# Armijo搜索下求解无约束优化问题的扰动BFGS方法The Perturbed BFGS Method for the Unconstrained Optimization with the Armijo Line Search

Abstract: In [1], a perturbed BFGS method was proposed to solve the unconstrained optimization and was proved to be globally convergent when the Wolfe line search was used. In this paper, we show that the perturbed BFGS method also possesses global convergence for nonconvex problems with the relatively weaker Armijo line search. Numerical results show that this method with the Armijo search is also promising.

1. 引言

$\mathrm{min}f\left(x\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {R}^{n},$ (1.1)

${x}_{k+1}={x}_{k}+{\alpha }_{k}{d}_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k\ge 1,$ (1.2)

${B}_{k}{d}_{k}+\nabla f\left({x}_{k}\right)=0,$ (1.3)

${B}_{k+1}={B}_{k}-\frac{{B}_{k}{s}_{k}{s}_{k}^{\text{T}}{B}_{k}}{{s}_{k}^{\text{T}}{B}_{k}{s}_{k}}+\frac{{y}_{k}{y}_{k}^{\text{T}}}{{y}_{k}^{\text{T}}{s}_{k}},$ (1.4)

Powell在文献 [4] 中证明了求解凸优化问题的BFGS方法只有全局收敛性。但对非凸优化问题，Dai [5] 和Mascarenhas [6] 给出例子说明采用Wolfe型非精确线性搜索和精确线性搜索的BFGS方法不一定具有全局收敛性。在文献 [1] 中，Liu提出了一种扰动的BFGS方法，简称PBFGS，在该算法中，搜索方向 ${d}_{k}$ 由以下线性方程组确定。

$\left({B}_{k}+{\mu }_{k}Q\right){d}_{k}+\nabla f\left({x}_{k}\right)=0,$ (1.5)

${B}_{k+1}=\left\{\begin{array}{l}{B}_{k}-\frac{{B}_{k}{s}_{k}{s}_{k}^{\text{T}}{B}_{k}}{{s}_{k}^{\text{T}}{B}_{k}{s}_{k}}+\frac{{y}_{k}{y}_{k}^{\text{T}}}{{y}_{k}^{\text{T}}{s}_{k}}, \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}如果\text{ }{y}_{k}^{\text{T}}{s}_{k}>0\hfill \\ {B}_{k}, \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\hfill \end{array}$ (1.6)

2. 算法与收敛性分析

$\left({B}_{k}+{\mu }_{k}Q\right){d}_{k}+\nabla f\left({x}_{k}\right)=0,$ (2.1)

$f\left({x}_{k}+{\alpha }_{k}{d}_{k}\right)-f\left({x}_{k}\right)\le {\sigma }_{1}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k},$ (2.2)

${B}_{k+1}=\left\{\begin{array}{l}{B}_{k}-\frac{{B}_{k}{s}_{k}{s}_{k}^{\text{T}}{B}_{k}}{{s}_{k}^{\text{T}}{B}_{k}{s}_{k}}+\frac{{y}_{k}{y}_{k}^{\text{T}}}{{y}_{k}^{\text{T}}{s}_{k}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}如果\text{ }{y}_{k}^{\text{T}}{s}_{k}>0\hfill \\ {B}_{k}, \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其他\hfill \end{array}$ (2.3)

${\mu }_{k+1}=\left\{\begin{array}{l}{\epsilon }_{k+1}{‖{B}_{k+1}‖}_{F},如果{‖{B}_{k+1}‖}_{F}\ge {M}_{k+1}\hfill \\ {\epsilon }_{k+1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}其它\hfill \end{array}$ (2.4)

${\epsilon }_{k+1}=\left\{\begin{array}{l}\tau {\epsilon }_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}如果‖\nabla f\left({x}_{k+1}\right)‖/\delta \le \eta \\ {\epsilon }_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}否则\end{array}$ (2.5)

$\underset{k\to \infty }{\mathrm{lim}\mathrm{inf}}‖\nabla f\left({x}_{k}\right)‖=0$ (2.6)

$‖\nabla f\left(x\right)-\nabla f\left(y\right)‖\le L‖x-y‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\forall x,y\in \Omega \right).$ (2.7)

$\begin{array}{l}f\left({x}_{1}+{\alpha }_{1}{d}_{1}\right)-f\left({x}_{1}\right)\le {\sigma }_{1}{\alpha }_{1}\nabla f{\left({x}_{1}\right)}^{\text{T}}{d}_{1}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ f\left({x}_{k}+{\alpha }_{k}{d}_{k}\right)-f\left({x}_{k}\right)\le {\sigma }_{1}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}\end{array}$

$f\left({x}_{k}+{\alpha }_{k}{d}_{k}\right)-f\left({x}_{1}\right)\le {\sigma }_{1}\underset{k=1}{\overset{\infty }{\sum }}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}.$

