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.
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.
The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problem...
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The metric regularity of multifunctions plays a crucial role in modern variational analysis and optimization. This property is a key to study the stability of solutions of generalized equations. Many practical problems lead to generalized equations associated to the sum of multifunctions. This paper is devoted to study the metric regularity of the sum of multifunctions. As the sum of closed multifunctions is not necessarily closed, almost all known results in the literature on the metric regularity for one multifunction (which is assumed usually to be closed) fail to imply regularity properties of the sum of multifunctions. To avoid this difficulty, we use an approach based on the metric regularity of so-called epigraphical multifunctions and the theory of error bounds to study the metric regularity of the sum of two multifunctions, as well as some related important properties of variational systems. Firstly, we establish the metric regularity of the sum of a regular multifunction and a pseudo-Lipschitz multifunction with a suitable Lipschitz modulus. These results subsume some recent results by Durea and Strugariu. Secondly, we derive coderivative characterizations of the metric regularity of epigraphical multifunctions associated with the sum of multifunctions. Applications to the study of the behavior of solutions of variational systems are reported.
This article is devoted to the study of a nonsmooth multiobjective bilevel optimization problem, which involves the vector-valued objective functions in both levels of the considered program. We first formulate a rela...
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This article is devoted to the study of a nonsmooth multiobjective bilevel optimization problem, which involves the vector-valued objective functions in both levels of the considered program. We first formulate a relaxation multiobjective formulation for the multiobjective bilevel problem and examine the relationships of solutions between them. We then establish Fritz John (FJ) and Karush-Kuhn-Tucker (KKT) necessary conditions for the nonsmooth multiobjective bilevel optimization problem via its relaxation. This is done by studying a related multiobjective optimization problem with operator constraints.
In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this...
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In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886-911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.
This paper introduces a kind of sub-Lipschitz continuity for set-valued mappings based on the cosmic metric. This type of Lipschitz behavior has applications with regards to necessary optimality conditions, the Hamilt...
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This paper introduces a kind of sub-Lipschitz continuity for set-valued mappings based on the cosmic metric. This type of Lipschitz behavior has applications with regards to necessary optimality conditions, the Hamilton-Jacobi equation, and invariance of unbounded differential inclusions. Cosmically Lipschitz assumptions allow for broader applications than previously allowed under Lipschitz assumptions. It is also shown that a cosmically Lipschitz mapping can be characterized by the normal cones to its graph using the coderivative, and various rules are presented in order to more easily identify such a mapping.
In this paper, we pursue two goals. First, we find exact relationships between the three concepts of semismooth sets, functions, and maps. Then, we consider the nonsmooth calculus of these notions. Particularly, we pr...
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In this paper, we pursue two goals. First, we find exact relationships between the three concepts of semismooth sets, functions, and maps. Then, we consider the nonsmooth calculus of these notions. Particularly, we prove that the Mordukhovich and linear subdifferentials (coderivatives) are equal for the semismooth functions (maps). Several examples are presented to illustrate the results of the paper.
In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of t...
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In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the sense of loffe. Clarke and Mordukhovich. As applications we obtain sufficient conditions for F to have a Henig proper minimizer or a super minimizer under the Palais-Smale type conditions. The techniques are essentially based on the characterizations of Henig proper efficient points and super efficient points by mean of the Henig dilating cones and the Hiriart-Urruty signed distance function. (C) 2009 Elsevier Inc. All rights reserved.
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.
In this paper, higher order strongly convex-graph multifunctions are introduced. Using the normal cones, we give some characterizations for higher order strongly convex-graph multifunctions. Moreover, in terms of the ...
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In this paper, higher order strongly convex-graph multifunctions are introduced. Using the normal cones, we give some characterizations for higher order strongly convex-graph multifunctions. Moreover, in terms of the coderivatives, we present some sufficient conditions for multifunctions with set constraints to have global error bounds with exponents in Banach spaces.
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