李動(dòng)鋒,邱根勝

(南昌航空大學(xué)應(yīng)用數(shù)學(xué)系,江西南昌 330063)

一種新廣義凸多目標(biāo)分式規(guī)劃的最優(yōu)性充分條件

李動(dòng)鋒,邱根勝

(南昌航空大學(xué)應(yīng)用數(shù)學(xué)系,江西南昌 330063)

提出了(F,α,ρ,θ)-b-凸函數(shù)的概念,它是一類新的廣義凸函數(shù),并給出了這類廣義凸函數(shù)的性質(zhì).在此基礎(chǔ)上,討論了目標(biāo)函數(shù)和約束函數(shù)均為(F,α,ρ,θ)-b-凸函數(shù)的多目標(biāo)分式規(guī)劃,利用廣義K-T條件,得到了這類多目標(biāo)規(guī)劃有效解和弱有效解的幾個(gè)充分條件,推廣了已有文獻(xiàn)的相關(guān)結(jié)果.

多目標(biāo)分式規(guī)劃;廣義凸函數(shù);最優(yōu)性條件

眾所周知,凸性概念在優(yōu)化理論中起著關(guān)鍵性的作用,因此各種凸性概念的拓廣以及相應(yīng)廣義凸性下目標(biāo)規(guī)劃解的最優(yōu)性條件的研究受到了廣泛關(guān)注.2001年,Liang[1]提出(F,α,ρ,d)-凸函數(shù)的概念,并就單目標(biāo)分式規(guī)劃問(wèn)題給出了一些最優(yōu)性充分條件.本文在文[1]的基礎(chǔ)上,給出一類新的廣義凸函數(shù)-(F,α,ρ,θ)-b-凸函數(shù),并在此假設(shè)下討論了多目標(biāo)分式規(guī)劃的最優(yōu)性充分條件.

本文考慮如下的多目標(biāo)分式規(guī)劃問(wèn)題

1 定義與引理

定義1.1給定一個(gè)非空開集X?Rn,函數(shù)F:X×X×Rn→R稱為次線性的,如果對(duì)任意的x1,x2∈X

顯然,對(duì)于次線性函數(shù)F,由(1-2)式可得F(x1,x2;0)=F(x1,x2;0α)=0F(x1,x2;α)=0.

定義1.2設(shè)X?Rn,X為非空開集,函數(shù)F:X×X×Rn→R為次線性的,f:X→R在ˉx處可微,α:X×X→R++,b:X×X→R+,θ:X×X→Rn,ρ∈R

2 最優(yōu)性條件

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[2]Liang Z A,Huang H X,Pardalos P M.Efficiency conditions and duality for a class of multiobjective fractional programming problems[J].Global Optim.,2003,27:447-471.

[3]林銼云,董加禮.多目標(biāo)優(yōu)化的方法與理論[M].遼寧:吉林教育出版社,1992.

[4]邱根勝.廣義凸多目標(biāo)分式規(guī)劃解的充分條件及其對(duì)偶定理[J].南昌航空工業(yè)學(xué)院學(xué)報(bào),2001,15(1):68-73.

[5]劉三明,馮恩民.具有凸的分式規(guī)劃問(wèn)題的最優(yōu)性條件和對(duì)偶性[J].數(shù)學(xué)的實(shí)踐與認(rèn)識(shí),2005,35(12):174-182.

[6]馮芙葉,賈繼紅,聶成.一類非光滑規(guī)劃問(wèn)題的最優(yōu)性和對(duì)偶[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2005,3:246-249.

Optimality conditions for multi-objective fractional programming with a new formulation of generalized convexity

LI Dong-feng,QIU Gen-sheng

(Department of Applied Mathematics,Nanchang Hangkong University,Nanchang330063,China)

In this paper,we present a new formulation of generalized convex function(F,α,ρ,θ)-b-convex function,obtain some properties;and a class of the multi-objective fractional programming is studied,where the involved functions are(F,α,ρ,θ)-b-convex function,using the generalized K-T condition,some sufficient conditions for a efficient solution and a weak efficient solution for the multi-objective fractional programming are obtained,and these results generalize some current results.

multi-objective fractional programming,generalized convex function,optimality condition

O221.6

A

1008-5513(2009)04-0807-09

2007-09-18.

李動(dòng)鋒(1980-),碩士,研究方向：最優(yōu)化理論與算法.

2000MSC:49K99