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  数学物理学报  2015, Vol. 35 Issue (4): 824-832   PDF (345 KB)    
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张运胜
高雷阜
对称锥互补问题的一种非精确光滑牛顿算法
张运胜, 高雷阜     
辽宁工程技术大学理学院, 辽宁 阜新 123000
摘要: 基于一个光滑函数, 就单调对称锥互补问题, 给出了一种解决高维对称锥互补问题的非精确光滑牛顿算法.在适当条件下, 证明了该算法具有全局收敛性和局部二次收敛性.数值试验证实了算法对大规 模对称锥互补问题的可行性和有效性.
关键词: 对称锥互补问题     非精确光滑牛顿法     大规模问题    
A Smoothing Inexact Newton Method for Symmetric Cone Complementarity Problems
Zhang Yunsheng, Gao Leifu    
College of Science, Liaoning Technical University, Liaoning Fuxin 123000
Abstract: Based on a smoothing function, an inexact smoothing Newton algorithm for large-scale symmetric cone complementarity problems is proposed. The algorithm is proved to be globally as well as locally quadratic convergence under proper conditions. Numerical experiments demonstrate that the algorithm is effective and feasible for large-scale problems.
Key words: Symmetric cone complementarity problems     Inexact smoothing Newton algorithm     Large-scale problems    

1 引言

考虑对称锥互补问题(SCCP),寻找x,yJ,使得

xK,y=F(x)K,x,y=0, (1)
其中J是赋有内积 的有限维实向量空间,KJ上的对称锥, F(x)JJ的一个连续映射,问题(1)包含了半定锥互补问题(SDCP)、 二阶锥互补问题(SOCCP)和线性和非线性互补问题(LCP/NCP). 对称锥互补问题具有非常广的理论和实际应用,近年来逐渐引起人们的重视, 对称锥互补问题的算法成为优化领域研究的热点之一,学者们对如何求解对称 锥互补问题作了大量的工作,把线性互补问题(LCP)和非线性互补问题(NCP), 二阶锥互补问题和半定锥互补问题(SDCP)的算法,推广到对称锥互补问题, 光滑牛顿法已经成功地推广到对称锥互补问题[1, 2, 3, 4, 5, 6]. 然而, 光滑牛顿法每次迭代都要求解一次线性方程组,对于规模较大的问题效率不高, 非精确光滑牛顿法是解决这一问题的一种好的选择[7, 8, 9].

本文考虑基于一个光滑函数,提出一个求解SCCP(1)的非精确光滑牛顿法, 数值实验表明,当维数变大非精确光滑牛顿法的效果好于光滑牛顿法.

2 初步知识

简要介绍一下Euclidean-Jordan代数知识,详细的内容请参考文献[10].

欧几里德若当代数(J,,,)是指: (J,,,)是一个定义在实数域上的有限维内积空间, (J,,,)是一个满足下述条件的双线性映射

(i) xy=yx,x,yJ;

(ii) x(x2y)=x2(xy),x,yJ, 其中x2=xx;

(iii) xy,z=y,xz,x,y,zJ.

xy为元素xy的若当积.假设J总存在一个元素e满足xJ, 有xe=ex=x,则将该元素称为单位元. 如果 ee=e0, 称元素eJ为幂等元.如果幂等元ee满足ee=0,称ee是正交的. 如果非零幂等元e不能分解为两个非零幂等元之和,称幂等元e为基本幂等元. 定义J的秩r:=max{deg(x):xJ},,其中deg(x)为使得 {e,x,,xm}为线性相关的最小正整数m.

如果e1,e2,,er相互正交且e1+e2++er=e,则幂等元集合{e1,e2,,er} 称为幂等元正交完备系.如果幂等元正交完备系中的所有幂等元都是基本幂等元, 则称此基本幂等元正交完备系为欧几里德若当代数 的一个若当基底.

