Before revision:  Relying on this methodology, we provide three new methods with conceptually simple proofs: (i) an optimized relatively inexact proximal point method, (ii) an extension of the hybrid proximal extragradient method of Monteiro and Svaiter (SIAM Journal on Optimization, 2013), and (iii) an inexact accelerated forward-backward splitting supporting backtracking line-search, and both (ii) and (iii) supporting possibly strongly convex objectives . Finally, we use the methodology for studying a recent inexact variant of the Douglas-Rachford splitting due to Eckstein and Yao (Mathematical Programming, 2018). We showcase and compare the different variants of the accelerated inexact forward-backward methodon a factorization and a total variation problem .
After revision:  Relying on this methodology, we provide three new methods with conceptually simple proofs: (i) an optimized relatively inexact proximal point method, (ii) an extension of the hybrid proximal extragradient method of Monteiro and Svaiter (SIAM Journal on Optimization, 2013), and (iii) an inexact accelerated forward-backward splitting supporting backtracking line-search, and both (ii) and (iii) supporting possibly strongly convex objectives by studying a simple relatively inexact proximal minimization method .