## Abubakar Bakoji Muhammad, Christiane Tammer, Aliyu Muhammed Awwal, Rosalind Elster, A Dai-Liao-like projection method for solving convex constrained nonlinear monotone equations and minimizing the $\ell_1$-regularized problem

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DOI: 10.23952/asvao.3.2021.3.02
Volume 3, Issue 3, 31 December 2021, Pages 259-279

Abstract. In this paper, a three-term derivative-free method for solving a nonlinear system of equations with convex constraints is proposed. In addition, by reformulating an $\ell_1$-regularized problem into a nonlinear system of equations, the proposed method is applicable to solving signal recovery and image deblurring problems. Our method is based on the projection technique of Solodov and Svaiter (1998) by incorporating a quasi-Newton-like direction with the Dai-Liao conjugate gradient parameter. The proposed method is matrix-free and the search direction satisfies a certain descent condition. Under the assumption that the underlying function is monotone and Lipschitzian, the global convergence of the proposed method is established. Preliminary numerical experiments on some large-scale nonlinear system of equations with convex constraints show that the proposed method is efficient. Furthermore, we apply the proposed method to the $\ell_1$-regularization problem in compressive sensing.

Abubakar Bakoji Muhammad, Christiane Tammer, Aliyu Muhammed Awwal, Rosalind Elster, A Dai-Liao-like projection method for solving convex constrained nonlinear monotone equations and minimizing the $\ell_1$-regularized problem, Appl. Set-Valued Anal. Optim. 3 (2021), 259-279.