定理1 (谱分解类型I [10])设J是一个秩为r的欧几里得若当代数, 则对于每一个元素XJ,必存在一个若当基底{c1,c2,,cr} 和一组实数λ1,λ2,,λr,使得

x=λ1c1++λrcr. (2)
式(2)称为元素x的一个谱分解,实数λi=λi(x)(i=1,,r)称为元素x的特征值,ci(i=1,,r} 称为元素x的相应于特征值λi的特征向量.

平方集合K:={x2:xJ}称为对称锥,即K是一个自对偶的非空凸集. 对任意两元素x,yK,存在一个可逆的线性变换Γ:JJ使得 Lx(y)=xyΓ(x)y.

定义 1 若当积是双线性的,对xJ, 存在线性映射Lx:JJ, 使得: 对yJLx(y)=xy, 称该线性映射为Lyapunov变换,Lx的逆记为L1x.

定义 2 设变换F:JJ,任取x,yJxy,F(x)F(y)0,则称FJ上单调.

定义 3 对于非光滑函数g:RnRm,含有参数μ的函数gμ:RnRm有下列性质

(1) 对于任意的 μ>0,gμ是可微的;

(2) 对于任意的xRn,limμ0gμ(x)=g(x),则称函数gμg 的光滑函数.

3 光滑函数及性质

文献[13]中证明了最小值函数ϕmin:J×JJ ϕmin:(x,y)=x+y(xy)2 满足ϕmin(x,y)=0xK,yK,xy=0, 而最小值函数是非光滑函数. 文献[2, 14]中光滑化极小值函数得到如下的光滑函数

ϕ(μ,x,y):=(cosμ+sinμ)(x+y)(cosμsinμ)2(xy)2+2μ2e. (3)

引理1[2, 14]ϕ:R++×J×JJ 定义如式(3),则

(1) xK,yK,xy=0xK,yK,x,y=0;

(2) ϕ(0,x,y)=ϕmin(x,y)=0xK,yK,xy=0.

定理2ϕ:R++×J×JJ定义如式(1)则

(1) ϕ处处全局Lipschitz连续,强半光滑,并且ϕ在任意z=(μ,x,y)R++×J×J 处连续可微,且其雅克比矩阵为

ϕ(u,x,y)=(ϕμϕxϕy)=((cosμsinμ)(x+y)+L1w[cos2μ(xy)22μe](cosμ+sinμ)I(cosμsinμ)2L1wL(xy)(cosμ+sinμ)I+(cosμsinμ)2L1wL(xy)), (4)
其中
w:=w(μ,x,y)=(cosμsinμ)2(xy)2+2μ2e. (5)

(2) 对于任意(x,y)J×J, 有limμ0ϕ(μ,x,y)=ϕmin(x,y)成立,因此,ϕmin(x,y)ϕ(μ,x,y) 的光滑逼近函数.

(1) 根据文献[11,定理3.2]的推论,可知ϕ处处全局Lipschitz连续,强半光滑, 且在任意z=(μ,x,y)R++×J×J处连续可微.下证式(4)成立,对于任意z=(μ,x,y)R++×J×J, 通过式(5)有 wintK,因此Lw可逆,由w的定义可以得到

w2=(cosμsinμ)2(xy)2+2μ2e, (6)
对式(6)两边同时求导,通过求微分的链式法则得出
w(μ,x,y)=(L1w[cos2μ(xy)22μe](cosμsinμ)2L1wL(xy)(cosμsinμ)L1wL(xy)), (7)
因此得到ϕ的雅克比矩阵.

(2) 由定理1,设x+y=ri=1aiui(xy)2=ri=1βivi因此得到ψ(0,x,y)=ri=1aiuiri=1βivi 所以 ϕ(μ,x,y)=ri=1((cosμ+sinμ)αi)uiri=1(βi(cosμsinμ)2+2μ2)vi=ri=1ai(μ)uiri=1βi(μ)vi.limμ0αi(μ)=αilimμ0βi(μ)=βi得到 limμ0ϕ(μ,x,y)=ψ(0,x,y)成立,因此, ψ(0,x,y)ϕ(μ,x,y)的光滑逼近函数.

