鄔月月，胡艷紅

(哈爾濱師范大學(xué))

無(wú)限維空間中強(qiáng)對(duì)偶定理在潤(rùn)滑問(wèn)題上的應(yīng)用*

鄔月月，胡艷紅

(哈爾濱師范大學(xué))

研究在無(wú)限維空間中，強(qiáng)對(duì)偶定理在潤(rùn)滑問(wèn)題上的應(yīng)用，并找到了潤(rùn)滑問(wèn)題的對(duì)偶問(wèn)題的無(wú)限維lagrange乘子.

強(qiáng)對(duì)偶定理;潤(rùn)滑問(wèn)題;lagrange乘子

0 引言

該文主要研究的是介于無(wú)限維凸優(yōu)化問(wèn)題和它的lagrange對(duì)偶問(wèn)題之間的強(qiáng)對(duì)偶定理及其它的應(yīng)用.文獻(xiàn)[1]中，作者通過(guò)假設(shè) 給出無(wú)限維空間中凸優(yōu)化問(wèn)題的強(qiáng)對(duì)偶定理，并把它應(yīng)用到雙障礙問(wèn)題上.在文獻(xiàn)[2]中，作者研究了強(qiáng)對(duì)偶定理在彈縮扭轉(zhuǎn)問(wèn)題上的應(yīng)用.以上作者是把這些實(shí)際問(wèn)題轉(zhuǎn)換成變分不等式，進(jìn)而轉(zhuǎn)化為無(wú)限維凸優(yōu)化問(wèn)題，再應(yīng)用強(qiáng)對(duì)偶定理找到研究問(wèn)題的對(duì)偶問(wèn)題的無(wú)限維lagrange乘子.

筆者研究的問(wèn)題是一個(gè)完整軸頸軸承的潤(rùn)滑劑薄膜的壓強(qiáng)分配問(wèn)題，所以說(shuō)這是一個(gè)力學(xué)問(wèn)題中的潤(rùn)滑問(wèn)題[3].主要結(jié)果是把強(qiáng)對(duì)偶定理應(yīng)用到這個(gè)力學(xué)問(wèn)題上.通過(guò)把原問(wèn)題轉(zhuǎn)變成變分不等式，再把強(qiáng)對(duì)偶定理應(yīng)用到變分不等式上，同時(shí)找到了原問(wèn)題的對(duì)偶問(wèn)題的無(wú)限維lagrange乘子.

1 預(yù)備知識(shí)

定義1[1]X是線性拓?fù)淇臻g,Y是由賦序錐C定義的實(shí)賦范空間,Z是實(shí)賦范空間,S是X的一個(gè)凸子集.f是S到X的給定泛函，g是S到Y(jié)的映射，考慮的約束集合是K={x∈S:g(x)∈-C}.

該文研究的優(yōu)化問(wèn)題為：找到x0∈K,使得

(1)

通常它的Lagrange對(duì)偶問(wèn)題為：

(2)

定義3[2]假設(shè)S在一點(diǎn)x0∈K滿足是指

推論5[2]如果問(wèn)題(1)與問(wèn)題(2)之間的強(qiáng)對(duì)偶成立，那么假設(shè)S也是滿足的.

2 主要內(nèi)容

文獻(xiàn)[4]中的潤(rùn)滑問(wèn)題可以寫成以下變分不等式

(3)

ν(0,z)=ν(2π,z),|z|

ν(b,θ)=ν(-b,θ)=0,0≤θ≤2π.

Ω={(θ,z):0<θ<2π,|z|

這里提到的ω,η,ε都是文獻(xiàn)[3]中潤(rùn)滑問(wèn)題提到的變量.

x1=(a+b+z)cosθ,x2=(a+b+z)sinθ,令x=(x1,x2)

則O={x∈R2:a<|x|

問(wèn)題(3)可等價(jià)的表示為如下變分不等式問(wèn)題：

(4)

(5)

(6)

下面給出該文研究的主要結(jié)果：

(7)

(-φ)dx≥0.

另一個(gè)方向，要證假設(shè)S成立，下面讓(7)式成立，所以有

故假設(shè)S成立，綜上所述，定理得到證明.

[1] Evans L C. The infinite dimensional Lagrange multiplier rule for convex optimization problems[J].Journal of Functional Analysis,2011(261):2083-2093.

[2] Daniele P, Maugeri A,Raciti F.Duality Theory and Applications to Unilateral Problems[J],J Optim Theory Appl,2014,162:718-734.

[3] David Kinderlehrer,Guido Stampacchia.An Introduction to Variational Inequalities and Their Applications. Springer in Applied Mathematics, New York, 1980.

[4] Mauger A,Puglisi D. A new necessary and sufficient condition for the strong duality and the infinite dimensional Lagrange multiplier rule[J]. J Math Anal Appl, 2014,415:661-676.

(責(zé)任編輯：季春陽(yáng))

Strong Duality Theory on Application of Lubrication Problem in Infinite Dimensional Space

Wu Yueyue, Hu Yanhong

(Harbin Normal University)

In this paper, concerned with the problem in infinite dimensional space, strong duality theory of the lubrication problem are applied and the dual problem of the lubrication problem and its infinite dimensional lagrange multiplier are found.

Strong duality theory; The lubrication problem; Lagrange multiplier

2016-12-23

*黑龍江省教育廳項(xiàng)目(12521147)

O189

A

1000-5617(2016)05-0013-03