Suppose $X\subseteq\mathbb{R}^n$ is a convex set and that a function $g(x,y):X\times X\rightarrow\mathbb{R}\_+$ is smooth, "strictly biconvex" (strictly convex in $x$ and $y$ independently but not jointly), and satisfies $g(x,y)=g(y,x)$.


Alternating optimization on $g$ in this case takes a nice form: $x\_{i+1}\gets \arg\min\_x g(x,x\_i).$


I know alternating optimization doesn't have to converge in general. But, is there any chance to prove that $x\_i\rightarrow x^\ast$ that under the symmetry and biconvexity assumptions?



Fixed point iteration on symmetric biconvex function
The paper cited in [my answer here](https://mathoverflow.net/a/179167/8430) provides a detailed proof of the two-block case of alternating minimization (block coordinate descent). In particular, as mentioned in my comment, the convergence follows thanks ultimately to having unique subproblem solutions.


