Convergence of Nonmonotone Proximal Gradient Methods under the Kurdyka-Łojasiewicz Property without a Global Lipschitz Assumption

Christian Kanzow () and Leo Lehmann ()
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Christian Kanzow: University of Würzburg
Leo Lehmann: University of Würzburg

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 1, No 4, 26 pages

Abstract: Abstract We consider the composite minimization problem with the objective function being the sum of a continuously differentiable and a merely lower semicontinuous and extended-valued function. The proximal gradient method is probably the most popular solver for this class of problems. Its convergence theory typically requires that either the gradient of the smooth part of the objective function is globally Lipschitz continuous or the (implicit or explicit) a priori assumption that the iterates generated by this method are bounded. Some recent results show that, without these assumptions, the proximal gradient method, combined with a monotone stepsize strategy, is still globally convergent with a suitable rate-of-convergence under the Kurdyka-Łojasiewicz property. For a nonmonotone stepsize strategy, there exist some attempts to verify similar convergence results, but, so far, they need stronger assumptions. This paper is the first which shows that nonmonotone proximal gradient methods for composite optimization problems share essentially the same nice global and rate-of-convergence properties as its monotone counterparts, still without assuming a global Lipschitz assumption and without an a priori knowledge of the boundedness of the iterates.

Keywords: Composite optimization; Nonsmooth optimization; Proximal gradient method; Kurdyka-Łojasiewicz property; Nonmonotone line search; Global convergence; Linear convergence; 49J52; 90C30 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02762-w

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