An Inertial Forward-Backward Algorithm with Moving Point Projection Method for Solving Monotone Inclusion Problems

In this paper, we propose and analyze an inertial forward-backward algorithm integrated with a moving point projection method to solve the inclusion problem for the sum of two monotone operators in real Hilbert spaces. By leveraging the properties of the resolvent of a maximally monotone operator, the inverse strongly monotone operator, and carefully controlled scalar parameters alongside the novel moving point projection method, we derive a sequence of nonempty, closed, and convex sets containing the set of zeros of the sum of the two monotone operators. These constructed sets enable the generation of new iterative elements through the proposed moving point projection method, ensuring the resulting sequence remains well-defined. We prove that the sequence generated by this method converges weakly to a zero of the sum of the two monotone operators. This moving point projection method introduces a new approach that enhances convergence efficiency compared to existing methods.

Furthermore, to demonstrate the practical utility of the proposed method, we conduct numerical experiments to compare the convergence rates of our proposed method against existing conventional methods. The results illustrate the computational advantages and improved performance of our approach.