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arXiv · 2301.05002

Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka-{\L}ojasiewicz Property without Global Lipschitz Assumptions

Xiaoxi Jia·Christian Kanzow·Patrick Mehlitz
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Abstract

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-{\L}ojasiewicz 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 non-quadratic 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-{\L}ojasiewicz property is sufficient to recover these convergence results.

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Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz. 2023-01-12. Convergence Analysis of the Proximal Gradient Method in the Presence of the Kurdyka-{\L}ojasiewicz Property without Global Lipschitz Assumptions. https://arxiv.org/abs/2301.05002

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