On the Weak and Strong Convergence of a Conceptual Algorithm for Solving Three Operator Monotone Inclusions

Abstract: In this paper, a conceptual algorithm modifying the forward-backward-half-forward (FBHF) splitting method for solving three operator monotone inclusion problems is investigated. The FBHF splitting method adjusts and improves Tseng's forward-backward-forward (FBF) splitting method when the inclusion problem has a third-part operator that is cocoercive. The FBHF method recovers the FBF iteration (when this aforementioned part is zero), and it also works without using the widely used Lipschitz continuity assumption. The conceptual algorithm proposed in this paper also has those advantages, and it derives two variants (called Method 1 and Method 2) by choosing different projection (forward) steps. Both proposed methods also work efficiently without assuming the Lipschitz continuity and without directly using the cocoercive constant. Moreover, they have the following desired features: (i) very general iterations are derived for both methods, recovering the FBF and the FBHF iterations and allowing possibly larger stepsizes if the projection steps are over-relaxing; and (ii) strong convergence to the best approximation solution of the problem is proved for Method 2. To the best of our knowledge, this is the first time that an FBF-type method converges strongly for finding the best approximation solution of the three operator monotone inclusion.