We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence b...
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We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality. We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting. (C) 2011 Elsevier Ltd. All rights reserved.
This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization called the optimistic bilevel problem and its initial more difficult formulation that we c...
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This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization called the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not-to a large extent-differ from those known for the conventional problem. It has already been well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This is related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem;this function is labeled here as the two-level value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data.
In this paper we investigate the Lipschitz-like property of the solution mapping of parametric variational inequalities over perturbed polyhedral convex sets. By establishing some lower and upper estimates for the cod...
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In this paper we investigate the Lipschitz-like property of the solution mapping of parametric variational inequalities over perturbed polyhedral convex sets. By establishing some lower and upper estimates for the coderivatives of the solution mapping, among other things, we prove that the solution mapping could not be Lipschitz-like around points where the positive linear independence condition is invalid. Our analysis is based heavily on the Mordukhovich criterion (Mordukhovich in Variational Analysis and Generalized Differentiation. vol. I: Basic Theory, vol. II: Applications. Springer, Berlin, 2006) of the Lipschitz-like property for set-valued mappings between Banach spaces and recent advances in variational analysis. The obtained result complements the corresponding ones of Nam (Nonlinear Anal 73:2271-2282, 2010) and Qui (Nonlinear Anal 74:1674-1689, 2011).
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, 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.
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 considering the coderivatives of the generalized perturbation maps in general Banach spaces. Under some mild conditions, the upper estimate of coderivatives of the generalized perturbation map...
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This paper is devoted to considering the coderivatives of the generalized perturbation maps in general Banach spaces. Under some mild conditions, the upper estimate of coderivatives of the generalized perturbation maps are obtained. Their exact calculus rules are obtained under some additional conditions. Furthermore, the generalized perturbation maps are shown to be differentiably regular under some strong conditions.
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