一類(lèi)非光滑分式優(yōu)化問(wèn)題的最優(yōu)性條件和對(duì)偶

王國(guó)棟1,陳林2

(1.重慶水利電力職業(yè)技術(shù)學(xué)院,重慶402160;2.四川大學(xué)數(shù)學(xué)學(xué)院,成都610065)

研究了一類(lèi)非光滑多目標(biāo)分式優(yōu)化問(wèn)題,利用變分分析和廣義微分中的工具,在新的凸性假設(shè)下,建立了此類(lèi)優(yōu)化問(wèn)題有效解的必要條件和充分條件.這些結(jié)果都是用極限次微分來(lái)刻畫(huà)的,這在非光滑多目標(biāo)分式優(yōu)化問(wèn)題的研究中是一個(gè)比較新的結(jié)果,而對(duì)于極限次微分的研究是近年來(lái)國(guó)內(nèi)外優(yōu)化領(lǐng)域的研究學(xué)者比較關(guān)注的一個(gè)課題.此外,文中第二部分提出了此類(lèi)優(yōu)化問(wèn)題的Mond-Weir對(duì)偶模型,并研究了弱對(duì)偶、強(qiáng)對(duì)偶的結(jié)果.

非光滑;極限次微分;廣義凸;對(duì)偶

1 預(yù)備知識(shí)

本文使用文獻(xiàn)［1］中常用的記號(hào).除非特殊說(shuō)明,所有空間都假設(shè)為Asplund空間,其中的范數(shù)記為‖·‖.空間X和它的對(duì)偶空間X*的對(duì)偶積記為〈·,·〉.集合Ω?X的拓?fù)溟]包和拓?fù)鋬?nèi)部分別記為clΩ和intΩ.記是Rm的非負(fù)卦限.

定義1［1］稱(chēng)Banach空間X是Asp lund空間,如果X的可分子空間具有可分對(duì)偶空間.

定義2［2］稱(chēng)函數(shù)f:X→R在附近是局部Lipschitz的,如果存在的鄰域U及正常數(shù)K,使得對(duì)任意x1,x2∈U都有|f(x1)-f(x2)|≤K‖x1-x2‖,其中常數(shù)K稱(chēng)為f在附近的Lipschitz常數(shù).如果f在X中每一點(diǎn)都是局部Lipschitz的,則稱(chēng)f在X上是(全局)Lipschitz的.

定義3［3］集合Ω?X的極錐定義為Ω°:={x*∈X*|〈x*,x〉≤0,?x∈Ω}.

定義4［1］給定集值映射F:X→2X*,記

定義5［1］稱(chēng)集合Ω?X在附近是閉的,如果存在的領(lǐng)域U使得Ω∩cl U是閉的.稱(chēng)集合Ω?X是局部閉的,如果Ω在每一點(diǎn)x∈Ω附近都是閉的.

定義6［1］Ω在x∈Ω處的Frechet法錐定義為

2 一類(lèi)非光滑分式優(yōu)化問(wèn)題的最優(yōu)性條件

本文考慮如下半無(wú)限多目標(biāo)分式規(guī)劃問(wèn)題

定理得證.

定義12［6］設(shè)f:=(f1,f2,···,fm),g:=(g1,g2,···,gm),hT:=(ht)t∈T.

(i)稱(chēng)(f,-g,hT)在x*∈Ω處是廣義凸的,如果對(duì)任意x∈Ω,ξi∈?fi(x*),i∈M, ηj∈［?gj(x*)∪?(-gj)(x*)］,j∈M,γt∈?ht(x*),t∈T,存在v∈N(x*;Ω)°,使得

(ii)稱(chēng)(f,-g,hT)在x*∈Ω處是嚴(yán)格廣義凸的,如果對(duì)任意x∈Ω{x*},ξi∈?fi(x*), i∈M,ηj∈［?gj(x*)∪?(-gj)(x*)］,j∈M,γt∈?ht(x*),t∈T,存在v∈N(x*;Ω)°,使得

注:按定義12定義的廣義凸函數(shù)是經(jīng)典凸函數(shù)的真推廣,參見(jiàn)文獻(xiàn)［6］中例3.2.