$-\underset{k=1}{\overset{\infty }{\sum }}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}\le -\frac{{C}_{1}}{{\delta }_{1}}<+\infty .$

$-\underset{k\to +\infty }{\mathrm{lim}}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}=-\underset{k\to +\infty }{\mathrm{lim}}\nabla f{\left({x}_{k}\right)}^{\text{T}}{s}_{k}=0.$ (2.8)

${K}_{\tau }=\left\{k:{\epsilon }_{k+1}=\tau {\epsilon }_{k}\right\},$ ${K}_{B}=\left\{k:{‖{B}_{k}‖}_{F}\ge {M}_{k}\right\}.$ (2.9)

$-\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}={d}_{k}^{\text{T}}\left({B}_{k}+{\mu }_{k}I\right){d}_{k}\ge {\epsilon }_{k}{‖{B}_{k}‖}_{F}{‖{d}_{k}‖}^{2}\ge {\epsilon }_{k}\mathrm{min}\left\{1,{M}_{B}\right\}{‖{d}_{k}‖}^{2}.$ (2.10)

$f\left({x}_{k}+{\rho }^{-1}{\alpha }_{k}{d}_{k}\right)-f\left({x}_{k}\right)>{\sigma }_{1}{\rho }^{-1}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}.$ (2.11)

$\begin{array}{l}f\left({x}_{k}+{\rho }^{-1}{\alpha }_{k}{d}_{k}\right)-f\left({x}_{k}\right)\\ ={\rho }^{-1}{\alpha }_{k}\nabla f{\left({x}_{k}+{\theta }_{k}{\rho }^{-1}{\alpha }_{k}{d}_{k}\right)}^{\text{T}}{d}_{k}\\ ={\rho }^{-1}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}+{\rho }^{-1}{\alpha }_{k}\left[\nabla f{\left({x}_{k}+{\theta }_{k}{\rho }^{-1}{\alpha }_{k}{d}_{k}\right)}^{\text{T}}-\nabla f\left({x}_{k}\right)\right]{d}_{k}\\ \le {\rho }^{-1}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}+L{\rho }^{-2}{\alpha }_{k}^{2}{‖{d}_{k}‖}^{2}\end{array}$(2.12)

$\begin{array}{l}{\alpha }_{k}\ge \frac{-\left(1-{\sigma }_{1}\right)\rho \nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}}{L{‖{d}_{k}‖}^{2}}=\frac{\left(1-{\sigma }_{1}\right)\rho {d}_{k}^{\text{T}}\left({B}_{k}+{\mu }_{k}I\right){d}_{k}}{L{‖{d}_{k}‖}^{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\ge \frac{\left(1-{\sigma }_{1}\right)\rho {\epsilon }_{k}{}_{{}_{1}}}{L}\mathrm{min}\left\{1,{M}_{B}\right\}>0.\end{array}$ (2.13)

${\alpha }_{k}=1$ 时，显然可得 ${\alpha }_{k}>0$ 。又由(2.8)，(2.10)可得

$0=-\underset{k\to +\infty }{\mathrm{lim}}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}=\underset{k\to +\infty }{\mathrm{lim}}{\alpha }_{k}{d}_{k}^{\text{T}}\left({B}_{k}+{\mu }_{k}I\right){d}_{k}\ge \underset{k\to +\infty }{\mathrm{lim}}{\alpha }_{k}{\epsilon }_{k}\mathrm{min}\left\{1,{M}_{B}\right\}{‖{d}_{k}‖}^{2}>0,$

$\underset{k\to \infty }{\mathrm{lim}}\mathrm{inf}‖\nabla f\left({x}_{k}\right)‖=0.$ (2.14)

${B}_{k}{d}_{k}+{\epsilon }_{{k}_{1}}{‖{B}_{k}‖}_{F}{d}_{k}+\nabla f\left({x}_{k}\right)=0.$

$-\nabla f{\left({x}_{k}\right)}^{\text{T}}{\stackrel{^}{d}}_{k}={d}_{k}^{\text{T}}{B}_{k}{\stackrel{^}{d}}_{k}+{\epsilon }_{{k}_{1}}{‖{\stackrel{^}{d}}_{k}‖}^{2}.$ (2.15)

$L{\rho }^{-2}{\alpha }_{k}^{2}{‖{d}_{k}‖}^{2}\ge \left(\sigma -1\right){\rho }^{-1}{\alpha }_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{d}_{k}.$

$L{\rho }^{-2}{\stackrel{^}{\alpha }}_{k}^{2}{‖{\stackrel{^}{d}}_{k}‖}^{2}\ge \left(\sigma -1\right){\rho }^{-1}{\stackrel{^}{\alpha }}_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{\stackrel{^}{d}}_{k},$

