A Barzilai–Borwein-Like Iterative Half Thresholding Algorithm for the $$L_{1/2}$$L1/2 Regularized Problem

In this paper, we propose a Barzilai–Borwein-like iterative half thresholding algorithm for the $$L_{1/2}$$L1/2 regularized problem. The algorithm is closely related to the iterative reweighted minimization algorithm and the iterative half thresholding algorithm. Under mild conditions, we verify that any accumulation point of the sequence of iterates generated by the algorithm is a first-order stationary point of the $$L_{1/2}$$L1/2 regularized problem. We also prove that any accumulation point is a local minimizer of the $$L_{1/2}$$L1/2 regularized problem when additional conditions are satisfied. Furthermore, we show that the worst-case iteration complexity for finding an $$\varepsilon $$ε scaled first-order stationary point is $$O(\varepsilon ^{-2})$$O(ε-2). Preliminary numerical results show that the proposed algorithm is practically effective.