Novel Accelerated Cyclic Iterative Approximation for Hierarchical Variational Inequalities Constrained by Multiple-Set Split Common Fixed-Point Problems

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed-point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, with special cases being able to be found in many important engineering practical applications, such as image recognizing, signal processing, and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.