# American Institute of Mathematical Sciences

2014, 4(4): 327-340. doi: 10.3934/naco.2014.4.327

## Minimax problems for set-valued mappings with set optimization

 1 College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China 2 College of Public Foundation, Yunnan Open University, Kunming 650223, China

Received  September 2014 Revised  December 2014 Published  December 2014

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.
Citation: Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
##### References:
 [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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##### References:
 [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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