The paper concerns an optimization problem with a generalized equation among the constraints. This model includes standard mathematical programs with parameter-dependent variational inequalities or complementarity pro...
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The paper concerns an optimization problem with a generalized equation among the constraints. This model includes standard mathematical programs with parameter-dependent variational inequalities or complementarity problems as side constraints. Using Mordukhovich's generalized differential calculus, we derive necessary optimality conditions and apply them to problems, where the equilibria are governed by implicit complementarity problems and by hemivariational inequalities.
In this paper, adopting an admissible function phi, we consider a kind of generalized metric subregularity/regularity of a multifunction F with respect to phi, which is a natural generalization of the Holder metric re...
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In this paper, adopting an admissible function phi, we consider a kind of generalized metric subregularity/regularity of a multifunction F with respect to phi, which is a natural generalization of the Holder metric regularity. In the special case when F is the subdifferential mapping of a proper lower semicontinuous function f, it is known that such a generalized metric subregularity is very closely related to the well-posedness of f. Using the technique of variational analysis and in terms of the coderivative, we established some sufficient conditions for a multifunction to be metrically subregular/regular with respect to an admissible function phi. In particular, we extend some existing results on the metric regularity and Holder metric regularity.
This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.
This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.
We introduce concepts of metric regularity and metric subregularity of a positive-order for an implicit multifunction and provide new sufficient conditions for the implicit multifunctions to achieve the addressed prop...
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We introduce concepts of metric regularity and metric subregularity of a positive-order for an implicit multifunction and provide new sufficient conditions for the implicit multifunctions to achieve the addressed properties. The conditions provided are presented in terms of the Frechet/Mordukhovich coderivative of the corresponding parametric multifunction formulated the implicit multifunction. We show that such sufficient conditions are also necessary for the metric regularity/subregularity of a positive-order of the implicit multifunction when the corresponding parametric multifunction is (locally) convex and closed. In this way, we establish criteria ensuring that an implicit multifunction is Holder-like and calm of a positive-order at a given point. As applications, we derive sufficient conditions in terms of coderivatives for a multifunction (resp., its inverse multifunction) to have the open covering property and the metric regularity/subregularity of a positive-order (resp., the Holder-like/calm property).
This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equ...
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This paper concerns a generalized equation defined by a closed multifunction between Banach spaces, and we employ variational analysis techniques to provide sufficient and/or necessary conditions for a generalized equation to have the metric subregularity (i.e., local error bounds for the concerned multifunction) in general Banach spaces. Following the approach of loffe [Trans. Amer. Math. Soc., 251 (1979), pp. 61-69] who studied the numerical function case, our conditions are described in terms of coderivatives of the concerned multifunction at points outside the solution set. Motivated by the existing modulus representation and point-based criteria for the metric regularity, we establish the corresponding results for the metric subregularity. In the Asplund space case, sharper results are obtained.
The paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory. New forms of the rule and further applications are presented....
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The paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory. New forms of the rule and further applications are presented. (C) 2011 Elsevier Ltd. All rights reserved.
In this paper, based on basic constraint qualification (BCQ) and strong BCQ for convex generalized equation, we are inspired to further discuss constraint qualifications of BCQ and strong BCQ for nonconvex generalized...
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In this paper, based on basic constraint qualification (BCQ) and strong BCQ for convex generalized equation, we are inspired to further discuss constraint qualifications of BCQ and strong BCQ for nonconvex generalized equation and then establish their various characterizations. As applications, we use these constraint qualifications to study metric subregularity of nonconvex generalized equation and provide necessary and/or sufficient conditions in terms of constraint qualifications considered herein to ensure nonconvex generalized equation having metric subregularity.
In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two...
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In this paper, we introduce and consider the concept of the prox-regularity of a multifunction. We mainly study the metric subregularity of a generalized equation defined by a proximal closed multifunction between two Hilbert spaces. Using proximal analysis techniques, we provide sufficient and/or necessary conditions for such a generalized equation to have the metric subregularity in Hilbert spaces. We also establish the results of Robinson-Ursescu theorem type for prox-regular multifunctions. (C) 2011 Elsevier Ltd. All rights reserved.
We obtain necessary and sufficient conditions for local Lipschitz-like property and sufficient conditions for local metric regularity in Robinson's sense of Karush-Kuhn-Tucker point setmaps of trust-region subprob...
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We obtain necessary and sufficient conditions for local Lipschitz-like property and sufficient conditions for local metric regularity in Robinson's sense of Karush-Kuhn-Tucker point setmaps of trust-region subproblems in trust-region methods. The main tools being used in our investigation are dual criteria for fundamental properties of implicit multifunctions which are established on the basis of generalized differentiation of normal cone mappings.
This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this pr...
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This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.
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