An inertial generalized iteratively reweighted $ \ell_1 $ algorithm for nonconvex and nonsmooth optimization

Myeongmin Kang; Myeongmin Kang

doi:10.3934/math.2026674

AIMS Mathematics

2026, Volume 11, Issue 6: 16414-16447. doi: 10.3934/math.2026674

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Research article

An inertial generalized iteratively reweighted $ \ell_1 $ algorithm for nonconvex and nonsmooth optimization

Myeongmin Kang ^,

Department of Mathematics, Chungnam National University, Daejeon, 34134, Korea

Received: 27 February 2026 Revised: 12 May 2026 Accepted: 18 May 2026 Published: 09 June 2026
MSC : 90C26, 65K10, 49M37

We study a class of nonconvex and nonsmooth optimization problems arising in sparse recovery and related applications, which are often addressed using iteratively reweighted $ \ell_1 $ (IRL1)-type algorithms. Classical IRL1 methods are typically developed under a Lipschitz gradient assumption, which may limit their applicability. In this paper, we propose a generalized iteratively reweighted $ \ell_1 $ algorithm with inertial extrapolation (GIRL1E), where the generalization is based on a co-coercivity condition imposed on the smooth component, thereby allowing a broader class of problems to be treated. By integrating inertial extrapolation into the generalized IRL1 framework, we establish global convergence of the proposed algorithm to a critical point of the objective function. In particular, we prove a sufficient descent property of an associated Lyapunov function and the convergence of the entire sequence without assuming convexity. Numerical experiments on compressive sensing-based signal recovery and image deblurring demonstrated that GIRL1E consistently achieves improved practical performance compared with existing IRL1 methods.
- iteratively reweighted $ \ell_1 $ algorithm,
- inertial extrapolation,
- co-coercivity,
- nonconvex optimization,
- Kurdyka–Łojasiewicz property
Citation: Myeongmin Kang. An inertial generalized iteratively reweighted $ \ell_1 $ algorithm for nonconvex and nonsmooth optimization[J]. AIMS Mathematics, 2026, 11(6): 16414-16447. doi: 10.3934/math.2026674

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Abstract

We study a class of nonconvex and nonsmooth optimization problems arising in sparse recovery and related applications, which are often addressed using iteratively reweighted $ \ell_1 $ (IRL1)-type algorithms. Classical IRL1 methods are typically developed under a Lipschitz gradient assumption, which may limit their applicability. In this paper, we propose a generalized iteratively reweighted $ \ell_1 $ algorithm with inertial extrapolation (GIRL1E), where the generalization is based on a co-coercivity condition imposed on the smooth component, thereby allowing a broader class of problems to be treated. By integrating inertial extrapolation into the generalized IRL1 framework, we establish global convergence of the proposed algorithm to a critical point of the objective function. In particular, we prove a sufficient descent property of an associated Lyapunov function and the convergence of the entire sequence without assuming convexity. Numerical experiments on compressive sensing-based signal recovery and image deblurring demonstrated that GIRL1E consistently achieves improved practical performance compared with existing IRL1 methods.

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