I have read these two papers by Svaiter and Solodov. The first one, published in 1999 (http://pages.cs.wisc.edu/~solodov/solsva99Teps.pdf) presents an error criterion for the hybrid proximal extragradient method, and it has the $\varepsilon$-enlargement on the criterion condition.
The second one, published later, in 2000 (http://pages.cs.wisc.edu/~solodov/solsva00Bregman.pdf) expands the theory for Bregman functions, however doesn't take the $\varepsilon$-enlargement in consideration.
I'm trying to find the first reference which treats the hybrid proximal extragradient method taking both the enlargement and the Bregman functions in consideration, but had no luck.
I loked in Svaiter and Solodov's pages, also looked the 2000 paper on the internet to see which papers cited that but didn't find what I want... I'm running out of ideas here.
Anyone happens to know when was the first paper which took those two things in concern released? Or any idea of how is the best way to look for it?
The paper:
An accelerated non-Euclidean hybrid proximal extragradient-type algorithm for convex-concave saddle-point problems
http://www.optimization-online.org/DB_HTML/2015/09/5113.html
Deals with the HPE for Bregmain with epsilon-enlargements.
Best regards, Benar F. Svaiter