Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coordinate-wise Armijo's condition: General case   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 $0<\alpha <1$. For a point $(x,y) \in \mathbb{R}^{m_1}\times
\mathbb{R}^{m_2}$, a number $\delta >0$ satisfies Armijo's condition at $(x,y)$
if the following inequality holds: \begin{eqnarray*} f(x-\delta \partial
_xf,y-\delta \partial _yf)-f(x,y)\leq -\alpha \delta (||\partial
_xf||^2+||\partial _yf||^2). \end{eqnarray*}
  In one previous paper, we proposed the following {\bf coordinate-wise}
Armijo's condition. Fix again $0<\alpha <1$. A pair of positive numbers $\delta
_1,\delta _2>0$ satisfies the coordinate-wise variant of Armijo's condition at
$(x,y)$ if the following inequality holds: \begin{eqnarray*} [f(x-\delta
_1\partial _xf(x,y), y-\delta _2\partial _y f(x,y))]-[f(x,y)]\leq -\alpha
(\delta _1||\partial _xf(x,y)||^2+\delta _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 $\delta _1$ and $\delta _2$
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.

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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc
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The categories of this paper are: nlin