An inexact regularized proximal Newton method without line search

Simeon vom Dahl () and Christian Kanzow ()
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Simeon vom Dahl: University of Würzburg
Christian Kanzow: University of Würzburg

Computational Optimization and Applications, 2024, vol. 89, issue 3, No 2, 585-624

Abstract: Abstract In this paper, we introduce an inexact regularized proximal Newton method (IRPNM) that does not require any line search. The method is designed to minimize the sum of a twice continuously differentiable function f and a convex (possibly non-smooth and extended-valued) function $$\varphi $$ φ . Instead of controlling a step size by a line search procedure, we update the regularization parameter in a suitable way, based on the success of the previous iteration. The global convergence of the sequence of iterations and its superlinear convergence rate under a local Hölderian error bound assumption are shown. Notably, these convergence results are obtained without requiring a global Lipschitz property for $$ \nabla f $$ ∇ f , which, to the best of the authors’ knowledge, is a novel contribution for proximal Newton methods. To highlight the efficiency of our approach, we provide numerical comparisons with an IRPNM using a line search globalization and a modern FISTA-type method.

Keywords: Nonsmooth and nonconvex optimization; Global and local convergence; Regularized proximal Newton method; Hölderian local error bound; 49M15; 65K10; 90C26; 90C30; 90C55 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-024-00600-9

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