Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka–Łojasiewicz Property Without Global Lipschitz Assumptions

We consider a composite optimization problem where the sum of a continuously differentiable and a merely lower semicontinuous function has to be minimized. The proximal gradient algorithm is the classical method for solving such a problem numerically. The corresponding global convergence and local rate-of-convergence theory typically assumes, besides some technical conditions, that the smooth function has a globally Lipschitz continuous gradient and that the objective function satisfies the Kurdyka–Łojasiewicz property. Though this global Lipschitz assumption is satisfied in several applications where the objective function is, e.g., quadratic, this requirement is very restrictive in the nonquadratic case. Some recent contributions therefore try to overcome this global Lipschitz condition by replacing it with a local one, but, to the best of our knowledge, they still require some extra condition in order to obtain the desired global and rate-of-convergence results. The aim of this paper is to show that the local Lipschitz assumption together with the Kurdyka–Łojasiewicz property is sufficient to recover these convergence results.