On the Generalized Fritz John Optimality Conditions of Vector Optimization With Set-Valued Maps Under Benson Proper Efficiency

SHENG Bao-huai; LIU San-yang

Volume 23 Issue 12

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2002 > 23(12): 1289-1295

SHENG Bao-huai, LIU San-yang. On the Generalized Fritz John Optimality Conditions of Vector Optimization With Set-Valued Maps Under Benson Proper Efficiency[J]. Applied Mathematics and Mechanics, 2002, 23(12): 1289-1295.

Citation:

PDF( 304 KB)

On the Generalized Fritz John Optimality Conditions of Vector Optimization With Set-Valued Maps Under Benson Proper Efficiency

SHENG Bao-huai^1,2,
LIU San-yang¹

1.
Department of Applied Mathematics, Xidian University, Xi'an 710071, P R China;
2.
Institute of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P R China

Received Date: 2000-03-29
Rev Recd Date: 2002-05-14
Publish Date: 2002-12-15

Abstract

Abstract

A kind of tangent derivative and the concepts of strong and weak * pseudoconvexity for a set-valued map are introduced. By the standard separation theorems of the convex sets and cones the optimality Fritz John condition of set-valued optimization under Benson proper efficiency is established,its sufficience is discussed. The form of the optimality conditions obtained here completely tally with the classical results when the set-valued map is specialized to be a single-valued map.
- Contingent tangent cone,
- set-valued map,
- Benson proper efficiency,
- Fritz John condition

FullText(HTML)

References(15)

References

[1]	胡毓达.多目标规划有效性理论[M].上海:上海科学技术出版社,1994,126-127.
[2]	Luc D T.Theory of Vector Optimization,Lecture Notes in Economics and Mathematical Systems[M].Berlin:Springer-Verlag,1989:140-164.
[3]	Corley H W.Optimality conditions for maximizations of set-valued functions[J].J Optim Theory Appl,1988,58(1):1-10.
[4]	CHEN Guang-ya,Jahn J.Optimality conditions for set-valued optimization problems[J].Math Methods Oper Res,1998,48(2):187-200.
[5]	孟志青.集值映射的Hahn-Banach定理[J].应用数学和力学,1998,19(1):55-61.
[6]	Baier J,Jahn J.On subdifferential of set-valued maps[J].J Optim Theory Appl,1999,100(1):233-240.
[7]	WANG Shou-yang,LI Zhong-fei.Scalarization and Lagrange duality in multiobjective optimization[J].Optimization,1992,26(2):315-324.
[8]	CHEN Guang-ya,RONG Wei-dong.Characterizations of the Benson proper efficiency for nonconvex vector optimization[J].J Optim Theory Appl,1998,98(2):365-384.
[9]	LI Zhong-fei.Benson proper efficiency in the vector optimization of set-valued maps[J].J Optim Theory Appl,1998,98(3):623-649.
[10]	盛宝怀,刘三阳,熊胜君.Benson 真有效意义下向量集值优化的广义Fritz John条件[J].经济数学,2000,17(1):59-65.
[11]	Aubin J P,Frankowska H.Set-Valued Analysis[M].Boston:Birkhauser,1990,121-138.
[12]	Borwein J M.Proper efficient points for maximizations with respect to cone[J].SIAM J Control Optim,1977,15(1):57-63.
[13]	LI Ze-min.A theorem of the alternative and its application to the optimization of set-valued maps[J].J Optim Theory Appl,1999,100(2):365-375.
[14]	Dauer J P,Saleh O A.A characterization of proper minimal problem as a solutions of sublinear optimization problems[J].J Math Anal Appl,1993,178(2):227-246.
[15]	Zowe J.A remark on a regularity assumption for the mathematical programming problem in Banach spaces[J].J Optim Theory Appl,1978,25(3):375-381.

