论文标题
协调Armijo的状况：一般案例
Coordinate-wise Armijo's condition: General case
论文作者
论文摘要
令$ z =（x，y）$为产品空间$ \ mathbb {r}^{m_1} \ times \ times \ mathbb {r}^{m_2} $。令$ f：\ mathbb {r}^{m_1} \ times \ mathbb {r}^{m_2} {m_2} \ rightarrow \ Mathbb {r} $ be a $ c^1 $ function，$ \ nabla f =（\ nabla f =（\ partial _xf，partial _xf，part \ partial _f）修复$ 0 <α<1 $。对于点$（x，y） _xf，y-δ\ partial _yf）-f（x，y）\ leq-αδ（|| \ partial _xf ||^2+|| \ partial _yf ||^2）。 \ end {eqnarray*} 在上一篇论文中，我们提出了以下{\ bf坐标} Armijo的状况。再次修复$ 0 <α<1 $。如果以下不平等存在：\ begin {eqnarray*} [f（x-Δ_1\ partial _xf（x，xf（x，x，x，x，x，x，x，x，y-δ__2），y-em-Δ_2\ partial part（y-Δ_2feq）， -α（Δ_1|| \ partial _xf（x，y）||^2+δ_2|| \ partial _yf（x，y）||^2）。 \ end {eqnarray*}以前我们将此条件应用于$ f（x，y）= f（x）+g（y）$的功能，并证明了它们的各种收敛结果。对于一般功能，对于能够进行实际计算至关重要 - 具有系统的算法来获得$Δ_1$和$δ_2$满足Armijo状况的坐标版本，就像为通常的Armijo的状况进行回溯。在本文中，我们提出了这种算法，并根据收敛结果证明。 然后，我们对某些功能进行分析并提出实验结果，例如$ f（x，y）= a | x | + y $（由ASL和Overton与Wolfe的方法相关），$ f（x，y）= x^3 sin（1/x） + y^3 sin（1/y）$和Rosenbrock的函数。
Let $z=(x,y)$ be coordinates for the product space $\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}$. Let $f:\mathbb{R}^{m_1}\times \mathbb{R}^{m_2}\rightarrow \mathbb{R}$ be a $C^1$ function, and $\nabla f=(\partial _xf,\partial _yf)$ its gradient. Fix $0<α<1$. For a point $(x,y) \in \mathbb{R}^{m_1}\times \mathbb{R}^{m_2}$, a number $δ>0$ satisfies Armijo's condition at $(x,y)$ if the following inequality holds: \begin{eqnarray*} f(x-δ\partial _xf,y-δ\partial _yf)-f(x,y)\leq -αδ(||\partial _xf||^2+||\partial _yf||^2). \end{eqnarray*} In one previous paper, we proposed the following {\bf coordinate-wise} Armijo's condition. Fix again $0<α<1$. A pair of positive numbers $δ_1,δ_2>0$ satisfies the coordinate-wise variant of Armijo's condition at $(x,y)$ if the following inequality holds: \begin{eqnarray*} [f(x-δ_1\partial _xf(x,y), y-δ_2\partial _y f(x,y))]-[f(x,y)]\leq -α(δ_1||\partial _xf(x,y)||^2+δ_2||\partial _yf(x,y)||^2). \end{eqnarray*} Previously we applied this condition for functions of the form $f(x,y)=f(x)+g(y)$, and proved various convergent results for them. For a general function, it is crucial - for being able to do real computations - to have a systematic algorithm for obtaining $δ_1$ and $δ_2$ satisfying the coordinate-wise version of Armijo's condition, much like Backtracking for the usual Armijo's condition. In this paper we propose such an algorithm, and prove according convergent results. We then analyse and present experimental results for some functions such as $f(x,y)=a|x|+y$ (given by Asl and Overton in connection to Wolfe's method), $f(x,y)=x^3 sin (1/x) + y^3 sin(1/y)$ and Rosenbrock's function.
