廣義強(qiáng)非線性擬變分不等式組的迭代算法

佘珺彤，　夏福全

(四川師范大學(xué) 數(shù)學(xué)與軟件科學(xué)學(xué)院, 四川 成都 610066)

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廣義強(qiáng)非線性擬變分不等式組的迭代算法

佘珺彤，夏福全*

(四川師范大學(xué) 數(shù)學(xué)與軟件科學(xué)學(xué)院, 四川 成都 610066)

摘要：研究一類新的廣義強(qiáng)非線性擬變分不等式組解的存在性及算法.首先建立廣義強(qiáng)非線性擬變分不等式組與不動(dòng)點(diǎn)問(wèn)題的等價(jià)關(guān)系.利用這一等價(jià)關(guān)系討論廣義強(qiáng)非線性擬變分不等式組解的存在性與唯一性.然后給出一個(gè)含有誤差的投影迭代算法.最后證明了該算法產(chǎn)生的迭代序列收斂到廣義強(qiáng)非線性擬變分不等式組的唯一解.

關(guān)鍵詞：廣義強(qiáng)非線性擬變分不等式組; 強(qiáng)單調(diào); Lipschitz條件; 投影動(dòng)態(tài)系統(tǒng)

變分不等式理論,早在20世紀(jì)60年代就已經(jīng)出現(xiàn)[1],并在最優(yōu)控制、非線性規(guī)劃、對(duì)策理論、物理學(xué)、經(jīng)濟(jì)與工程學(xué)等眾多領(lǐng)域有著重要應(yīng)用.因此許多專家學(xué)者對(duì)變分不等式問(wèn)題作了深入的研究,獲得了豐富的結(jié)果[2-4]等.同時(shí),R. U. Verma[4]首先在變分不等式問(wèn)題的基礎(chǔ)上介紹并研究了變分不等式組并且利用投影算法尋找變分不等式組的近似解.在這之后文獻(xiàn)[5-7]對(duì)變分不等式組問(wèn)題作了推廣.而擬變分不等式問(wèn)題出現(xiàn)的時(shí)間相對(duì)較晚,它在納什博弈及運(yùn)輸網(wǎng)絡(luò)平衡等問(wèn)題上有著廣泛的應(yīng)用,參見文獻(xiàn)[8-9]及其參考文獻(xiàn).此外,擬變分不等式問(wèn)題的結(jié)果也可以應(yīng)用到一些經(jīng)濟(jì)與商業(yè)模型中[10].但相較于更為成熟的變分不等式問(wèn)題,擬變分不等式問(wèn)題在理論分析與算法應(yīng)用上得到的結(jié)果都是有限的,擬變分不等式組的結(jié)果就更少了.因而近年來(lái),擬變分不等式以及擬變分不等式組問(wèn)題引起了越來(lái)越多的專家學(xué)者的關(guān)注和研究.

由于投影動(dòng)態(tài)系統(tǒng)有良好的分析性質(zhì)，并且其的穩(wěn)定點(diǎn)集合與變分不等式問(wèn)題的解集相一致，因此，在近幾年,人們考慮應(yīng)用投影動(dòng)態(tài)系統(tǒng)來(lái)分析解決變分不等式問(wèn)題及其相關(guān)問(wèn)題,獲得了豐碩的成果,參見文獻(xiàn)[11-15]等.

本文主要利用投影動(dòng)態(tài)系統(tǒng)研究了一類新的廣義強(qiáng)非線性擬變分不等式組解的存在性及算法.首先建立了廣義強(qiáng)非線性擬變分不等式組與不動(dòng)點(diǎn)問(wèn)題的等價(jià)關(guān)系.通過(guò)這一等價(jià)關(guān)系討論了廣義強(qiáng)非線性擬變分不等式解的存在性與唯一性.然后給出了一個(gè)含有誤差的投影迭代算法.最后證明了該算法產(chǎn)生的迭代序列收斂到廣義強(qiáng)非線性擬變分不等式組的唯一解.

