Considering a closed multifunction \Psi between two Banach spaces, it is known that metric regularity and strong metric subregularity of \Psi are, respectively, stable with respect to ``small Lipschitz perturbations&q...
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Considering a closed multifunction \Psi between two Banach spaces, it is known that metric regularity and strong metric subregularity of \Psi are, respectively, stable with respect to ``small Lipschitz perturbations"" and ``small calm perturbations,"" but the corresponding results are no longer true for metric subregularity of \Psi . This paper further deals with the stability issues of metric subregularity with respect to these two kinds of perturbations. We prove that either metric regularity or strong metric subregularity of \Psi at (x=\, y=\) is sufficient for the stability of metric subregularity of \Psi at (x=\, y=\) with respect to small calm subsmooth perturbations and that, under the convexity assumption on \Psi , it is also necessary for the stability of metric subregularity of \Psi at (x=\, y=\) with respect to small calm subsmooth (or Lipschitz) perturbations. Moreover, in terms of the coderivative of \Psi , we provide some sufficient and necessary conditions for metric subregularity of \Psi to be stable with respect to small calm perturbations. Some results obtained in this paper improve and generalize the corresponding results for error bounds in the literature.
In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fr,chet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these con...
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In this paper, we study several types of basic constraint qualifications in terms of Clarke/Fr,chet coderivatives for generalized equations. Several necessary and/or sufficient conditions are given to ensure these constraint qualifications. It is proved that basic constraint qualification and strong basic constraint qualification for convex generalized equations can be obtained by these constraint qualifications, and the existing results on constraint qualifications for the inequality system can be deduced from the given conditions in this paper. The main work of this paper is an extension of the study on constraint qualifications from inequality systems to generalized equations.
We investigate full Lipschitzian and full Holderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where the cost functional, the state equation, and the admi...
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We investigate full Lipschitzian and full Holderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where the cost functional, the state equation, and the admissible control set of the control problems all undergo perturbations. We establish explicit characterizations of both Lipschitzian and Holderian full stability for the class of control problems. We show that for this class of control problems the two full stability properties are equivalent. In particular, the two properties are always equivalent in general when the admissible control set is an arbitrary fixed nonempty, closed, and convex set.
In Part 1 of this paper, we have estimated the Frechet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint u...
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In Part 1 of this paper, we have estimated the Frechet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.
This paper deals with the Holder metric subregularity property of a certain constraint system in Asplund space. Using the techniques of variational analysis, its main part is devoted to establish new sufficient condit...
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This paper deals with the Holder metric subregularity property of a certain constraint system in Asplund space. Using the techniques of variational analysis, its main part is devoted to establish new sufficient conditions in dual spaces for Holder metric subregularity and estimate its modulus, which are derived in terms of coderivatives and normal cones. As an application, the results are applied to study the relationship between higher order growth condition of an unconstraint minimization problem and Holder metric subregularity property of the related constraint system.
The purpose of this paper is to discuss some of the highlights of the theory of metric regularity relative to a cone. For example, we establish a slope and some coderivative characterizations of this concept, as well ...
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The purpose of this paper is to discuss some of the highlights of the theory of metric regularity relative to a cone. For example, we establish a slope and some coderivative characterizations of this concept, as well as some stability results with respect to a Lipschitz perturbation.
This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral con...
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This paper studies solution stability of generalized equations over polyhedral convex sets. An exact formula for computing the Mordukhovich coderivative of normal cone operators to nonlinearly perturbed polyhedral convex sets is established based on a chain rule for the partial second-order subdifferential. This formula leads to a sufficient condition for the local Lipschitz-like property of the solution maps of the generalized equations under nonlinear perturbations.
In this paper, we follow Kuroiwa's set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type dist...
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In this paper, we follow Kuroiwa's set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.
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