# What's the terminology for this alternative minimization algorithm?

Say the model is $$F(x_1)G(x_2)Z(x_3) = y \in \mathbb{R}^N$$, with $$F,G,Z$$ explicitly known, we are given observation of $$y$$ as $$y_b \in \mathbb{R}^N$$ to find the value of $$x_1$$, $$x_2$$, $$x_3$$ for each sample. Clearly this is an underdetermined inverse problem.

I have seen some iterative algorithm like this,

1. start with random number of $$x_1$$, $$x_2, x_3$$.
2. choose $$x_1$$, keep $$x_2$$, $$x_3$$ constant as change $$x_1$$ to $$x_1^*$$ such that

$$F(x_1^*)G(x_2)Z(x_3) = y_b$$

1. update $$x_1 = x_1^*$$

2. same thing for $$x_2$$, but fix $$x_1$$, $$x_3$$, and update $$x_2 = x_2^*$$.

3. repeat for $$x_3$$, and do the whole process several iterations until it doesn't change

4. final result is taken as the solution to the inverse problem.

I have seen this philosophy many times across different areas of computer science, just hope to know what is the big picture behind it. like ADMM?

## 1 Answer

A general name for this approach is "Block Coordinate Descent." It's important to understand that convergence isn't guaranteed without additional hypotheses.

ADMM is not simply block coordinate descent- it's a more complicated method that is optimizing with respect to primal variables $$x$$ and $$z$$ in each iteration and then adjusting the Lagrange multiplier and penalty parameter.

• just found in a 1977 paper, it is called nonlinear Gauss Seidel method. oh yeah, now it looks more familiar. Commented Aug 23, 2019 at 3:54