A modified Peaceman-Rachford splitting method with correction steps for three-block convex problem

In this paper we consider a three-block separable convex minimization problem, where the objective function is the sum of three individual convex functions, each with separate variables. Motivated by the prediction-correction framework, we propose a modified Peaceman-Rachford splitting method (PRSM) with correction steps. The most recent variables from the PRSM decomposition are utilized to update all variables, while specific correction steps are employed to refine the output. Through theoretical analysis, we guarantee the global convergence of the proposed algorithm and establish its worst-case convergence rates, both in the ergodic and non-ergodic sense. Finally, we apply our method to robust principal component analysis (PCA) model using both synthetic data and real-world data, demonstrating the feasibility and effectiveness of the proposed algorithm.