陳喬

(重慶師范大學(xué)生命科學(xué)學(xué)院,重慶 401331)

嚴(yán)格r-預(yù)不變凸函數(shù)

陳喬

(重慶師范大學(xué)生命科學(xué)學(xué)院,重慶 401331)

引入了嚴(yán)格r-預(yù)不變凸函數(shù)的概念,并證明了嚴(yán)格r-預(yù)不變凸函數(shù)與r-預(yù)不變凸函數(shù)有關(guān)的一個(gè)充分條件.同時(shí),討論得到了嚴(yán)格r-預(yù)不變凸性和半嚴(yán)格r-預(yù)不變凸性的等價(jià)條件.

r-預(yù)不變凸函數(shù);嚴(yán)格r-預(yù)不變凸函數(shù);半嚴(yán)格r-預(yù)不變凸函數(shù);

1 引言

在數(shù)學(xué)規(guī)劃、最優(yōu)化等領(lǐng)域中,凸性及廣義凸性起著十分重要的作用.因此,對(duì)凸性及廣義凸性的研究具有十分重要的意義.1999年,文獻(xiàn)[1]引入了r-不變凸函數(shù)的定義并討論了它的一些性質(zhì)特征.2005年,文獻(xiàn)[2]在預(yù)不變凸性和r-凸性的基礎(chǔ)上,結(jié)合r-不變凸函數(shù)的定義給出了r-(嚴(yán)格)預(yù)不變凸函數(shù)的定義.此外,文獻(xiàn)[3]引入了(半)嚴(yán)格預(yù)不變凸函數(shù)的定義,研究了預(yù)不變凸函數(shù)和半嚴(yán)格預(yù)不變凸函數(shù)的聯(lián)系.并在文獻(xiàn)[4-5]中討論了凸函數(shù),半嚴(yán)格凸函數(shù),嚴(yán)格凸函數(shù)間的關(guān)系.文獻(xiàn)[6]把預(yù)不變凸函數(shù)的性質(zhì)運(yùn)用到了多目標(biāo)優(yōu)化問題中.文獻(xiàn)[7-8]研究了廣義凸不等式的一些性質(zhì).

本文進(jìn)一步研究了r-預(yù)不變凸函數(shù),r-嚴(yán)格預(yù)不變凸函數(shù),r-半嚴(yán)格預(yù)不變凸函數(shù),得到了關(guān)于嚴(yán)格r-預(yù)不變凸函數(shù)的一些性質(zhì).

2 相關(guān)定義及例子

3 嚴(yán)格r-預(yù)不變凸函數(shù)的性質(zhì)

[1]Atczakt T.(p,r)-Invex Sets and Functions[M].Lodz:University of Lodz,1999.

[2]Antczak T.r-preinvexity and r-invexity in mathematical programming[J].Computers and Mathematics with Applications,2005,50:551-556.

[3]Yang X M,Li D.Semi-strictly preinvex functions[J].Journal of Mathematical Analysis and Applications, 2001,258:287-308.

[4]Yang X M,Li D.On properties of preinvex functions[J].Journal of Mathematical Analysis and Applications, 2001,256:229-241.

[5]Yang X M.Semi-strictly convex functions[J].Opsearch,1994,31(1):15-27.

[6]Weir T,Bond B.Preinvex functions in multiple objective optimization[J].Journal of Mathematical Analysis and Applications,1988,136:29-38.

[7]宋振云,涂瓊霞.關(guān)于P-凸函數(shù)的Hadamard型不等式[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2011,27(3):313-317.

[8]宋振云.關(guān)于P-凸函數(shù)的積分型Jensen不等式[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2012,28(1):36-40.

[9]Avriel M.r-convex functions[J].Mathematical Programming,1972,2:309-323.

The properties of strictly r-preinvex functions

Chen Qiao

(College of Life Sciences,Chongqing Normal University,Chongqing401331,China)

In this paper,we introduce a new conception of the strictly r-preinvex function and prove some related properties.These properties include a sufcient condition in terms of strict r-preinvexity and r-preinvexity.A new equivalent condition between strict r-preinvexity and semistrict r-preinvexity are also presented.

r-preinvex functions,strictly r-preinvex functions,semistrictly r-preinvex functions, strictly r-convex functions

O224

A

1008-5513(2013)06-0621-06

10.3969/j.issn.1008-5513.2013.06.011

2013-01-18.

國(guó)家自然科學(xué)基金(11201240).

陳喬(1981-),碩士,助理研究員,研究方向：運(yùn)籌學(xué)與控制論.

嚴(yán)格r-凸函數(shù)

2010 MSC:90C25