Analysis of a variable metric block coordinate method under proximal errors

This paper focuses on an inexact block coordinate method designed for nonsmooth optimization, where each block-subproblem is solved by performing a bounded number of steps of a variable metric proximal–gradient method with linesearch. We improve on the existing analysis for this algorithm in the nonconvex setting, showing that the iterates converge to a stationary point of the objective function even when the proximal operator is computed inexactly, according to an implementable inexactness condition. The result is obtained by introducing an appropriate surrogate function that takes into account the inexact evaluation of the proximal operator, and assuming that such function satisfies the Kurdyka–Łojasiewicz inequality. The proof technique employed here may be applied to other new or existing block coordinate methods suited for the same class of optimization problems.