ..."
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Cited by 6 (5 self)
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. We discuss a smooth variational principle for partially smooth viscosity subdifferentials and explore its applications in nonsmooth analysis. Keywords: Smooth variational principle, fuzzy sum rules, mean value inequalities and partially smooth spaces. Short Title: Partially smooth variational principles. AMS (1991) subject classification: 49J50, 49J52. 1 Introduction Smooth variational analysis [7] has been highly successful in providing tools for the study of non smooth analysis and optimization problems: especially when married to viscosity concepts [10, 17]. Outside of smoothable Banach spaces (thus, notably in / L 1 spaces) general constructions such as those of Ioffe [25, 28, 29] require a largely nonconstructive intersection over smooth or finite dimensional subspaces. Equally, outside of Asplund or Fr'echet spaces the most puissant results [41, 42] fail. Nonetheless, many problems inevitably lie in large (nonsmooth or nonFr'echet) spaces, X. In such settings the ...
Amenable functions in optimization
 IN NONSMOOTH OPTIMIZATION METHODS AND APPLICATIONS
, 1992
"... ..."
Hamiltonian Necessary Conditions for a Multiobjective Optimal Control Problem with Endpoint Constraints
 SIAM J. CONTR. AND OPTIM
, 2000
"... This paper discusses Hamiltonian necessary conditions for a nonsmooth multiobjective optimal control problem with endpoint constraints related to a general preference. The transversality condition in our necessary conditions is stated in terms of a normal cone to the level sets of the preference. E ..."
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Cited by 5 (2 self)
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This paper discusses Hamiltonian necessary conditions for a nonsmooth multiobjective optimal control problem with endpoint constraints related to a general preference. The transversality condition in our necessary conditions is stated in terms of a normal cone to the level sets of the preference. Examples for a number of useful preferences are discussed.
Implicit Multifunction Theorems
"... . 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 pseudoL ..."
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Cited by 4 (0 self)
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. 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 pseudoLipschitzian 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 RobinsonUrsescu open mapping theorem. As a tool for this alternative proof of the RobinsonUrsescu theorem we also establish a refined version of the multidirectional mean value inequality which is of independent interest. Key Words. Nonsmooth analysis, subdifferentials, coderivatives, implicit function theorem, solvability, stability, open mapping theorem, metric regularity, multidirectional mean value inequality. AMS (1991) subject classification: 26B05. 1 Research...
The Equivalence of Several Basic Theorems for Subdifferentials
 CECM Research Report
"... . Several different basic properties are used for developing a system of calculus for subdifferentials. They are a nonlocal fuzzy sum rule in [5, 25], a multidirectional mean value theorem in [7, 8], local fuzzy sum rules in [14, 15] and an extremal principle in [19, 21]. We show that all these basi ..."
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Cited by 3 (1 self)
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. Several different basic properties are used for developing a system of calculus for subdifferentials. They are a nonlocal fuzzy sum rule in [5, 25], a multidirectional mean value theorem in [7, 8], local fuzzy sum rules in [14, 15] and an extremal principle in [19, 21]. We show that all these basic results are equivalent and discuss some interesting consequences of this equivalence. Keywords: Subdifferentials, mean value inequalities, local fuzzy sum rules, nonlocal fuzzy sum rules, extremal principles and Asplund spaces. AMS (1991) subject classification: Primary 26B05. 1 Introduction Smooth subdifferentials play important roles in nonsmooth analysis for two reasons. They characterize many important generalized differentials and results in terms of smooth subdifferentials often require very little technical assumptions. Currently there are several ways of developing a set of basic theorems for subdifferentials so that they can be conveniently applied to a wide range of problems. ...
Topological Properties of the Approximate Subdifferential
, 1994
"... The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This motivates to study some topological properties more in detai ..."
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Cited by 1 (1 self)
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The approximate subdifferential introduced by Mordukhovich has attracted much attention in recent works on nonsmooth optimization. Potential advantages over other concepts of subdifferentiability might be related to its nonconvexity. This motivates to study some topological properties more in detail. As the main result, it is shown that in a Hilbert space setting each weakly compact set may be obtained as the KuratowskiPainlev'e limit of the approximate subdifferentials of some family of Lipschitzian functions. As a consequence, apart from finiteness, there is no restriction on the number of connected components of the subdifferential. In the finite dimensional case, each topological type of a compact set may be realized by an approximate subdifferential of some Lipschitzian function. These are clear differences for instance to Clarke's subdifferential. The results stated above require the definition of Lipschitzian functions on a space which is enlarged by one extra dimension. Other...