Hybrid CG-Like Algorithm for Nonlinear Equations and Image Restoration.

This paper introduces a hybrid spectral-conjugate gradient (SCG) method to solve nonlinear monotone operator equations efficiently. The proposed method incorporates a hybrid parameter that encompasses the Polak--Ribière--Polyak (PRP), Liu-Storey (LS), Fletcher-Reeves (FR), and conjugate descent (CD) methods as particular instances. Additionally, we derive the spectral parameter to ensure that the search direction adheres to the sufficient descent condition. The search direction is also designed to be bounded, and under specific conditions, we demonstrate that the sequence produced by our hybrid SCG algorithm converges toward a solution. Furthermore, to underscore the effectiveness of our proposed method, we conducted extensive numerical experiments comparing its performance against that of existing algorithms. These experiments were based on a selection of benchmark nonlinear monotone operator equations, highlighting our proposed algorithm's superior efficiency and potential in practical applications. [ABSTRACT FROM AUTHOR]