Relative Articles

[1]	LIU Ying, FU Xiaoheng, TANG Liping. Asymptotic Characterization of Non-Emptiness and Boundedness of Efficient Solution Sets for Nonconvex Multi-Objective Optimization Problems[J]. Applied Mathematics and Mechanics, 2025, 46(4): 519-527. doi: 10.21656/1000-0887.450235
[2]	YANG Ming, LI Linting, GAO Ying. Relations Between Robust Efficient Solutions and Properly Efficient Solutions to Multiobjective Optimization Problems[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1364-1372. doi: 10.21656/1000-0887.400032
[3]	CHEN Wang, ZHOU Zhiang. Optimality Conditions for Set-Valued Vector Equilibrium Problems With Constraints Involving Improvement Sets[J]. Applied Mathematics and Mechanics, 2018, 39(10): 1189-1197. doi: 10.21656/1000-0887.390104
[4]	TANG Li-ping, YANG Yu-hong. Characterizations of E-Borwein Properly Efficient Solutions[J]. Applied Mathematics and Mechanics, 2017, 38(12): 1399-1404. doi: 10.21656/1000-0887.380238
[5]	LIU Dan-yang, JIANG Ya. Scalarization of Mixed Vector Variational Inequalities and Error Bounds of Gap Functions[J]. Applied Mathematics and Mechanics, 2017, 38(6): 715-726. doi: 10.21656/1000-0887.370292
[6]	LI Xiao-yan, GAO Ying. Optimality Conditions for Proximal Proper Efficiency in Multiobjective Optimization Problems[J]. Applied Mathematics and Mechanics, 2015, 36(6): 668-676. doi: 10.3879/j.issn.1000-0887.2015.06.011
[7]	ZHAO Yong, PENG Zai-yun, ZHANG Shi-sheng. Stability of the Sets of Efficient Points of Vector-Valued Optimization Problems[J]. Applied Mathematics and Mechanics, 2013, 34(6): 643-650. doi: 10.3879/j.issn.1000-0887.2013.06.010
[8]	XU Hai-li, GUO Xing-ming. Auxiliary Principle and Three-Step Iterative Algorithms for Generalized Set-Valued Strongly Nonlinear Mixed Variational-Like Inequalities[J]. Applied Mathematics and Mechanics, 2007, 28(6): 643-650.
[9]	SHENG Bao-huai, LIU San-yang. Kuhn-Tucker Condition and the Wolfe Duality of Preinvex Set-Valued Optimization[J]. Applied Mathematics and Mechanics, 2006, 27(12): 1447-1456.
[10]	LI Ting. Set Value Mapping and Its Application[J]. Applied Mathematics and Mechanics, 2006, 27(2): 237-242.
[11]	YAN Qing-you, XIONG Xi-wen. An Effcient and Stable Structure Preserving Algorithm for Computing the Eigenvalues of a Hamiltonian Matrix[J]. Applied Mathematics and Mechanics, 2002, 23(11): 1150-1168.
[12]	XUAN Zhao-cheng, LI Xing-si. Unilateral Contact Problems Using Quasi-Active Set Strategy[J]. Applied Mathematics and Mechanics, 2002, 23(8): 811-818.
[13]	J. M. Soriano. Fredholm and Compact Mappings Sharing a Value[J]. Applied Mathematics and Mechanics, 2001, 22(6): 609-612.
[14]	Amitabh Banerjee, Thakur Balwant Singh. A Fixed Point Theorem for Set-Valued Mappings[J]. Applied Mathematics and Mechanics, 2001, 22(12): 1255-1261.
[15]	Gu Guoqing, Yu Kinwah. A Theoretical Research to Effective Viscosity of Colloidal Dispersions[J]. Applied Mathematics and Mechanics, 2000, 21(3): 245-252.
[16]	Cao Chun. Fuzzy Approaching Set and Fuzzy Approaching Functional Mapping[J]. Applied Mathematics and Mechanics, 1998, 19(11): 1004-1013.
[17]	Ding Xieping, He Yiran. Best Approximation Theorem for Set-Valued Mappings without Convex Values and Continuity[J]. Applied Mathematics and Mechanics, 1998, 19(9): 773-778.
[18]	Meng Zhiqing. Hahn-Banach Theorem of Set-Valued Map[J]. Applied Mathematics and Mechanics, 1998, 19(1): 54-61.
[19]	Hu Ning. An Effective Boundary Method for the Analysis of Elastoplastic Problems[J]. Applied Mathematics and Mechanics, 1992, 13(8): 711-718.
[20]	Ding Xie-ping. Fixed Point Theorems of Random Set-Valued Mappings and Their Applications[J]. Applied Mathematics and Mechanics, 1984, 5(4): 561-575.