  The ultimate goal of any numerical scheme for partial differential equations
(PDEs) is to compute an approximation of user-prescribed accuracy at
quasi-minimal computational time. To this end, algorithmically, the standard
adaptive finite element method (AFEM) integrates an inexact solver and nested
iterations with discerning stopping criteria balancing the different error
components. The analysis ensuring optimal convergence order of AFEM with
respect to the overall computational cost critically hinges on the concept of
R-linear convergence of a suitable quasi-error quantity. This work tackles
several shortcomings of previous approaches by introducing a new proof
strategy. First, the algorithm requires several fine-tuned parameters in order
to make the underlying analysis work. A redesign of the standard line of
reasoning and the introduction of a summability criterion for R-linear
convergence allows us to remove restrictions on those parameters. Second, the
usual assumption of a (quasi-)Pythagorean identity is replaced by the
generalized notion of quasi-orthogonality from [Feischl, Math. Comp., 91
(2022)]. Importantly, this paves the way towards extending the analysis to
general inf-sup stable problems beyond the energy minimization setting.
Numerical experiments investigate the choice of the adaptivity parameters.