${\stackrel{^}{\alpha }}_{k}\ge \frac{-\left(1-{\sigma }_{1}\right)\rho \nabla f{\left({x}_{k}\right)}^{\text{T}}{\stackrel{^}{d}}_{k}}{L{‖{\stackrel{^}{d}}_{k}‖}^{2}}=\frac{\left(1-{\sigma }_{1}\right)\rho \left({d}_{k}^{\text{T}}{B}_{k}{\stackrel{^}{d}}_{k}+{\epsilon }_{{k}_{1}}{‖{\stackrel{^}{d}}_{k}‖}^{2}\right)}{L{‖{\stackrel{^}{d}}_{k}‖}^{2}}\ge \frac{\left(1-{\sigma }_{1}\right)\rho {\epsilon }_{{k}_{1}}}{L}>0.$

$0=-\underset{k\to +\infty }{\mathrm{lim}}{\stackrel{^}{\alpha }}_{k}\nabla f{\left({x}_{k}\right)}^{\text{T}}{\stackrel{^}{d}}_{k}=\underset{k\to +\infty }{\mathrm{lim}}\stackrel{^}{\alpha }\left({d}_{k}^{\text{T}}{B}_{k}{\stackrel{^}{d}}_{k}+{\epsilon }_{{k}_{1}}{‖{\stackrel{^}{d}}_{k}‖}^{2}\right)\ge \underset{k\to +\infty }{\mathrm{lim}}{\stackrel{^}{\alpha }}_{k}{\epsilon }_{{k}_{1}}{‖{\stackrel{^}{d}}_{k}‖}^{2}>0.$

$\underset{k\to \infty ,k\in {K}_{B}}{\mathrm{lim}}{\stackrel{^}{d}}_{k}=0.$ (2.16)

$‖\nabla f\left({x}_{k}\right)‖\le ‖{B}_{k}‖‖{\stackrel{^}{d}}_{k}‖/{‖{B}_{k}‖}_{F}+{\epsilon }_{{k}_{1}}‖{\stackrel{^}{d}}_{k}‖\le \left(1+{\epsilon }_{{k}_{1}}\right)‖{\stackrel{^}{d}}_{k}‖.$

$\underset{k\to \infty }{\mathrm{lim}}\mathrm{inf}‖\nabla f\left({x}_{k}\right)‖=0.$

$\underset{k\to \infty }{\mathrm{lim}}\mathrm{inf}‖\nabla f\left({x}_{k}\right)‖=0.$

${‖{B}_{k}‖}_{F}<\mathrm{max}\left\{{M}_{B},1/‖\nabla f\left({x}_{k}\right)‖\right\}\le \mathrm{max}\left\{{M}_{B},1/\gamma \right\}.$ (2.17)

${K}_{\tau }$ 为有限集时，对所有的 $k\ge {k}_{1}$ 有(2.10)式成立。由引理1和(2.8)可知，当 $k\to \infty$ 时， ${d}_{k}\to 0$ 。进一步由修正后的拟牛顿公式(2.1)以及(2.17)式可得，对所有的 $k\notin {K}_{B}$ 有下面不等式

$‖\nabla f\left({x}_{k}\right)‖\le \left(‖{B}_{k}‖+{\epsilon }_{k}\right)‖{d}_{k}‖\le \left({‖{B}_{k}‖}_{F}+{\epsilon }_{1}\right)‖{d}_{k}‖\le \left(\mathrm{max}\left\{{M}_{B},1/\gamma \right\}+{\epsilon }_{1}\right)‖{d}_{k}‖$

$\underset{k\to \infty ,k\notin {K}_{B}}{\mathrm{lim}}\mathrm{inf}‖\nabla f\left({x}_{k}\right)‖=0,$

$\underset{k\to \infty }{\mathrm{lim}}\mathrm{inf}‖\nabla f\left({x}_{k}\right)‖=0.$ (2.18)

3. 数值试验及结果分析

$\theta \left({x}_{1},{x}_{2}\right)=\left\{\begin{array}{l}\frac{1}{\text{2π}}\mathrm{arctan}\left(\frac{{x}_{2}}{{x}_{1}}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}>0\\ \frac{1}{\text{2π}}\mathrm{arctan}\left(\frac{{x}_{2}}{{x}_{1}}\right)+0.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{1}<0\end{array}$

Table 1. Experimental results of algorithm

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https://doi.org/10.1137/S1052623499354242

[8] Armijo, L. (1966) Minimization of Functions Having Lipschitz Continuous Partial Derivatives. Pacific Journal of Mathematics, 16, 1-3.
https://doi.org/10.2140/pjm.1966.16.1

[9] Li, D.H. and Fukushima, M. (2001) A Modified BFGS Method and Its Global Convergence in Nonconvex Minimization. Journal of Computational and Applied Mathematics, 192, 15-35.
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