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.

In this paper, a novel variational model is proposed for image segmentation via joint restoration of images corrupted by blurring and Rician noise. The proposed model is built upon the piecewise constant Mumford–Shah framework and combines an appropriate data fidelity term with nonconvex total variation (NTV) regularization. The NTV regularization effectively denoises homogeneous regions while accurately preserving object boundaries to facilitate robust segmentation. To solve the resulting nonconvex optimization problem, a proximal alternating minimization algorithm is employed. In addition, an iteratively reweighted ${\ell }_1$ algorithm and the alternating direction method of multipliers are adopted to efficiently handle the corresponding subproblems. Numerical experiments demonstrate the effectiveness of the proposed model in achieving accurate and robust segmentation performance when compared with several state-of-the-art methods.