A New Algorithm for Split Minimization Using a Relaxed Inertial Gradient Direction Method With Conjugate Gradient and Self‐Adaptive Step Size

ABSTRACT This artcile presents a novel algorithm for solving the split minimization problem that incorporates conjugate gradient direction, self‐adaptive step size, and an alternative inertial method. We propose a generalized step size formulation that eliminates the need for prior knowledge of operator norms and establishes strong convergence results for the algorithm under suitable parameter conditions. The effectiveness of the proposed method is validated through various applications and numerical experiments. Furthermore, we provide comprehensive comparisons with existing image restoration algorithms. The numerical results demonstrate the superior performance of our algorithm compared to other methods.