scholarly journals Sharp Efficiency for Vector Equilibrium Problems on Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Si-Huan Li ◽  
Qiang Wang ◽  
Shu Xu ◽  
Jun-Xiang Wang
Keyword(s):  
Optimality Conditions ◽  
Banach Spaces ◽  
Normal Cone ◽  
Equilibrium Problems ◽  
Necessary Optimality Conditions ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Sufficient Optimality Conditions ◽  
Vector Equilibrium Problems ◽  
Vector Equilibrium

The concept of sharp efficient solution for vector equilibrium problems on Banach spaces is proposed. Moreover, the Fermat rules for local efficient solutions of vector equilibrium problems are extended to the sharp efficient solutions by means of the Clarke generalized differentiation and the normal cone. As applications, some necessary optimality conditions and sufficient optimality conditions for local sharp efficient solutions of a vector optimization problem with an abstract constraint and a vector variational inequality are obtained, respectively.

scholarly journals Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

2019 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Maria B. Donato ◽  
Rafaela Osuna-Gómez ◽  
Monica Milasi
Keyword(s):  
Optimality Conditions ◽  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Efficient Solutions ◽  
Vector Optimization Problem ◽  
Hadamard Manifolds ◽  
Vector Equilibrium Problems ◽  
Topological Vector Spaces ◽  
Weakly Efficient Solutions ◽  
Vector Equilibrium

The aim of this paper is to obtain Karush-Kuhn-Tucker optimality conditions for weakly efficient solutions to vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient solutions to the constrained vector optimization problem are presented. As well as some examples. The results presented in this paper generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces and real Banach spaces to Hadamard manifolds, respectively.

scholarly journals Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1037 ◽  
Author(s):  
Gabriel Ruiz-Garzón ◽  
Rafaela Osuna-Gómez ◽  
Jaime Ruiz-Zapatero
Keyword(s):  
Optimality Conditions ◽  
Symmetric Spaces ◽  
Equilibrium Problems ◽  
Sufficient Conditions ◽  
Sufficient Optimality Conditions ◽  
Hadamard Manifolds ◽  
Vector Equilibrium Problems ◽  
Pareto Points ◽  
Vector Equilibrium ◽  
Necessary And Sufficient

The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds.

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