Inexact proximal methods for weakly convex functions

Pham Duy Khanh (), Boris S. Mordukhovich (), Vo Thanh Phat () and Dat Ba Tran ()
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Pham Duy Khanh: Ho Chi Minh City University of Education
Boris S. Mordukhovich: Wayne State University
Vo Thanh Phat: University of North Dakota
Dat Ba Tran: Wayne State University

Journal of Global Optimization, 2025, vol. 91, issue 3, No 8, 646 pages

Abstract: Abstract This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term. A general framework for finding zeros of a continuous mapping is derived from our previous paper on this subject to establish convergence properties of the inexact proximal point method when the smooth term is vanished and of the inexact proximal gradient method when the smooth term satisfies a descent condition. The inexact proximal point method achieves global convergence with constructive convergence rates when the Moreau envelope of the objective function satisfies the Kurdyka–Łojasiewicz (KL) property. Meanwhile, when the smooth term is twice continuously differentiable with a Lipschitz continuous gradient and a differentiable approximation of the objective function satisfies the KL property, the inexact proximal gradient method achieves the global convergence of iterates with constructive convergence rates.

Keywords: Inexact proximal methods; Weakly convex functions; Forward-backward envelopes; Kurdyka–Łojasiewicz property; Global convergence; Linear convergence rates; Proximal points; 90C30; 90C52; 49M05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10898-024-01460-7

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