3 非精确光滑牛顿算法及其适定性

基于光滑函数(3),给出求解对称锥互补问题的非精确光滑牛顿法. 记z=(μ,x,y)R++×J×J,将对称锥互补问题的非光滑等价方程组光滑为如下光滑方程组

H(z):=(μF(x)yϕ(z))=0. (8)
由互补函数的定义知道对称锥互补问题(1)的解和方程的解是一致的.

算法 3.1 (非精确光滑牛顿法)

初始步:任取{z^0} = \left( {{\mu ^0},{x^0},{y^0}} \right) \in {R_{ + + }} \times J \times J,选取常数\delta \in \left( {0,1} \right),\sigma \in \left( {0,1} \right)选取常数\gamma \in \left( {0,1} \right)使得\gamma {\mu ^0} < 1,另取序列 \left\{ {{\eta _k}} \right\}满足{\eta _k} \in \left[{0,\eta } \right], 这里\eta \in \left[{0{\rm{,1 - }}\gamma {\mu ^0}} \right]为一个常数, k: = 0.

步骤1\quad 若\left\| {H\left( {{z^k}} \right)} \right\| = 0,则终止.

步骤2\quad 由下列式子计算\Delta {z^k} = \left( {\Delta {\mu ^k},\Delta {x^k},\Delta {y^k}} \right)

\begin{equation} H\left( {{z^k}} \right) + \nabla H\left( {{z^k}} \right)\Delta {z^k} = \left( \begin{array}{c} {\beta _k}{\mu _0} \\ 0 \\ {r^k} \end{array} \right), \end{equation} (9)
这里{\beta _k} = \beta \left( {{z^k}} \right) : = \gamma \left\| {H\left( {{z^k}} \right)} \right\|\min \left\{ {1,\left\| {H\left( {{z^k}} \right)} \right\|} \right\} ,{r^k} \in {R^n}满足\left\| {{r^k}} \right\| \le {\eta _k}\left\| {H\left( {{z^k}} \right)} \right\|.

步骤3\quad 让{\lambda _k}1,\delta ,{\delta ^2},\cdots 中使得下式成立的最大值. \|H(z^k+\lambda_k\Delta {z^k})\|\leq [1-\sigma (1-\gamma \mu_0 -\eta_k)\lambda_k]\|H(z^k)\|.

步骤4\quad 置{z^{k + 1}}:= {z^k} + {\lambda _k}\Delta {z^k}, k=k+1,返回步骤1.

{r^k} = 0时就成为精确的牛顿算法.

引理2[3] 对于任意的a \in K,b \in K,u \in {R^n},v \in {R^n}, 如果a \succ 0,b \succ 0,a \circ b \succ 0,〈 {u,v} 〉 \ge 0a \circ u + b \circ v = 0成立,则u = v = 0.

定理3z = \left( {\mu ,x,y} \right) \in {R_{ ++ }} \times J \times J,由式(8)定义,则得到如下结果

(1) H在任意z = \left( {\mu ,x,y} \right) \in {R_{ ++ }} \times J \times J处连续可微,其雅克比矩阵为

\begin{equation} \nabla {H_\tau }\left( z \right):= \left( {\begin{array}{*{20}{c}} 1 & 0 & 0 \\ 0 & {\nabla F\left( x \right)} & { - I} \\ {\left( {{{\phi '}_\mu }} \right)} & {\left( {{{\phi '}_x}} \right)} & {\left( {{{\phi '}_y}} \right)} \\ \end{array}} \right), \end{equation} (10)
其中{\left( {{{\phi '}_\mu }} \right)},{\left( {{{\phi '}_x }} \right)}, {\left( {{{\phi '}_y }} \right)}的表达式见式(4).

(2) 如果函数F连续可微且单调,则\nabla H\left( z \right)z = \left( {\mu ,x,y} \right) \in {R_{ ++ }} \times J \times J处可逆.

由定理2可以得到结论(1),下面证明结论(2),任取\mu>0,令\Delta z = \left( {\Delta \mu ,\Delta x,\Delta y} \right) \in R \times J \times J.

