集值映射的一种锥凸性及标量化

doi:10.15960/j.cnki.issn.1007-6093.2015.04.003

Operations Research Transactions

Previous Articles Next Articles

A kind of cone-convexity for set-valued maps and its scalarization

LI Fei^1,* TANG Liping² YANG Xinmin³

1. School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China; 2. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China; 3. School of Mathematical Sciences, Chongqing Normal University, Chongqing 400047, China

Received:2015-11-12 Online:2016-12-15 Published:2016-12-15

Abstract

Abstract:

In this paper, we firstly introduce the notion of scalar cone-quasiconvexity for set-valued maps and discuss the relationships among several cone-convexities. A characterization for proper cone-quasiconvexity of set-valued maps is also given in the sense of a type of level set. Meanwhile, the composition rule of cone-convexity of set-valued maps is established by scalar increasing convex functions. We obtain a characterization for cone-quasiconvexity of set-valued maps by Gerstewitz functional finally.

Key words: set-valued maps, cone-convexity, Gerstewitz functional, scalarizations

LI Fei, TANG Liping, YANG Xinmin. A kind of cone-convexity for set-valued maps and its scalarization[J]. Operations Research Transactions, doi: 10.15960/j.cnki.issn.1007-6093.2015.04.003.

References

[1] Yu P L. Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives [J]. Journal of Optimization Theory and Applications, 1974, 14(3): 319-377.

[2] Jeyakumar V. A generalization of a minimax theorem of Fan via a theorem of the alternative [J]. Journal of Optimization Theory and Applications, 1986, 48(3): 525-533.

[3] Tanaka T. Generalized quasiconvexities, cone saddle points, and minimax theorems for vector-valued functions [J]. Journal of Optimization Theory and Applications, 1994, 81(2): 355-377.

[4] Aubin J-P, Frankowska H. Set-Valued Analysis [M]. Boston: Birkh\"{a}user, 1990.

[5] Khan A A, Tammer C, Z\u{a}linescu C. Set-Valued Optimization-An Introduction with Applications [M]. Berlin: Springer, 2015.

[6] Yang X M, Yang X Q, Chen G Y. Theorems of the alternative and optimization with set-valued maps [J]. Journal of Optimization Theory and Applications, 2000, 107(3): 627-640.

[7] Yang X M, Li D, Wang S Y. Near-subconvexlikeness in vector optimization with set-valued functions [J]. Journal of Optimization Theory and Applications, 2001, 110(2): 413-427.

[8] Sach P H. New generalized convexity notion for set-valued maps and application to vector optimization [J]. Journal of Optimization Theory and Applications, 2005, 125(1): 157-179.

[9] Kuroiwa D, Tanaka T, Ha T X D. On cone convexity of set-valued maps [J]. Nonlinear Analysis: Theory, Methods $\&$ Applications, 1997, 30(3): 1487-1496.

[10] Kuroiwa D. Convexity for set-valued maps [J]. Applied Mathematics Letters, 1996, 9(2): 97-101.

[11] Benoist J, Popovici N. Characterizations of convex and quasiconvex set-valued maps [J]. Mathematical Methods of Operations Research, 2003, 57(3): 427-435.

[12] Luc D T. Theory of Vector Optimization [M]. Berlin: Springer, 1989.

[13] Lin Y C, Ansari Q H, Lai H C. Minimax theorems for set-valued mappings under cone-convexities [J]. Abstract and Applied Analysis, 2012, DOI: 10.1155/2012/310818.

[14] G\"{o}pfert A, Riahi H, Tammer C, et al. Variational Methods in Partially Ordered Spaces [M]. New York: Springer, 2003.

[15] Jeyakumar V, Oettli W, Natividad M. A solvability theorem for a class of quasiconvex mappings with applications to optimization [J]. Journal of Mathematical Analysis and Applications}, 1993, 179(2): 537-546.

[16] Li S J, Chen G Y, Teo K L, et al. Generalized minimax inequalities for set-valued mappings [J]. Journal of Mathematical Analysis and Applications, 2003, 281(2): 707-723.

[17] Chen G Y, Huang X X, Yang X Q. Vector Optimization-Set-Valued and Variational Analysis [M]. Berlin: Springer, 2005.

A kind of cone-convexity for set-valued maps and its scalarization

PDF

Knowledge

Abstract

Cite this article

share this article

References

Related Articles 1

Recommended Articles

Metrics

Comments