On Boyd et al.'s convergence analysis of ADMM: What are the additional assumptions needed to show primal variable convergence?

Question

Please refer to Boyd et al.'s convergence analysis of ADMM (Chapter 3 and Appendix A). In Section 3.2.1, it say that $x^k$ and $z^k$ need not converge to optimal values, although such results can be shown under additional assumptions.

I have not been able to find these assumptions within the text or elsewhere.

I need the explicit assumptions required for the primal variable convergence, and preferably also a reference to where they are used to show primal variable convergence in ADMM.

Thanks.

Answer 1

I am encoutering the same problem. I find two materials discussing such "additional assumption". In these two materials, their assumptions are in similar form so I think they should be correct (I didn't check the proof). One is in a book of Theorem 16.1, with link in https://www.cis.upenn.edu/~cis5150/ws-book-IIb.pdf

The other one is in a PhD thesis of Assumption 2.3, with link in https://www.research-collection.ethz.ch/handle/20.500.11850/311505