假设

\begin{equation} \nabla H\left( z \right)\Delta z = 0. \end{equation} (11)
则只需要证明\Delta z = 0,由式(10)(11)得到
\begin{equation} \begin{array}{l} \Delta \mu = 0 ,\\ \nabla F\left( x \right)\Delta x - \Delta y = 0,\\ \left( {\cos \mu + \sin \mu } \right)\left( {\Delta x + \Delta y} \right) - L_w^{ - 1}\left[{{{\left( {\cos \mu - \sin \mu } \right)}^2}\left( {x - y} \right)\left( {\Delta x - \Delta y} \right)} \right] + {{\phi '}_\mu }\Delta \mu = 0 . \end{array} \end{equation} (12)

由函数F的单调性和式(12)中第2式可以得到

\begin{equation} 〈 {\Delta x,\Delta y} 〉 = 〈 {x,\nabla F\left( x \right)\Delta x} 〉 \ge 0. \end{equation} (13)
由式(12)其余2式得到 {L_w}\left[{\left( {\cos \mu + \sin \mu } \right) \left( {\Delta x + \Delta y} \right)} \right] - {\left( {\cos \mu - \sin \mu } \right)^2}\left( {x - y} \right)\left( {\Delta x - \Delta y} \right) = 0, \begin{array}{l} \left[{w - \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)} \right]\left( {\Delta x\cos \mu + \Delta y\sin \mu } \right) \\ + \left[{w + \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)} \right]\left( {\Delta x\sin \mu + \Delta y\cos \mu } \right) = 0 . \end{array}
\begin{equation} \left( {\Delta x\sin \mu + \Delta y\cos \mu } \right) = - {\rm{L}}_{w + \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)}^{ - 1}{{\rm{L}}_{w - \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)}}\left( {\Delta x\cos \mu + \Delta y\sin \mu } \right) \end{equation} (14)
\begin{eqnarray} && 〈 {{\rm{L}}_{w + \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)}^{ - 1}{{\rm{L}}_{w - \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)}}\left( {\Delta x\cos \mu + \Delta y\sin \mu } \right),\Delta x\cos \mu + \Delta y\sin \mu } 〉 \nonumber\\ & =& - 〈 {\Delta x\sin \mu + \Delta y\cos \mu , \Delta x\cos \mu + \Delta y\sin \mu } 〉 \nonumber\\ &=& - \sin \mu \cos \mu \left( {{{\left\| {\Delta x} \right\|}^2} + {{\left\| {\Delta y} \right\|}^2}} \right) - \left( {{{\sin }^{\rm{2}}}\mu + {{\cos }^{\rm{2}}}\mu } \right)〈 {\Delta x\Delta y} 〉 \le {\rm{0}} . \end{eqnarray} (15)
通过w - \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right) \in K, w + \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right) \in K
\begin{equation} \left[{w - \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)} \right] \circ \left[{w + \left( {\cos \mu - \sin \mu } \right)\left( {x - y} \right)} \right] \ge 0, \end{equation} (16)
由引理2可得到: \Delta x\cos \mu + \Delta y\sin \mu = 0, 由式(14)可以得到\Delta x\sin \mu + \Delta y\cos \mu = 0 得到〈 {\Delta x,\Delta y} 〉 = 0, 所以\nabla H\left( z \right)可逆. 定理得证.

由式(9)可以得到{\mu _{k + 1}} = \left( {1 - {\lambda _k}} \right) {\mu _k} + {\lambda _k}{\beta _k}{\mu _0} > 0,{\mu _0} > 0, 在由定理3知\nabla H\left( z \right)可逆,因此步骤2是适定的. 对于任意\alpha \in \left[{0,1} \right],令

\begin{equation} R\left( \alpha \right) = H\left( {{z^k} + \alpha \Delta {z^k}} \right) - H\left( {{z^k}} \right) - \alpha {\left( {\nabla H\left( {{z^k}} \right)} \right)^T}\Delta {z^k}, \end{equation} (17)
\left( {{\mu ^k} + \alpha \Delta {\mu ^k}} \right) > 0. 由定理3知H在任意z = \left( {\mu ,x,y} \right) \in {R_{ ++ }} \times J \times J处连续可微,故 \left\| {R\left( \alpha \right)} \right\| = o\left( \alpha \right). {\beta _k} = \beta \left( {{z^k}} \right):= \gamma \left\| {H\left( {{z^k}} \right)} \right\|\min \left\{ {1,\left\| {H\left( {{z^k}} \right)} \right\|} \right\}, 得到当\left\| {H\left( {{z^k}} \right)} \right\| > 1时, {\beta _k} = \gamma \left\| {H\left( {{z^k}} \right)} \right\| \left\| {H\left( {{z^k}} \right)} \right\|\leq 1时,{\beta _k} \leq \gamma \left\| {H\left( {{z^k}} \right)} \right\|,故有
\begin{equation} {\beta _k} \le \gamma \left\| {H\left( {{z^k}} \right)} \right\|. \end{equation} (18)

由式(9),(14),(18)知,对于\alpha \in \left[{0,1} \right], {\eta _k} \in \left[{0{\rm{,1 - }}\gamma {\mu ^0}} \right]容易得出

\begin{eqnarray} \left\| {H\left( {{z^k} + \alpha \Delta {z^k}} \right)} \right\| &=& \left\| {H\left( {{z^k}} \right) + \alpha {{\left( {\nabla H\left( {{z^k}} \right)} \right)}^T}\Delta {z^k} + R\left( \alpha \right)} \right\| \nonumber\\ & =& \left\| {H\left( {{z^k}} \right) - \alpha H\left( {{z^k}} \right) + \alpha \left( {{\beta _k}{\mu ^0},0,{r^k}} \right) + R\left( \alpha \right)} \right\| \nonumber\\ & \le& \left( {1 - \alpha } \right)\left\| {H\left( {{z^k}} \right)} \right\| + \alpha \left( {{\beta _k}{\mu ^0}} \right) + \alpha \left\| {{r^k}} \right\| + \left\| {R\left( \alpha \right)} \right\| \nonumber\\ &\le& \left( {1 - \alpha } \right)\left\| {H\left( {{z^k}} \right)} \right\| + \alpha \left( {\gamma \left\| {{H_\tau }\left( {{z^k}} \right)} \right\|{\mu ^0}} \right) + \alpha {\eta _k}\left\| {H\left( {{z^k}} \right)} \right\| + o\left( \alpha \right) \nonumber\\ &\le& \left[{\left( {1 - \alpha } \right) + \alpha \gamma {\mu ^0} + \alpha {\eta _k}} \right]\left\| {H\left( {{z^k}} \right)} \right\| + o\left( \alpha \right) . \end{eqnarray} (19)
因此由式(19)知,存在数\bar \alpha \in \left( {0,1} \right], 对所有的\alpha \in \left[{0,\bar \alpha } \right], {\eta _k} \in \left[{0{\rm{,1 - }}\gamma {\mu ^0}} \right]\left\| {H\left( {{z^k} + \alpha \Delta {z^k}} \right)} \right\| \le \left[{1 - \sigma \left( {1 - \gamma {\mu _0} - {\eta _k}} \right)\alpha } \right]\left\| {H\left( {{z^k}} \right)} \right\|. 这样,步骤3也是适定的.

4 算法的收敛性分析

引理3 设函数F 连续可微且单调,函数H如(10)式定义,则函数Hz = \left( {\mu ,x,y} \right) \in {R_{ ++ }} \times J \times J处是强制的,即 \mathop {\lim }\limits_{\left\| {\left( {\mu ,x,y} \right)} \right\| \to \infty } \left\| {H\left( {\mu ,x,y} \right)} \right\|{\rm{ = + }}\infty . 其证明可参见文献[3].

算法3.1中,当迭代序列的的聚点满足一个非奇异假设时, 迭代序列全局线性收敛且局部二次收敛于该聚点. 由函数F单调连续可微, 则算法3.1生成的迭代点列满足

\begin{equation} \left\{ {{z^k}} \right\} \subset \Omega := \left\{ {z = \left( {\mu ,x,y} \right) \in {R_{ ++ }} \times {R^n} \times {R^n}{\rm{|}}\mu \ge \beta \left( z \right){\mu _0}} \right\}. \end{equation} (20)
事实上{z^0} \in \Omega ,假设{z^k} \in \Omega ,则有 {\mu _{k + 1}} = {\lambda _k}{\beta _k}{\mu _0} + \left( {1 - {\lambda _k}} \right){\mu _k} \ge \left( {1 - {\lambda _k}} \right){\beta _k}{\mu _0} + {\lambda _k}{\beta _k}{\mu _0} = {\beta _k}{\mu _0}. 又由\left\| {H\left( {{z^{k + 1}}} \right)} \right\| \le \left\| {H\left( {{z^k}} \right)} \right\|,有{\beta _k} \ge {\beta _{k + 1}},所以有 {\mu _{k + 1}} \ge {\beta _{k + 1}}{\mu ^0}.

定理4 设函数F单调连续可微,且{z^ * } = \left( {{\mu ^ * },{x^ * },{y^ * }} \right)是算法3.1所生成的迭代点列 [\left\{ {{z^k}} \right\}的一个聚点,则

(1) {z^ * } = \left( {{\mu ^ * },{x^ * },{y^ * }} \right) H\left( {{z^k}} \right) = 0的一个解;

(2) 如果所有的V\in \partial H\left( {{z^ * }} \right)可逆,且\nabla F Lipchitz 连续,则\left\{ {{z^k}} \right\}局部二阶收敛于{z^ * }, 即\left\| {{z^{k + 1}} - {z^ * }} \right\| = O\left( {{{\left\| {{z^{k + 1}} - {z^ * }} \right\|}^2}} \right){\mu _{k + 1}} = O\left( {\mu _k^2} \right).

(1) 由定理3.1和算法3.1知{z^k} \in \Omega , \left\| {H\left( {{z^{k + 1}}} \right)} \right\| \le \left\| {H\left( {{z^k}} \right)} \right\| ,因此\left\| {H\left( {{z^k}} \right)} \right\|{\beta _k} = \beta \left( {{z^k}} \right)是单调递减的, 由于\left\| {{H_\tau }\left( {{z^k}} \right)} \right\| \ge 0, \beta \left( {{z^k}} \right) \ge 0,则当k \to \infty 时, 由连续性知\left\| {H\left( {{z^k}} \right)} \right\| \to \left\| {H\left( {{z^ * }} \right)} \right\| \ge 0 ,\beta \left( {{z^k}} \right) \to \beta \left( {{z^ * }} \right) \ge 0.

用反正法证明{H_\tau }\left( {{z^k}} \right) = 0,假设\left\| {H\left( {{z^ * }} \right)} \right\| > 0,且{z^ * } = \left( {{\mu ^ * },{x^ * },{y^ * }} \right) 是算法3.1所生成的迭代点列\left\{ {{z^k}} \right\}的聚点,由定理3,存在{z^ * } 的一个闭邻域N\left( {{z^ * }} \right),和一个正数\bar \alpha \in \left( {0,1} \right],使得z \in N\left( {{z^ * }} \right)\alpha \in \left[{0,\bar \alpha } \right],有\left\| {H\left( {z + \alpha \Delta {z^k}} \right)} \right\| \le \left[{1 - \sigma \left( {1 - \gamma {\mu _0} - {\eta _k}} \right)\alpha } \right]\left\| {H\left( z \right)} \right\|成立. 对任意非负整数l满足{\delta ^l} \in \left( {0,\bar \alpha } \right], \left\| {H\left( {{z^k} + {\delta ^l}\Delta {z^k}} \right)} \right\| \le \left[{1 - \sigma \left( {1 - \gamma {\mu _0} - {\eta _k}} \right){\delta ^l}} \right]\left\| {H\left( {{z^k}} \right)} \right\|,对于充分大的k, 有{\delta ^l} \le {\lambda _k},从而对于充分大的k

