Unifying abstract inexact convergence theorems for descent methods and block coordinate variable metric iPiano

An abstract convergence theorem for a class of descent method that explicitly models relative errors is proved. The convergence theorem generalizes and unifies several recent abstract convergence theorems, and is applicable to possibly non-smooth and non-convex lower semi-continuous functions that satisfy the Kurdyka– Lojasiewicz inequality, which comprises a huge class of problems. The descent property is measured with respect to a function that is allowed to change along the iterations, which makes block coordinate and variable metric methods amenable to the abstract convergence theorem. As a particularly algorithm, the convergence of a block coordinate variable metric version of iPiano (an inertial forward– backward splitting algorithm) is proved. The newly introduced algorithms perform favorable on an inpainting problem with a Mumford–Shah-like regularization from image processing.