Minimax theorems for scalar set-valued mappings with nonconvex domains and applications

In this paper, by virtue of the separation theorem of convex sets, we prove a minimax theorem, a cone saddle point theorem and a Ky Fan minimax theorem for a scalar set-valued mapping under nonconvex assumptions of its domains, respectively. As applications, we obtain an existence result for the gen...

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Bibliographic Details
Published inJournal of global optimization Vol. 57; no. 4; pp. 1359 - 1373
Main Authors Zhang, Y., Li, S. J.
Format Journal Article
LanguageEnglish
Published Boston Springer US 01.12.2013
Springer
Springer Nature B.V
Subjects
Computer Science
Control theory
Equilibrium
Game theory
Inequality
Mapping
Mathematical analysis
Mathematical functions
Mathematics
Mathematics and Statistics
Maximization
Minimax technique
Neighborhoods
Operations Research/Decision Theory
Optimization
Real Functions
Scalars
Studies
Theorems
Vector space
Vectors (mathematics)
49J35
Minimax theorem
Set-valued mapping
90C47
Vector optimization
Cone saddle point
49K35
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Summary:In this paper, by virtue of the separation theorem of convex sets, we prove a minimax theorem, a cone saddle point theorem and a Ky Fan minimax theorem for a scalar set-valued mapping under nonconvex assumptions of its domains, respectively. As applications, we obtain an existence result for the generalized vector equilibrium problem with a set-valued mapping. Simultaneously, we also obtain some generalized Ky Fan minimax theorems for set-valued mappings, in which the minimization and the maximization of set-valued mappings are taken in the sense of vector optimization.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-012-9992-2