I am trying to apply coordinate descent to the function in two variables $$f(x,y)=100(y-x^2)^2+(1-x)^2$$
When trying to do this, I first attempted to apply exact line search over each component. If I try to first look at the y component and consider my step direction $p=(0,1)$, when solving for alpha such that $\phi(\alpha)=f((x,y)+\alpha*p)$ is minimized, I get $\alpha=x^2-y$. However, when trying a similar approach over the x component, I don't find it possible to solve for $\alpha$. I would like to know if someone can provide an idea over which other optimization method besides exact line search should I attempt over each coordinate when applying the coordinate descent method.
You should first have some starting point $[x_0,y_0]$. Then, from this point you need to minimize in the direction $-(\nabla f(x,y))(x_0,y_0)$ and in this direction to proceed a line search. Repeatedly.