定理2設(shè)x*∈C,并且滿(mǎn)足(1)式.

(i)如果(f,-g,hT)在x*∈Ω處是廣義凸的,則x*∈Sw(M F).

(ii)如果(f,-g,hT)在x*∈Ω處是嚴(yán)格廣義凸的,則x*∈S(M F).

3 相應(yīng)的對(duì)偶結(jié)果

本節(jié)考慮上一節(jié)中半無(wú)限多目標(biāo)分式規(guī)劃問(wèn)題(M F)的M ond-Weir型對(duì)偶問(wèn)題(D),得到了相應(yīng)的弱對(duì)偶定理和強(qiáng)對(duì)偶定理.

定理3(弱對(duì)偶定理)設(shè)x∈C,(z,α,λ)∈F.

(i)如果(f,-g,hT)在z∈Ω處是廣義凸的,則下列結(jié)果不可能成立:

(ii)如果(f,-g,hT)在z∈Ω處是嚴(yán)格廣義凸的,則下列結(jié)果不可能成立:

4 結(jié)論

本文給出了一類(lèi)非光滑多目標(biāo)分式優(yōu)化問(wèn)題的最優(yōu)性條件和對(duì)偶結(jié)果,主要工具是極限次微分,這在非光滑半無(wú)限多目標(biāo)分式優(yōu)化問(wèn)題的研究中是一個(gè)比較新的結(jié)果.

［1］MORDUKHOVICH B S. Variational Analysis and Generalized Differentiation I: Basic Theory ［M］. Berlin:Springer, 2006.

［2］CLARKE F H. Optimization and Nonsmooth Analysis ［M］. New York: Wiley-Interscience, 1983.

［3］ROCKAFELLAR R T. Convex Analysis ［M］. princeton: princeton University press, 1970.

［4］CHEN G Y,HUANG X X,YANG X Q.Vector Optimization:Set-Valued and Variational Analysis［M］.Berlin: Sp ringer-Verlag,2005.

［5］SOGHORA N. Optimatlity and duality for nonsmooth multiobjective fractional programming with mixed constraints［J］. J Glob Optim, 2008, 41: 103-115.

［6］CHUONG T D, KIM D S. Nonsmooth semi-infinite multiobjective optimization problems ［J］. J Optim TheoryAppl, 2014, 160: 748-762.

［7］CHUONG T D, KIM D S. Optimality conditions and duality in nonsmooth multiobjective optimization problems［J］. Annals of Operations Research, 2014, 217(1): 117-136.

(責(zé)任編輯林磊)

Optimality conditions and duality for a class of non-smooth fractional optimization problems

WANG Guo-dong1,CHEN Lin2
(1.Chongqing Water Resources and Electric Engineering College,Chongqing 402160,China; 2.College of Mathematics,Sichuan University,Chengdu 610065,China)

This paper studies a class of non-smooth multi-objective fractional optimization problems,using the tools in variational analysis and the generalized differential,and establishes necessary conditions and sufficient conditions under some new convexity.These results,which are relatively new in the study of non-smooth multi-objective fractional optimization problems,are characterized by limiting subdifferential.And the study of limiting subdifferential is a pretty hot subject in recent years.In addition,the weak duality and the strong duality results have been obtained in Mond-Weir type duality.

non-smooth;limiting subdifferential;generalized convexity;duality

O221.2

A

10.3969/j.issn.1000-5641.2016.01.006

1000-5641(2016)01-0043-08

2014-11

王國(guó)棟,男,碩士,講師,研究方向?yàn)閮?yōu)化理論及應(yīng)用.E-mail:wangguodong love@163.com.

第二作者:陳林,男,博士研究生,研究方向?yàn)閮?yōu)化理論及應(yīng)用.E-mail:chinaallenchen@126.com.