\begin{equation} \left\| {H\left( {{z^{k + 1}}} \right)} \right\| \le \left[{1 - \sigma \left( {1 - \gamma {\mu _0} - {\eta _k}} \right){\lambda _k}} \right]\left\| {H\left( {{z^k}} \right)} \right\| \le \left[{1 - \sigma \left( {1 - \gamma {\mu _0} - {\eta _k}} \right){\delta ^l}} \right]\left\| {H\left( {{z^k}} \right)} \right\|. \end{equation} (21)
对(21)式两边取极限与\left\| {H\left( {{z^ * }} \right)} \right\| > 0矛盾, 因此结论(1)得证.

(2) 因为所有V \in \partial H\left( {{z^ * }} \right)可逆, {z^ * } = \left( {{\mu ^ * },{x^ * },{y^ * }} \right)是算法3.1所生成的迭代点列 \left\{ {{z^k}} \right\}的一个聚点,根据文献[12,定理3]可知, 对于充分靠近{z^ * } = \left( {{\mu ^ * },{x^ * },{y^ * }} \right){z^k}, 存在常数C > 0使得\left\| {\nabla H{{\left( {{z^k}} \right)}^{{\rm{ - }}1}}} \right\| < C,对于任意k \ge 0 成立. 由定理3 和\nabla F Lipchitz 连续知{H_\tau }\left( \cdot \right){z^ * } = \left( {{\mu ^ * }, {x^ * },{y^ * }} \right)处是强半光滑的,因此,在 {z^*} 的一个很小的邻域内存在{z^k},有

\begin{equation} \left\| {H\left( {{z^k}} \right) - H\left( {{z^ * }} \right)} \right\| = O\left( {\left\| {{z^k} - {z^ * }} \right\|} \right) \end{equation} (22)
\begin{equation} \left\| {H\left( {{z^k}} \right) - H\left( {{z^ * }} \right) - \nabla H\left( {{z^k}} \right)\left( {{z^k} - {z^ * }} \right)} \right\| = O\left( {{{\left\| {{z^k} - {z^ * }} \right\|}^2}} \right). \end{equation} (23)
得到
\begin{eqnarray} && \left\| {{z^k} + \Delta {z^k} - {z^ * }} \right\| \nonumber\\ &=& \left\| {{z^k} + \nabla H{{\left( {{z^k}} \right)}^{ - 1}}\left[{ - H\left( {{z^k}} \right) + {{\left( {{\beta _k}{\mu ^0},0,{r^k}} \right)}^T}} \right] - {z^ * }} \right\| \nonumber\\ & = &\left\| {\nabla H{{\left( {{z^k}} \right)}^{ - 1}}} \right\|\left( {\left\| {H\left( {{z^k}} \right) - H\left( {{z^ * }} \right) - \nabla H\left( {{z^k}} \right)\left( {{z^k} - {z^ * }} \right) + {{\left( {{\beta _k}{\mu ^0},0,{r^k}} \right)}^T}} \right\|} \right) \nonumber\\ & \le& \left\| {\nabla H{{\left( {{z^k}} \right)}^{ - 1}}} \right\|\left( {\left\| {H\left( {{z^k}} \right) - H\left( {{z^ * }} \right) - \nabla H\left( {{z^k}} \right)\left( {{z^k} - {z^ * }} \right)} \right\| + {\beta _k}{\mu ^0} + \left\| {{r^k}} \right\|} \right) \nonumber\\ & =& O\left( {{{\left\| {{z^k} - {z^ * }} \right\|}^2}} \right) . \end{eqnarray} (24)
第一步等式根据式(9)而得,最后一步不等式由式(23)和{\beta _k} 的定义及{r^k} \in {R^n}满足\left\| {{r^k}} \right\| \le {\eta _k}\left\| {H\left( {{z^k}} \right)} \right\|而得. 由式(22)(24)得到
\begin{equation} \left\| {H\left( {{z^k} + \Delta {z^k}} \right)} \right\| = O\left( {{{\left\| {H\left( {{z^k}} \right)} \right\|}^2}} \right). \end{equation} (25)
由式(25)知当k 充分大时{\lambda _k} = 1,因此 在{z^ * }的一个很小的邻域内存在{z^k},有{z^{k + 1}} = {z^k} + \Delta {z^k} 得到\left\| {{z^{k + 1}} - {z^ * }} \right\| = O\left( {{{\left\| {{z^{k + 1}} - {z^ * }} \right\|}^2}} \right) 此外 {\mu _{k + 1}} = {\mu _k} + \Delta {\mu _k} = {\beta _k}{\mu _0} = O\left( {{{\left\| {H\left( {{z^k}} \right)} \right\|}^2}} \right) = O\left( {{{\left\| {{z^{k + 1}} - {z^ * }} \right\|}^2}} \right) = O\left( {{{\left( {{\mu _k}} \right)}^2}} \right). 所以则\left\{ {{z^k}} \right\}局部二阶收敛于{z^ * }.

