Xiangyu Yang

Hao Wang, Xiangyu Yang, Yichen Zhu

This paper presents a novel hybrid algorithm for minimizing the sum of a continuously differentiable loss function and a nonsmooth, possibly nonconvex, sparse regularization function. The proposed method alternates between solving a reweighted $\ell_1$-regularized subproblem and performing an inexact subspace Newton step. The reweighted $\ell_1$-subproblem allows for efficient closed-form solutions via the soft-thresholding operator, avoiding the computational overhead of proximity operator calculations. As the algorithm approaches an optimal solution, it maintains a stable support set, ensuring that nonzero components stay uniformly bounded away from zero. It then switches to a perturbed regularized Newton method, further accelerating the convergence. We prove global convergence to a critical point and, under suitable conditions, demonstrate that the algorithm exhibits local linear and quadratic convergence rates. Numerical experiments show that our algorithm outperforms existing methods in both efficiency and solution quality across various model prediction problems.

Hao Wang, Xiangyu Yang, Wei Jiang

This paper is intended to solve the nonconvex $\ell_{p}$-ball constrained nonlinear optimization problems. An iteratively reweighted method is proposed, which solves a sequence of weighted $\ell_{1}$-ball projection subproblems. At each iteration, the next iterate is obtained by moving along the negative gradient with a stepsize and then projecting the resulted point onto the weighted $\ell_{1}$ ball to approximate the $\ell_{p}$ ball. Specifically, if the current iterate is in the interior of the feasible set, then the weighted $\ell_{1}$ ball is formed by linearizing the $\ell_{p}$ norm at the current iterate. If the current iterate is on the boundary of the feasible set, then the weighted $\ell_{1}$ ball is formed differently by keeping those zero components in the current iterate still zero. In our analysis, we prove that the generated iterates converge to a first-order stationary point. Numerical experiments demonstrate the effectiveness of the proposed method.