Sufficient conditions for a norm-to-weak* continuous mapping f : X -> X* being monotone or submonotone are established by its Frechet and normal coderivatives, where X is an Asplund space with its dual space X*. Un...
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Sufficient conditions for a norm-to-weak* continuous mapping f : X -> X* being monotone or submonotone are established by its Frechet and normal coderivatives, where X is an Asplund space with its dual space X*. Under some additional assumptions, they are also necessary conditions. Among other things, we obtain a criterion for the monotonicity of continuous mappings which extends the following classical result: a differentiable mapping F : R-n -> R-n is monotone if and only if for each x is an element of R-n the Jacobian matrix del F(x) is positive semi-definite;see [22, Proposition 12.3]. As a by-product, sufficient conditions for a function being convex or approximately convex are given.
In this paper, error bounds for gamma-paraconvex multifunctions are considered. Characterizations of a gamma-paraconvex multifunction are given. In terms of normal cone and coderivative, some results on the existence ...
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In this paper, error bounds for gamma-paraconvex multifunctions are considered. Characterizations of a gamma-paraconvex multifunction are given. In terms of normal cone and coderivative, some results on the existence of error bounds are presented.
The explicit representation of Mordukhovich coderivative of a solution mapping to a second-order cone constrained parametric variational inequality is established by the reduction approach. The result obtained is used...
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The explicit representation of Mordukhovich coderivative of a solution mapping to a second-order cone constrained parametric variational inequality is established by the reduction approach. The result obtained is used to obtain a necessary and sufficient condition for the Lipschitz-like property of the solution mapping to the parametric variational inequality and global optimality conditions for a bilevel programming with a second-order cone constrained lower level problem.
This paper deals with a kind of nonconvex optimistic bilevel optimization programs. In some process of dealing this kind of bilevel programs, difficulties are essentially moved to estimating for coderivative of the so...
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This paper deals with a kind of nonconvex optimistic bilevel optimization programs. In some process of dealing this kind of bilevel programs, difficulties are essentially moved to estimating for coderivative of the solution map. To deal with these difficulties, we use value function of the lower level problem and its modifications as implicit functions to describe the solution map. By applying techniques in variational analysis, we give estimates for coderivative of the solution map. Then, we will show the applications in optimality conditions for these bilevel programs which we derived by using the extremal principle.
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 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 variational ineq...
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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 variational inequality with equality and inequality constraints. The notion of integrated deviation is introduced to characterize the outer limit of a sequence of sets. It is demonstrated that, under suitable conditions, both the cosmic deviation and the integrated deviation between the coderivative of the solution mapping to SAA problem and that of the solution mapping to the parametric stochastic variational inequality 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 stochastic variational inequality 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 bilevel program.
In this study, we examine the Generalized Equations' subregularity in Asplund spaces utilizing a novel approach. We obtain sufficient conditions for a family of multifunctions to be metrically subregular which are...
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In this study, we examine the Generalized Equations' subregularity in Asplund spaces utilizing a novel approach. We obtain sufficient conditions for a family of multifunctions to be metrically subregular which are stronger than the known sufficient conditions thanks to a modification of the well-known coderivative concept and of the partial sequential normal compactness.
In the original article a mistake has been introduced into the abstract. To put matters right we are now reprinting the abstract as it should have appeared: The limiting (Mordukhovich) coderivative of the metric proje...
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In the original article a mistake has been introduced into the abstract. To put matters right we are now reprinting the abstract as it should have appeared: The limiting (Mordukhovich) coderivative of the metric projection onto the second-order cone in Rn is computed. This result is used to obtain a sufficient condition for the Aubin property of the solution map of a parameterized secondorder cone complementarity problem and to derive necessary optimality conditions for a mathematical program with a second-order cone complementarity problem among the constraints. Everything else in the paper remains correct.
This paper is devoted to strict efficiency in set optimization studied with the set approach. Strict efficient solutions are defined with respect to the l-type less order relation and the possibly less order relation....
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This paper is devoted to strict efficiency in set optimization studied with the set approach. Strict efficient solutions are defined with respect to the l-type less order relation and the possibly less order relation. Scalar characterization and necessary and/or sufficient conditions for such solutions are obtained. In particular, we establish some conditions expressed in terms of a high-order directional derivative of set-valued maps and the (convex or limiting) subdifferentials, normal cones and coderivatives. Various illustrating examples are presented.
Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, ...
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Variational convexity, together with ist strong counterpart, of extended-real-valued functions has been recently introduced by Rockafellar. In this paper we present second-order characterizations of these properties, i.e., conditions using first-order generalized derivatives of the subgradient mapping. Up to now, such characterizations are only known under the assumptions of prox-regularity and subdifferential continuity and in this paper we discard the latter. To this aim we slightly modify the definitions of the generalized derivatives to be compatible with the f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f$\end{document}-attentive convergence appearing in the definition of subgradients. We formulate our results in terms of both coderivatives and subspace containing derivatives. We also give formulas for the exact bound of variational convexity and study relations between variational strong convexity, tilt-stable local minimizers and strong metric regularity of some truncation of the subgradient mapping.
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