二階漸近切上圖導(dǎo)數(shù)及應(yīng)用

張愛紅,徐義紅,涂相求

(南昌大學(xué) 數(shù)學(xué)系,南昌 330031)

(,)∈graph F(,,,)=(epi F,(,),(,)).(7)

(F(Q1∩U1)-)∩(-intcone(Θ+V0))≠?.

(F(Q1∩U1)-)∩(intcone(V0-Θ))≠?.

二階漸近切上圖導(dǎo)數(shù)及應(yīng)用

張愛紅,徐義紅,涂相求

(南昌大學(xué) 數(shù)學(xué)系,南昌 330031)

在實(shí)賦范線性空間中利用新定義的二階漸近切上圖導(dǎo)數(shù)研究集值優(yōu)化問(wèn)題的嚴(yán)有效性.通過(guò)二階漸近切錐引進(jìn)一種新的二階漸近切上圖導(dǎo)數(shù),給出一個(gè)例子說(shuō)明它的存在條件比二階漸近切導(dǎo)數(shù)存在條件更弱,并利用此導(dǎo)數(shù)及擴(kuò)張錐的性質(zhì)給出了集值優(yōu)化問(wèn)題取得局部嚴(yán)有效元的必要條件.

二階漸近切錐; 二階漸近切上圖導(dǎo)數(shù); 局部嚴(yán)有效元

在集值優(yōu)化理論研究中,有效性理論一直是人們關(guān)注的課題[1-3].由于(弱)有效解范圍較大,因此收縮解的范圍成為向量?jī)?yōu)化研究的主要內(nèi)容.傅萬(wàn)濤等[4-6]提出了嚴(yán)有效點(diǎn)的概念,每個(gè)嚴(yán)有效點(diǎn)都能用嚴(yán)格正泛函標(biāo)量化,因而嚴(yán)有效點(diǎn)引起了人們的廣泛關(guān)注[7-8].

目前,借助切錐定義的切導(dǎo)數(shù)廣泛用于研究集值優(yōu)化問(wèn)題的最優(yōu)性條件[9-12].Li等[9]借助高階切導(dǎo)數(shù)研究了當(dāng)目標(biāo)函數(shù)和約束函數(shù)均為錐凹時(shí),集值優(yōu)化問(wèn)題取得弱最大有效解的充分和必要條件; Khan等[10]利用二階漸近切錐定義了二階漸近導(dǎo)數(shù)并給出了集值優(yōu)化問(wèn)題取得弱有效解的最優(yōu)化條件.本文引進(jìn)一種新的二階切上圖導(dǎo)數(shù),并給出其應(yīng)用.

1 預(yù)備知識(shí)

設(shè)X,Y,Z為實(shí)賦范線性空間,0X,0Y,0Z分別表示X,Y,Z中的零元.設(shè)S是Y中的非空子集,分別用intS和clS表示集合S的內(nèi)部和閉包.集合S的生成錐定義為coneS={λs:λ≥0,s∈S}.設(shè)C和D分別為Y和Z中的閉凸點(diǎn)錐,若0Y?clΘ且C=coneΘ， 稱凸集Θ為錐C的一個(gè)基.

設(shè)P∶={t∈|t>0|},為正整數(shù)集.

設(shè)F:X→2Y表示從X到Y(jié)的集值映射.F的有效域、 圖及上圖分別定義為

F的剖面映射(profile map)F+:X→2Y定義為

F+(x)∶=F(x)+C, ?x∈domF.

F關(guān)于S的弱逆像(weak-inverse image)F[S]-定義為

F[S]-∶={x∈X|F(x)∩S≠?|}.

下面引進(jìn)一種新的二階切導(dǎo)數(shù).

例1設(shè)X=Y=,(0,1).

直接計(jì)算得:

(1)

設(shè)F:X→2Y,G:X→2Z,考慮集值優(yōu)化問(wèn)題

F(x1)∩V?F(x2)+L‖x1-x2‖BY

2 局部嚴(yán)有效元的最優(yōu)性條件

證明: 反證法.假設(shè)

{(sn,tn)}?P×P, (sn,tn)→(0+,0+),sn/tn→0,

使得

由

得

設(shè)

由式(7)得

于是

令

由式(9)得

由式(8),(11),(12)得

于是存在bn∈BY,使得

從而

由式(14)知,存在wn∈F(un),使得

于是

由C?cone(Θ+V0)得

又由cone(Θ+V0)是凸錐得

cone(Θ+V0)+intcone(Θ+V0)?intcone(Θ+V0),

再由式(17)得

C+intcone(Θ+V0)?intcone(Θ+V0).

由式(16)得

于是

由式(4),(5)得

由式(10)得

由式(19)～(21)得

于是

又由V0=-V0得

下證0?intcone(V0-Θ).

由式(22),(23)得

另一方面,

由式(24)及λ2+λ3>0得

再由式(25)得0∈V0-Θ,與式(2)矛盾,故0?intcone(V0-Θ).

取

與式(2)矛盾.故式(3)成立.證畢.

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(責(zé)任編輯： 趙立芹)

Second-OrderAsymptoticEpiderivativeandApplication

ZHANG Aihong,XU Yihong,TU Xiangqiu
(DepartmentofMathematics,NanchangUniversity,Nanchang330031,China)

The strict efficiency of set-valued optimization was considered in real normed linear space by a new second-order asymptotic epiderivative.With the help of second-order asymptotic tangent cone,a new second-order asymptotic epiderivative was introduced.At the same time,an example was given to show that its existence condition is weaker than that of second-order asymptotic tangent derivative.By applying the derivative and properties of a dilating cone,an optimality necessary condition of locally strictly efficient element for set-valued optimization was established.

second-order asymptotic tangent cone; second-order asymptotic epiderivative; locally strictly efficient element

2013-12-20.

張愛紅(1988—),女,漢族,碩士研究生,從事多目標(biāo)規(guī)劃的研究,E-mail: 498863904@qq.com.通信作者: 徐義紅(1969—),男,漢族,博士,教授,從事多目標(biāo)規(guī)劃的研究,E-mail: xuyihong@ncu.edu.cn.

國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào): 61175127)、 江西省自然科學(xué)基金(批準(zhǔn)號(hào): 20122BAB201003)和江西省教育廳科技項(xiàng)目(批準(zhǔn)號(hào): GJJ12010).

O221.6

A

1671-5489(2014)05-0943-06