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An abstract convergence theorem for a class of generalized descent methodsthat explicitly models relative errors is proved. The convergence theoremgeneralizes and unifies several recent abstract convergence theorems. It isapplicable to possibly non-smooth and non-convex lower semi-continuousfunctions that satisfy the Kurdyka--Lojasiewicz (KL) inequality, whichcomprises a huge class of problems. Most of the recent algorithms thatexplicitly prove convergence using the KL inequality can cast into the abstractframework in this paper and, therefore, the generated sequence converges to astationary point of the objective function. Additional flexibility compared torelated approaches is gained by a descent property that is formulated withrespect to a function that is allowed to change along the iterations, a genericdistance measure, and an explicit/implicit relative error condition withrespect to finite linear combinations of distance terms. As an application ofthe gained flexibility, the convergence of a block coordinate variable metricversion of iPiano (an inertial forward--backward splitting algorithm) isproved, which performs favorably on an inpainting problem with aMumford--Shah-like regularization from image processing.

Unifying abstract inexact convergence theorems and block coordinate variable metric iPiano