Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems
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 noti...
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| Vydáno v: | Nonlinear analysis Ročník 75; číslo 3; s. 1305 - 1323 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Amsterdam
Elsevier Ltd
01.02.2012
Elsevier |
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| ISSN: | 0362-546X, 1873-5215 |
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| Abstract | 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. |
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| AbstractList | 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. 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. |
| Author | Ha, Truong Xuan Duc |
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| Keywords | Quasi weakly efficient solutions Sufficient optimality conditions Necessary optimality conditions Normal cone Pareto efficient solutions Coderivative Set-valued equilibrium problem Relative efficient solutions 90C29 Weakly efficient solutions 49J53 Properly efficient solutions Strongly efficient solutions 90C33 Vector equilibrium problem 49K27 Vector variational inequality Set-valued optimization problem Optimization method Variational inequality Mathematical model Nonlinear analysis Necessary condition Sufficient condition Optimality condition |
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| SubjectTerms | Approximation Banach space Calculus of variations and optimal control Coderivative Exact sciences and technology Geometric constraints Global analysis, analysis on manifolds Inequalities Mathematical analysis Mathematics Necessary optimality conditions Normal cone Optimization Pareto efficient solutions Pareto optimality Properly efficient solutions Quasi weakly efficient solutions Relative efficient solutions Sciences and techniques of general use Set-valued equilibrium problem Set-valued optimization problem Strongly efficient solutions Sufficient optimality conditions Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds Vector equilibrium problem Vector variational inequality Vectors (mathematics) Weakly efficient solutions |
| Title | Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems |
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