Convergence analysis of nonmonotone proximal gradient methods under local Lipschitz continuity and Kurdyka--Łojasiewicz property

Abstract: The proximal gradient method is a standard approach for solving composite minimization problems in which the objective function is the sum of a continuously differentiable function and a lower semicontinuous, extended-valued function. The traditional convergence theory for both monotone and nonmonotone variants replies heavily on the assumption of global Lipschitz continuity of the gradient of the smooth part of the objective function. Recent work has shown that monotone proximal gradient methods converge globally only when the local (rather than global) Lipschitz continuity is assumed, provided that the Kurdyka--{\L}ojasiewicz (KL) property holds. However, these results have not been extended to nonmonotone proximal gradient (NPG) methods. In this manuscript, we consider two types of NPG methods: those combined with the average line search and the max line search, respectively. By partitioning indices into two subsets, one of which aims to achieve a sufficient decrease in the functional sequence, we establish global convergence and rate-of-convergence results using the local Lipschitz continuity and the KL property, without requiring boundedness of the iterates. While finalizing this work, we noticed that [18] presented analogous results for the NPG method with average line search, but with a different partitioning strategy. Together, we confidently conclude that the convergence theory of the NPG method is independent on index partitioning choices.