Is there a convergence proof for ADMM applied to biconvex/bilinear problems?

Question

Ok, I've already asked this question in math.stackexchange, but I feel it is more appropriate to ask here (hopefully I am not violating any rules by repeating!). So here it is:

I wonder if there is a local convergence proof for ADMM applied to biconvex problems? More specifically, my problem is as follows:

$$\min_{x,y} f(x) + g(y) + \| y \circ Ax \|_2^2 $$ ,

where $\circ$ denotes Hadamard product and $f(x)$ and $g(y)$ are convex functions. This fits into biconvex setting mentioned in Boyd's paper on distributed ADMM, but without theoretical evidence.

Thank you!

Answer 1

To partially answer my question - there is this paper, but coming with a strong assumption on the multiplier value. Mingyi Hong, Zhi-Quan Luo, Meisam Razaviyayn "Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems"