Coordinate-wise Armijo's condition

Let $z=(x,y)$ be coordinates for the product space $$\mathbb{R}$^{m_1}\times $\mathbb{R}$^{m_2}$\mathrm{.}Let$f:$\mathbb{R}$^{m_1}\times $\mathbb{R}$^{m_2}\rightarrow $\mathbb{R}$$bea$C^1$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}$,anumber$\delta >0$satisfiesArmij{o}^{\mathrm{\prime }}sconditionat$(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} 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_1(x-\delta _1\nabla f_1(x))+f_2(y-\delta _2\nabla f_2(y))]-[f_1(x)+f_2(y)]\leq -\alpha (\delta _1||\nabla f_1(x)||^2+\delta _2||\nabla 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 Armijo’s condition over the usual Armijo’s condition.