  We propose an adaptive iteratively linearized finite element method (AILFEM)
in the context of strongly monotone nonlinear operators in Hilbert spaces. The
approach combines adaptive mesh-refinement with an energy-contractive
linearization scheme (e.g., the Ka\v{c}anov method) and a norm-contractive
algebraic solver (e.g., an optimal geometric multigrid method). Crucially, a
novel parameter-free algebraic stopping criterion is designed and we prove that
it leads to a uniformly bounded number of algebraic solver steps. Unlike
available results requiring sufficiently small adaptivity parameters to ensure
even plain convergence, the new AILFEM algorithm guarantees full R-linear
convergence for arbitrary adaptivity parameters. Thus, parameter-robust
convergence is guaranteed. Moreover, for sufficiently small adaptivity
parameters, the new adaptive algorithm guarantees optimal complexity, i.e.,
optimal convergence rates with respect to the overall computational cost and,
hence, time.
