Two inertial-relaxed hybridized CG projection-based algorithms for solving nonlinear monotone equations applied in image restoration

Abstract

In this paper, we introduce two inertial-relaxed hybridized conjugate gradient projection-based algorithms for solving nonlinear monotone equations. Compared to traditional inertial-relaxed algorithms for solving such nonlinear equations, our algorithms utilize three-step iterative information to generate inertial iterates. The search directions of two proposed algorithms each include a flexible non-zero vector and feature their own self-adaptive hybrid structure to enhance their adaptability. Both search directions satisfy the sufficient descent and trust region properties, eliminating the need for additional conditions. In the theoretical analysis of global convergence and convergence rate results, both proposed algorithms exhibit similarities. We begin by using the first proposed algorithm to establish the global convergence without requiring the Lipschitz continuity assumption. Furthermore, under additional assumptions, we demonstrate the convergence rate of this algorithm. To evaluate their effectiveness, we conduct comparative tests against existing algorithms using nonlinear equations. Moreover, the practicality of the two proposed algorithms is shown through applications on image restoration problems.