本文假設(shè)H為Hilbert空間,它的內(nèi)積和范數(shù)分別記為〈·,·〉和‖·‖.設(shè)Ti:H→H,Ai:H→H,hi:H→H和gi:H→H(i=1,2)都是單值映射.Si:H→2H(i=1,2)是一個(gè)非空閉凸集值映射.對(duì)于給定的常數(shù)ρ,η>0.考慮如下問(wèn)題:

求x*,y*∈H,使得(h1(x*),h2(y*))∈S1(x*)×S2(y*),并且

上述問(wèn)題稱為廣義強(qiáng)非線性擬變分不等式組,簡(jiǎn)稱SGQVI.所研究的問(wèn)題包含以下問(wèn)題為特例：

當(dāng)Ai=0(i=1,2)時(shí),SGQVI退化成一類新的廣義擬變分不等式組,簡(jiǎn)記為SQVI:求x*,y*∈H,使得(h1(x*),h2(y*))∈S1(x*)×S2(y*),并且

(2)

當(dāng)T1=T2=T,h1=h2=h,g1=g2=g,ρ=η時(shí)SQVI退化成Q. H. Ansari等在文獻(xiàn)[16]中研究的一類變分不等式問(wèn)題:求x*∈H,使得h(x*)∈S(x*),且

設(shè)K是H中的非空閉凸子集,當(dāng)hi=gi=I(i=1,2),w=v=x,S1(x*)=S2(y*)=K時(shí),SQVI退化成S. S. Chang等在文獻(xiàn)[5]中研究的一類變分不等式組問(wèn)題:求x*,y*∈K,使得

(3)

1預(yù)備知識(shí)

為了研究廣義強(qiáng)非線性擬變分不等式組問(wèn)題(1)解的存在性和迭代算法以及其算法的收斂性,首先介紹一些有用的定義及定理.

定義 1.1設(shè)T:H→H是單值映射,稱T是:

(a) 單調(diào)映射,如果對(duì)?x,y∈H有

(b)ξ-強(qiáng)單調(diào)映射,如果存在一個(gè)常數(shù)ξ>0使得?x,y∈H有

(c)μ-Lipschitz連續(xù)映射,如果存在一個(gè)常數(shù)μ>0使得?x,y∈H有

引理 1.1[17]若對(duì)任意x∈H,S(x)都是H的一個(gè)非空閉凸集,則對(duì)一個(gè)給定的z∈H,u∈S(x)滿足

的充分必要條件是u=PS(x)(z),其中PS(x)是H在非空閉凸集S(x)上的投影.進(jìn)一步,知道投影算子PS(x)(·)是非擴(kuò)張的,即

引理 1.2[18]令{an}、{bn}和{cn}是3個(gè)非負(fù)實(shí)數(shù)序列,且存在一個(gè)自然數(shù)n0使得

2廣義強(qiáng)非線性擬變分不等式組解的存在性與唯一性

定理 2.1設(shè)Ti:H→H,Ai:H→H,hi:H→H和gi:H→H(i=1,2)都是單值映射.Si:H→2H(i=1,2)是一個(gè)非空閉凸集值映射,ρ,η>0為給定的常數(shù),則x*,y*∈H是廣義強(qiáng)非線性擬變分不等式組(1)的解,當(dāng)且僅當(dāng)

(4)

證明令x*,y*∈H是廣義強(qiáng)非線性擬變分不等式組SGQVI的一個(gè)解,使得(h1(x*),h2(y*))∈S1(x*)×S2(y*),并且

由引理1.1,可知(5)式等價(jià)于

應(yīng)用上述引理,證明廣義強(qiáng)非線性擬變分不等式組(1)解的唯一存在性.

定理 2.2設(shè)Si:H→2H(i=1,2)是2個(gè)非空閉凸集值映射.對(duì)任意i=1,2,Ti:H→H是ξi-強(qiáng)單調(diào)且μi-Lipschitz連續(xù)映射,gi:H→H是ζgi-強(qiáng)單調(diào)且σgi-Lipschitz連續(xù)映射,hi:H→H是ζhi-強(qiáng)單調(diào)且σhi-Lipschitz連續(xù)映射Ai:H→H是θi-Lipschitz連續(xù)映射,并且存在一個(gè)常數(shù)τ>0,使得

如果存在常數(shù)ρ>0,η>0,對(duì)所有i=1,2都有

(6)

則廣義強(qiáng)非線性擬變分不等式組SGQVI有唯一解.

證明對(duì)?x,y∈H使得h1(x)∈S1(x),h2(y)∈S2(y),定義映射Ψ,Φ:H×H→H如下:

(7)

(8)

在H×H中定義‖·‖*如下：

易知(H×H,‖·‖*)是一個(gè)Banach空間.

