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.
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 this paper, by revisiting coderivative calculus rules for convex multifunctions in finite-dimensional spaces, we derive formulae for estimating/computing the basic subdifferential and the coderivative of the effici...
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In this paper, by revisiting coderivative calculus rules for convex multifunctions in finite-dimensional spaces, we derive formulae for estimating/computing the basic subdifferential and the coderivative of the efficient point multifunction of parametric convex vector optimization problems. These results are then applied to a broad class of conventional convex vector optimization problems with the presence of operator constraints and equilibrium ones. Examples are also designed to analyze and illustrate the obtained results.
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 sensitivity analysis for a parametric vector variational inequality problem in finite dimensional spaces by using advanced tools in modern variational analysis and generalized differentiation. We...
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This paper deals with sensitivity analysis for a parametric vector variational inequality problem in finite dimensional spaces by using advanced tools in modern variational analysis and generalized differentiation. We mainly focus on computing the coderivatives of the solution mapping in the parametric vector variational inequality problem and then apply them to establish verifiable conditions for the Aubin property of the solution mapping.
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.
This paper introduces and considers the concept of generalized subsmoothness of a multifunction, which is a generalization of both the prox-regularity property and the subsmoothness property of multifunctions. Subsequ...
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This paper introduces and considers the concept of generalized subsmoothness of a multifunction, which is a generalization of both the prox-regularity property and the subsmoothness property of multifunctions. Subsequently, it mainly deals with generalized metric subregularity (in particular, H & ouml;lder metric subregularity) for general set-valued mappings in Asplund spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive sufficient conditions for generalized metric subregularity, which extend even the known results for the conventional metric subregularity. In particular, our results improve/extend the main results established by Li and Mordukhovich (SIAM J. Optim. 22:1655-1684, 2012). Moreover, we also conduct local convergence analysis of an inexact quasi-Newton method for solving the generalized equation 0 is an element of f(x)+F(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\in f(x)+F(x)$\end{document} in Banach spaces, where the function 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} is continuous but not smooth and 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} is a set-valued mapping with closed graph.
In this paper, we present a novel concept of the Fenchel conjugate for set-valued mappings and investigate its properties in finite and infinite dimensions. After establishing some fundamental properties of the Fenche...
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In this paper, we present a novel concept of the Fenchel conjugate for set-valued mappings and investigate its properties in finite and infinite dimensions. After establishing some fundamental properties of the Fenchel conjugate for set-valued mappings, we derive its main calculus rules in various settings. Our approach is geometric and draws inspiration from the successful application of this method by B.S. Mordukhovich and coauthors in variational and convex analysis. Subsequently, we demonstrate that our new findings for the Fenchel conjugate of set-valued mappings can be utilized to obtain many old and new calculus rules of convex generalized differentiation in both finite and infinite dimensions.
In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the sepa...
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In this paper, we study some relationships between polyhedral convex sets and generalized polyhedral convex sets. In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a polyhedral convex set obtained by Ng et al. (Nonlinear Anal. 55:845-858, 2003;Theorem 3.1) are no longer valid when considering generalized polyhedral convex sets instead of polyhedral convex sets. We also introduce and study the notions of generalized polyhedral multifunctions and optimal value functions generated by generalized polyhedral convex multifunctions along with their generalized differentiation calculus rules.
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