  An abstract convergence theorem for a class of generalized descent methods
that explicitly models relative errors is proved. The convergence theorem
generalizes and unifies several recent abstract convergence theorems. It is
applicable to possibly non-smooth and non-convex lower semi-continuous
functions that satisfy the Kurdyka--Lojasiewicz (KL) inequality, which
comprises a huge class of problems. Most of the recent algorithms that
explicitly prove convergence using the KL inequality can cast into the abstract
framework in this paper and, therefore, the generated sequence converges to a
stationary point of the objective function. Additional flexibility compared to
related approaches is gained by a descent property that is formulated with
respect to a function that is allowed to change along the iterations, a generic
distance measure, and an explicit/implicit relative error condition with
respect to finite linear combinations of distance terms. As an application of
the gained flexibility, the convergence of a block coordinate variable metric
version of iPiano (an inertial forward--backward splitting algorithm) is
proved, which performs favorably on an inpainting problem with a
Mumford--Shah-like regularization from image processing.
