An inertial generalized iteratively reweighted ${\ell }_1$ algorithm for nonconvex and nonsmooth optimization

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.