Resolvent-Free Inclusion Problems with Applications

In this paper, we propose a resolvent-free and projection-free iterative algorithm to solve monotone inclusion problems. The proposed method employs a double inertial extrapolation strategy, in which two distinct inertial steps are used to construct two extrapolated points, together with a new self-adaptive step-size for selecting the inertial parameter in the proposed algorithm. This combination provides an effective strategy to incorporate information from two extrapolated directions without the metric projections and resolvent of an operator. Under suitable assumptions, we establish the strong convergence of the proposed sequence to a solution of the monotone inclusion problem. The obtained convergence result is further applied to minimax and critical point problems. Moreover, numerical experiments in finite and infinite dimensional spaces are presented to compare the proposed method with some existing resolvent-free and inertial schemes. The numerical results demonstrate that the proposed method achieves faster convergence in terms of the number of iterations and provides improved reconstruction performance, with higher SNR and lower MSE for the considered image restoration problems. Further, applications to image reconstruction and compressive sensing provide the practical effectiveness of the proposed approach.