单调Minkowski泛函与Henig真有效性的标量化

数学物理学报 ›› 2014, Vol. 34 ›› Issue (3): 581-592.

单调Minkowski泛函与Henig真有效性的标量化

张申媛¹|丘京辉^2*

1.上海电力学院数理学院上海 |200090;
2.苏州大学数学科学学院江苏苏州 215006

收稿日期:2012-08-09 修回日期:2013-03-11 出版日期:2014-06-25 发布日期:2014-06-25
通讯作者: 丘京辉，jhqiu@suda.edu.cn E-mail:jhqiu@suda.edu.cn
基金资助:
国家自然科学基金(10871141)资助

Monotone Minkowski Functionals and Scalarizations of Henig Proper Efficiency

ZHANG Shen-Yuan¹, QIU Jing-Hui^2*

1.School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai 200090;
2.School of Mathematical Sciences, Soochow University, Jiangsu Suzhou 215006

Received:2012-08-09 Revised:2013-03-11 Online:2014-06-25 Published:2014-06-25
Contact: QIU Jing-Hui，jhqiu@suda.edu.cn E-mail:jhqiu@suda.edu.cn
Supported by:
国家自然科学基金(10871141)资助

摘要/Abstract

摘要：

没有凸锥的闭性和点性假设, 该文考虑由一般凸锥生成的单调Minkowski泛函并研究其性质. 由此, 在偏序局部凸空间的框架下, 通过利用单调连续Minkowski泛函和单调连续半范, 该文分别获得了一般集合及锥有界集合的弱有效点的标量化. 利用此弱有效性的标量化, 该文分别推导出一般集合及锥有界集合的Henig真有效点的标量化. 进而, 当序锥具备有界基时, 该文获得局部凸空间中超有效性的一些标量化结果. 最后, 该文给出Henig真有效性和超有效性的稠密性结果. 这些结果推广并改进了有关的已知结果.

关键词: 局部凸空间, Minkowski泛函, 弱有效性, Henig真有效性, 标量化, 稠密性

Abstract:

Without the closedness and the pointedness of convex cones, we consider monotone Minkowski functionals generated
by general convex cones and investigate properties of those Minkowski functionals. From this, in the framework of partially ordered locally convex spaces we obtain scalarizations of weakly efficient points of a general set and of a cone-bounded set by using a monotone continuous Minkowski functional and a monotone continuous semi-norm,
respectively. Using the scalarizations of weak efficiency, we deduce scalarizations of Henig properly efficient points of a
general set and of a cone-bounded set, respectively. Moreover, when the ordering cone has a bounded base, we obtain some scalarization results on superefficiency in locally convex spaces. Finally we give some density results for Henig proper efficiency and superefficiency. These results generalize and improve the related known results.

Key words: Locally convex space, Minkowski functional, Weak efficiency, Henig proper efficiency, Scalarization, Density

中图分类号:

90C29

张申媛, 丘京辉. 单调Minkowski泛函与Henig真有效性的标量化[J]. 数学物理学报, 2014, 34(3): 581-592.

ZHANG Shen-Yuan, QIU Jing-Hui. Monotone Minkowski Functionals and Scalarizations of Henig Proper Efficiency[J]. Acta mathematica scientia,Series A, 2014, 34(3): 581-592.

参考文献

[1] Zheng X Y. {Scalarization of Henig proper efficient points in a normed space.} J Optim Theory Appl, 2000, 105: 233--247

[2] Zheng X Y. {Proper efficiency in locally convex topological vector spaces.} J Optim Theory Appl, 1997, 94: 469--486

[3] Qiu J H, Hao Y. {Scalarization of Henig properly efficient points in locally convex spaces.} J Optim Theory Appl, 2010, 147: 71--92

[4] Gerstewitz C. {Nichtkonvexe dualit\"{a}t in der vektoroptimierung.} Wiss Z - Tech Hochsch Ilmenau, 1983, 25: 357--364
[5] Gerstewitz C, Iwanow E. {Dualit\"{a}t f\"{u}r nichlkonvexe vektoroptimierungsprobleme.} Wiss Z - Tech Hochsch Ilmenau, 1985, 31: 61--81

[6] Qiu J H. {Superefficiency in locally convex spaces.} J Optim Theory Appl, 2007, 135: 19--35

[7] K\"{o}the G. {Topological Vector SpacesI.} Berlin: Springer-Verlag, 1969

[8] Holmes R B. {Geometric Functional Analysis and its Applications.} New York: Springer-Verlag, 1975

[9] Rudin W. {Functional Analysis.} New York: McGraw-Hill, 1991

[10] Chen G Y, Huang X X, Yang X Q. {Vector Optimization -- Set-Valued and Variational Analysis.} Berlin: Springer-Verlag, 2005

[11] Gerth C, Weidner P. {Nonconvex separation theorems and some applications in vector optimization.} J Optim Theory Appl, 1990, 67: 297--320

[12] u{S}mulian V. {On some geometrical properties of the unit sphere in the space of the type (B).} Mat Sbornik, 1939, 6: 77--94

[13] Borwein J M, Zhuang D. {Super efficiency in vector optimization.} Trans Amer Math Soc, 1993, 338: 105--122

[14] Liu J, Song W. {On proper efficiencies in locally convex spaces - a survey.} Acta Math Vietnam, 2001, 26: 301--312

[15] Fu W T, Cheng Y H. {On the super efficiency in locally convex spaces.} Nonlinear Anal, 2001, 44: 821--828

[16] Makarov E K, Rachkovski N N. {Density theorems for generalized Henig proper efficiency.} J Optim Theory Appl, 1996, 91: 419--437

[17] Zheng X Y. The domination property for efficiency in locally convex spaces. J Math Anal Appl, 1997, 213: 455--467