American Institute of Mathematical Sciences

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

Minimax problems for set-valued mappings with set optimization

Yu Zhang 1, and  Tao Chen 2,

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.
Keywords: set optimization, vector optimization, saddle point, Minimax theorem, set-valued mapping..
Mathematics Subject Classification: Primary: 49J35, 49K35; Secondary: 90C4.
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.

show all references

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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