5 数值实验

为了验证算法3.1的可行性和有效性,我们进行了对比实验,利用Matlab7.0编程, 在Intel(R),Pentium(R) Dual 1.73GHz,0.99GB的内存, Windows XP操作系统的hp笔记本上进行数学实验. 在测试中使用如下参数:

\gamma {\rm{ = 0}}.{\rm{01}}\min \left\{ {1,1 / \left\| {H\left( {{z^0}} \right)} \right\|} \right\},\eta = {{\rm{2}}^{ - k - 1}},\sigma = 0.001,\delta = 0, {r^k}\left[{0,1} \right)的随机n维向量,停机标准是\left\| {H\left( {{z^k}} \right)} \right\| \le {10^{ - 6}},CPU时间单位为秒. 初始点{z^0} = \left( {{\mu ^0},{x^0},{y^0}} \right) 随机选取.

r=0其他的参数不变,算法变为精确光滑牛顿法(记为: PSN),{r^k}\left[{0,1} \right)的随机n维向量,即不精确的光滑牛顿法 (记为:\!\!\!ISN),Iter,cpu分别表示500次数值实验平均的迭代次数和平均的cpu时间.

算例1 F(x) = Mx + q,其中 M \in {R^n} \times {R^n}为对称稀疏矩阵, 由库函数sprandsym(n,den,rc)随机生成,den表示矩阵的稀疏度, 条件数rc是由库函数unifrnd 随机生成的一个[0,1]中的数,它保证了矩阵M的半正定性. q \in {R^n}的每个元素从区间[-1,1]中Matlab的库函数rand随机生成.

表 1 算例1的数值结果比较

算例2 F(x) = {(0.02x_1^3,0.05x_2^3,0.09x_3^3, 0.01x_4^3,\cdots,0.01x_n^3)^T}, 其中K = {K^n}F:{R^n} \to {R^n}.由于 \nabla F(x) = \left( {\begin{array}{*{20}{c}} {0.06x_1^2} & {} & {} & {} & {} & {} \\ {} & {0.15x_2^2} & {} & {} & {} & {} \\ {} & {} & {0.27x_3^2} & {} & {} & {} \\ {} & {} & {} & {0.03x_4^2} & {} & {} \\ {} & {} & {} & {} & \cdots & {} \\ {} & {} & {} & {} & {} & {0.03x_n^2} \\ \end{array}} \right) 是半正定矩阵,因此F(x)单调.

表 2 算例2的数值结果比较

表 1表 2的对比实验可以看出算法3.1是有效的, 在维数较大的情况下非精确光滑算法cpu时间相对较小. 非精确光滑牛顿法对处理对称锥上复杂大规模问题是一种较好的算法.

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对称锥互补问题的一种非精确光滑牛顿算法
张运胜, 高雷阜