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.
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 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.
Using the Borwein-Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Holder metric subregularity and Holder error bounds in a class of smooth...
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Using the Borwein-Preiss variational principle and in terms of the proximal coderivative, we provide a new type of sufficient conditions for the Holder metric subregularity and Holder error bounds in a class of smooth Banach spaces. As an application, new characterizations for the tilt stability of Holder minimizers are established.
Metric subregularity is an important and active area in modem variational analysis and nonsmooth *** existing results on the metric suregularity were established in terms of coderivatives of the multifunctions *** not...
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Metric subregularity is an important and active area in modem variational analysis and nonsmooth *** existing results on the metric suregularity were established in terms of coderivatives of the multifunctions *** note tries to give a survey of the metric subregularity theory related to the coderivatives and normal cones.
The aim of this paper is to investigate the convergence properties for Mordukhovich's coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic generalized equa...
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The aim of this paper is to investigate the convergence properties for Mordukhovich's coderivative of the solution map of the sample average approximation (SAA) problem for a parametric stochastic generalized equation. It is demonstrated that, under suitable conditions, both the cosmic deviation and the rho-deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic generalized equation converge almost surely to zero as the sample size tends to infinity. Moreover, the exponential convergence rate of coderivatives of the solution maps to the SAA parametric generalized equations is established. The results are used to develop sufficient conditions for the consistency of the Lipschitz-like property of the solution map of SAA problem and the consistency of stationary points of the SAA estimator for a stochastic mathematical program with complementarity constraints.
The problem of minimizing a linear-quadratic function over the Euclidean ball is encountered frequently in the theory of trust-region methods in nonlinear programming. By some tools from Variational Analysis, we inves...
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The problem of minimizing a linear-quadratic function over the Euclidean ball is encountered frequently in the theory of trust-region methods in nonlinear programming. By some tools from Variational Analysis, we investigate the stability of the Karush-Kuhn-Tucker point set map of that problem with respect to total perturbations of its data. Verifiable sufficient conditions for the local Lipschitz-like property of the map are obtained, and the connection of our results with the existing criteria for the lower semicontinuity of this Karush-Kuhn-Tucker point set map is shown. (C) 2013 Elsevier Ltd. All rights reserved.
Metric subregularity and regularity of multifunctions are fundamental notions in variational analysis and optimization. Using the concept of strong slope, in this paper we first establish a criterion for metric subreg...
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Metric subregularity and regularity of multifunctions are fundamental notions in variational analysis and optimization. Using the concept of strong slope, in this paper we first establish a criterion for metric subregularity of multifunctions between metric spaces. Next, we use a combination of abstract coderivatives and contingent derivatives to derive verifiable first order conditions ensuring metric subregularity of multifunctions between Banach spaces. Then using second order approximations of convex multifunctions, we establish a second order condition for metric subregularity of mixed smooth-convex constraint systems, which generalizes a result established recently by Gfrerer
The equality type Mordukhovich coderivative rule for a solution mapping to a second-order cone constrained parametric variational inequality is derived under the constraint nondegenerate condition, which improves the ...
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The equality type Mordukhovich coderivative rule for a solution mapping to a second-order cone constrained parametric variational inequality is derived under the constraint nondegenerate condition, which improves the result published recently. The rule established is then applied to deriving a necessary and sufficient local optimality condition for a bilevel programming with a second-order cone constrained lower level problem.
This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of ...
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This paper focuses on the metric regularity of a positive order for generalized equations. More concretely, we establish verifiable sufficient conditions for a generalized equation to achieve the metric regularity of a positive order at its a given solution. The provided conditions are expressed in terms of the Frechet coderivative/or the Mordukhovich coderivative/or the Clarke one of the corresponding multifunction formulated the generalized equation. In addition, we show that such sufficient conditions turn out to be also necessary for the metric regularity of a positive order of the generalized equation in the case where the multifunction established the generalized equation is closed and convex.
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