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Minimax problems for set-valued mappings with set optimization

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  • In this paper, we introduce a class of set-valued mappings with some set order relations, which is called uniformly same-order. For this sort of mappings, we obtain some existence results of saddle points and depict the structures of the sets of saddle points. Moreover, we obtain a minimax theorem and establish an equivalent relationship between the minimax theorem and a saddle point theorem for the scalar set-valued mappings, in which the minimization and the maximization of set-valued mappings are taken in the sense of set optimization.
    Mathematics Subject Classification: Primary: 49J35, 49K35; Secondary: 90C47.

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  • [1]

    Q. H. Ansari, Y. C. Lin and J. C. Yao, General KKM theorem with applications to minimax and variational inequalities, J. Optim. Theory Appl., 104 (2000), 17-57.doi: 10.1023/A:1004620620928.

    [2]

    G. Y. Chen, A generalized section theorem and a minimax inequality for a vector-valued mapping, Optimization, 22 (1991), 745-754.doi: 10.1080/02331939108843716.

    [3]

    J. W. Chen, Z. P. Wang and Y. J. Cho, The existence of solutions and well-posedness for bilevel mixed equilibrium problems in Banach spaces, Taiwanese J. Math., 17 (2013), 725-748.doi: 10.11650/tjm.17.2013.2337.

    [4]

    Y. J. Cho, S. S. Chang, J. S. Jung, S. M. Kang and X. Wu, Minimax theorems in probabilistic metric spaces, Bull. Austral. Math. Soc., 51 (1995), 103-119.doi: 10.1017/S0004972700013939.

    [5]

    Y. J. Cho, M. R. Delavar, S. A. Mohammadzadeh and M. Roohi, Coincidence theorems and minimax inequalities in abstract convex spaces, J. Inequal. Appl., 2011 (2011), 1-14.

    [6]

    C. S. Chuang and L. J. Lin, New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces, J. Glob. Optim., 57 (2013), 533-547.doi: 10.1007/s10898-012-0004-3.

    [7]

    F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60 (1989), 19-31.doi: 10.1007/BF00938796.

    [8]

    F. Ferro, A minimax theorem for vector-valued functions, Part 2, J. Optim. Theory Appl., 68 (1991), 35-48.doi: 10.1007/BF00939934.

    [9]

    X. H. Gong, The strong minimax theorem and strong saddle points of vector-valued functions, Nonlinear Anal., 68 (2008), 2228-2241.doi: 10.1016/j.na.2007.01.056.

    [10]

    X. H. Gong, Strong vector equilibrium problems, J. Glob. Optim., 36 (2006), 339-349.doi: 10.1007/s10898-006-9012-5.

    [11]

    E. Hernández and L. Rodríguez-Marín, Nonconvex scalarization in set optimization with set-valued maps, J. Math. Anal. Appl., 325 (2007), 1-18.

    [12]

    J. Jahn and T. X. D. Ha, New order relations in set optimization, J. Optim. Theory Appl., 148 (2011), 209-236.doi: 10.1007/s10957-010-9752-8.

    [13]

    D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, in Nonlinear Analysis and Convex Analysis (eds. W. Takahashi and T. Tanaka), World Scientific, River Edge,, (1999), 221-228.

    [14]

    X. B. Li, S. J. Li and Z. M. Fang, A minimax theorem for vector valued functions in lexicographic order, Nonlinear Anal., 73 (2010), 1101-1108.doi: 10.1016/j.na.2010.04.047.

    [15]

    S. J. Li, G. Y. Chen and G. M. Lee, Minimax theorems for set-valued mappings, J. Optim. Theory Appl., 106 (2000), 183-200.doi: 10.1023/A:1004667309814.

    [16]

    S. J. Li, G. Y. Chen, K. L. Teo and X. Q. Yang, Generalized minimax inequalities for set-valued mappings, J. Math. Anal. Appl., 281 (2003), 707-723.doi: 10.1016/S0022-247X(03)00197-5.

    [17]

    Y. C. Lin, Q. H. Ansari and H. C. Lai, Minimax theorems for set-valued mappings under cone-convexities, Abstr. Appl. Anal., 2012 (2012), 1-26.

    [18]

    Y. C. Lin and H. J. Chen, Solving the set equilibrium problems, Fixed Point Theory Appl., 2011 (2011), 1-13.

    [19]

    Y. C. Lin, The hierarchical minimax theorems, Taiwan. J. Math., 18 (2014), 451-462.doi: 10.11650/tjm.18.2014.3503.

    [20]

    Y. C. Lin, On generalized vector equilibrium problems, Nonlinear Anal., 70 (2009), 1040-1048.doi: 10.1016/j.na.2008.01.030.

    [21]

    X. J. Long and J. W. Peng, Generalized B-well-posedness for set optimization, J. Optim. Theory Appl., 157 (2013), 612-623.doi: 10.1007/s10957-012-0205-4.

    [22]

    D. T. Luc, Theory of Vector Optimization: Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin Germany, 1989.

    [23]

    J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl., 40 (1983), 463-475.doi: 10.1007/BF00933511.

    [24]

    M. Patriche, Minimax theorems for set-valued maps without continuity assumptions, preprint, arXiv:1304.0339.

    [25]

    D. S. Shi and C. Ling, Minimax theorems and cone saddle points of uniformly same-order vector-valued functions, J. Optim. Theory Appl., 84 (1995), 575-587.doi: 10.1007/BF02191986.

    [26]

    M. G. Yang, J. P. Xu, N. J. Huang and S. J. Yu, Minimax theorems for vector-valued mappings in abstract convex spaces, Taiwanese J.Math., 14 (2010), 719-732.

    [27]

    Q. B. Zhang, C. Z. Cheng and X. X. Li, Generalized minimax theorems for two set-valued mappings, J. Ind. Manag. Optim., 9 (2013), 1-12.doi: 10.3934/jimo.2013.9.1.

    [28]

    Q. B. Zhang, M. J. Liu and C. Z. Cheng, Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces, Nonlinear Anal., 71 (2009), 212-218.doi: 10.1016/j.na.2008.10.040.

    [29]

    W. Y. Zhang, S. J. Li and K. L. Teo, Well-posedness for set optimization problems, Nonlinear Anal., 71 (2009), 3769-3778.doi: 10.1016/j.na.2009.02.036.

    [30]

    Y. Zhang, S.J. Li and S.K. Zhu, Mininax problems for set-valued mappings, Numer. Funct. Anal. Optim., 33 (2012), 239-253.doi: 10.1080/01630563.2011.610915.

    [31]

    Y. Zhang and S. J. Li, Minimax problems of uniformly same-order set-valued mappings, Bull. Korean Math. Soc., 50 (2013), 1639-1650.doi: 10.4134/BKMS.2013.50.5.1639.

    [32]

    Y. Zhang and S. J. Li, Minimax theorems for scalar set-valued mappings with nonconvex domains and applications, J. Glob. Optim., 57 (2013), 1359-1373.doi: 10.1007/s10898-012-9992-2.

    [33]

    Y. Zhang, S. J. Li and M. H. Li, Mininax inequalities for set-valued mappings, Positivity, 16 (2012), 751-770.doi: 10.1007/s11117-011-0144-6.

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