Coordinate-wise Armijos condition

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 $ abla 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 Armijos 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*} When $f(x,y)=f_1(x)+f_2(y)$ is a coordinate-wise sum map, we propose the following {bf coordinate-wise} Armijos condition. Fix again $0<alpha <1$. A pair of positive numbers $delta _1,delta _2>0$ satisfies the coordinate-wise variant of Armijos condition at $(x,y)$ if the following inequality holds: begin{eqnarray*} [f_1(x-delta _1 abla f_1(x))+f_2(y-delta _2 abla f_2(y))]-[f_1(x)+f_2(y)]leq -alpha (delta _1|| abla f_1(x)||^2+delta _2|| abla f_2(y)||^2). end{eqnarray*} We then extend results in our recent previous results, on Backtracking Gradient Descent and some variants, to this setting. We show by an example the advantage of using coordinate-wise Armijos condition over the usual Armijos condition.