An Iteratively Reweighted Method for Sparse Optimization on Nonconvex $\ell_{p}$ Ball
This addresses sparse optimization for researchers in computational mathematics, but it is incremental as it builds on existing reweighted methods for nonconvex constraints.
The paper tackles nonconvex optimization problems with ℓp-ball constraints by proposing an iteratively reweighted method that solves weighted ℓ1-ball projection subproblems, proving convergence to a first-order stationary point and demonstrating effectiveness in numerical experiments.
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.