再定義映射F:H×H→H×H如下:

(9)

(10)

因?yàn)閔1是ζh1-強(qiáng)單調(diào)且σh1-Lipschitz連續(xù),則有

(11)

因?yàn)間1是ζg1-強(qiáng)單調(diào)且σg1-Lipschitz連續(xù),則有

(12)

因?yàn)門1是ξ1-強(qiáng)單調(diào)且μ1-Lipschitz連續(xù),則有

(13)

根據(jù)A1是θ1-Lipschitz連續(xù)映射可得

(14)

由(10)～(14)式可得

(15)

其中

同理可得

(16)

其中

由(9)、(15)～(16)式可得

(17)

這里δ=max{κ1+λ2,κ2+λ1}.由條件(6)可得0≤δ<1.并且,由(17)可知F是一個(gè)壓縮映射.根據(jù)Banach不動(dòng)點(diǎn)定理可知,存在唯一的點(diǎn)(x*,y*)∈H×H,滿足(h1(x*),h2(y*))∈S1(x*)×S2(y*),使得F(x*,y*)=(x*,y*).由(7)～(9)式可推出

又由定理3.1.可知x*,y*∈H,使得(h1(x*),h2(y*))∈S1(x*)×S2(y*)是擬變分不等式組(1)的一個(gè)解.

注正如M. A. Noor[19]所說(shuō),若S(x)=m(x)+K,這里m(x):H→H是一個(gè)單值映射,K為H中的非空閉凸子集,則

再設(shè)m(x):H→H是γ-Lipschitz連續(xù),易知

(18)

從而,定理2.2中假設(shè)存在一個(gè)常數(shù)τ>0,使得

?x,y,z∈H

的條件成立.

3一個(gè)新的投影迭代算法及其收斂性

在這一部分,介紹SGQVI的一個(gè)含有誤差的投影迭代算法,并且證明了這一算法產(chǎn)生的迭代序列最終收斂到廣義強(qiáng)非線性擬變分不等式組SGQVI的解.

現(xiàn)將廣義強(qiáng)非線性擬變分不等式組的解的配對(duì)(x*,y*)所組成的集合記為sol(SGQIV).令Ti,Ai,hi,gi:H→H(i=1,2)都為單值映射,Si:H→2H(i=1,2)為集值映射,ρ>0,η>0為給定常數(shù).若(x*,y*)∈sol(SGQIV),則由定理2.1與定理2.2,可得(h1(x*),h2(y*))∈S1(x*)×S2(y*),且

由(19)式給出下列含有誤差的投影迭代算法.

易知,這一含有誤差的投影迭代算法包含了一般的投影迭代算法，即如下所述的算法2.

如果Ai=0(i=1,2),則由算法1得到廣義擬變分不等式組(2)的含有誤差的投影迭代算法.

證明由定理2.1和定理2.2可知,廣義強(qiáng)非線性擬變分不等式有唯一的解x*,y*∈H.因而sol(SGQIV)是單點(diǎn)集.可以推出

這里序列{αn}與{βn}滿足算法1中的假設(shè)條件.令

根據(jù)算法1中的假設(shè)，易知Γ≤0有界.由(20)、(24)式可知

(25)

因?yàn)門1:H→H是ξ1-強(qiáng)單調(diào)且μ1-Lipschitz連續(xù)映射,g1:H→H是ζg1-強(qiáng)單調(diào)且σg1-Lipschitz連續(xù)映射,h1:H→H是ζh1-強(qiáng)單調(diào)且σh1-Lipschitz連續(xù)映射,A1:H→H是θ1-Lipschitz連續(xù)映射.根據(jù)(11)～(14)式相同的證明方法可得

(26)

(27)

(28)

因此,由(26)～(29)式可得

(30)

同理,可得

(31)

這里κi、λi與(15)～(16)式中相同,由(30)～(31)式可得

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2010 MSC：47H05; 47J20; 49J40; 90C33

(編輯周俊)

Iterative Algorithms for a System of Extended General Strongly Nonlinear Quasi-Variational Inequalities

SHE Juntong,XIA Fuquan

(CollegeofMathematicsandSoftwareScience,SichuanNormalUniversity,Chengdu610066,Sichuan)

Abstract：In this paper, we introduce and study a new system of extended general strongly nonlinear quasi-variational inequalities. First, we establish the equivalences between this system and fixed point problems. By using these equivalences, we discuss the existence and uniqueness of solution to the system. And then, we define a new projection iterative algorithm with mixed errors for finding the unique solution of the system. Finally, we prove the convergence of the suggested iterative algorithm.

Key words：system of quasi-variational inequalities; strongly monotone; Lipschitz conditions; projection dynamical systems

doi:10.3969/j.issn.1001-8395.2016.01.002

中圖分類號(hào)：O176.3; O178

文獻(xiàn)標(biāo)志碼：A

文章編號(hào)：1001-8395(2016)01-0008-07

*通信作者簡(jiǎn)介：夏福全(1973—),男,教授,主要從事優(yōu)化理論及應(yīng)用的研究,E-mail:fuquanxia@sina.com

基金項(xiàng)目：教育部科學(xué)技術(shù)重點(diǎn)項(xiàng)目(212147)

收稿日期:2015-02-06