A novel memory gradient method for unconstrained multiobjective optimization problems with application to image restoration

In the field of image restoration, image data fidelity seeks to preserve details to the greatest extent possible, including noise, while edge-preserving image smoothing requires noise suppression. Excessive noise suppression, however, degrades fidelity. Striking an appropriate balance between the two is thus highly challenging. This paper introduces a memory gradient method for solving unconstrained multi-objective optimization problems. The search direction of the proposed method is a combination of the steepest descent directions from the previous two iterations and the search direction of the prior step, ensuring its sufficient descent property. Under the Wolfe line search framework, the global convergence of the method is still guaranteed even if the gradient of the objective function does not satisfy the commonly required Lipschitz continuity condition. Finally, in the first part of the numerical experiments, the proposed algorithm exhibits certain improvements in iterative performance compared with several classical multi-objective gradient algorithms. In the second part, we apply the method to image restoration tasks. The results demonstrate the effectiveness of applying multi-objective gradient algorithms to image restoration, primarily reflected in their ability to restore the quality of certain images in an extremely short time.