Luwei Bai

Kexin Li, Luwei Bai, Xiao Wang et al.

Anderson acceleration is an effective technique for enhancing the efficiency of fixed-point iterations; however, analyzing its convergence in nonsmooth settings presents significant challenges. In this paper, we investigate a class of nonsmooth optimization algorithms characterized by the active manifold identification property. This class includes a diverse array of methods such as the proximal point method, proximal gradient method, proximal linear method, proximal coordinate descent method, Douglas-Rachford splitting (or the alternating direction method of multipliers), and the iteratively reweighted $\ell_1$ method, among others. Under the assumption that the optimization problem possesses an active manifold at a stationary point, we establish a local R-linear convergence rate for the Anderson-accelerated algorithm. Our extensive numerical experiments further highlight the robust performance of the proposed Anderson-accelerated methods.

Luwei Bai, Yaohua Hu, Hao Wang et al.

In this paper, we consider a class of non-convex and non-smooth sparse optimization problems, which encompass most existing nonconvex sparsity-inducing terms. We show the second-order optimality conditions only depend on the nonzeros of the stationary points. We propose two damped iterative reweighted algorithms including the iteratively reweighted $\ell_1$ algorithm (DIRL$_1$) and the iteratively reweighted $\ell_2$ (DIRL$_2$) algorithm, to solve these problems. For DIRL$_1$, we show the reweighted $\ell_1$ subproblem has support identification property so that DIRL$_1$ locally reverts to a gradient descent algorithm around a stationary point. For DIRL$_2$, we show the solution map of the reweighted $\ell_2$ subproblem is differentiable and Lipschitz continuous everywhere. Therefore, the map of DIRL$_1$ and DIRL$_2$ and their inverse are Lipschitz continuous, and the strict saddle points are their unstable fixed points. By applying the stable manifold theorem, these algorithms are shown to converge only to local minimizers with randomly initialization when the strictly saddle point property is assumed.