We propose a novel algorithmic framework of Variable Metric Over-Relaxed Hybrid Proximal Extra-gradient (VMOR-HPE) method with a global convergence guarantee for the maximal monotone operator inclusion problem. Its iteration complexities and local linear convergence rate are provided, which theoretically demonstrate that a large over-relaxed step-size contributes to accelerating the proposed VMOR-HPE as a byproduct. Specifically, we find that a large class of primal and primal-dual operator splitting algorithms are all special cases of VMOR-HPE. Hence, the proposed framework offers a new insight into these operator splitting algorithms. In addition, we apply VMOR-HPE to the Karush-Kuhn-Tucker (KKT) generalized equation of linear equality constrained multi-block composite convex optimization, yielding a new algorithm, namely nonsymmetric Proximal Alternating Direction Method of Multipliers with a preconditioned Extra-gradient step in which the preconditioned metric is generated by a blockwise Barzilai-Borwein line search technique (PADMM-EBB). We also establish iteration complexities of PADMM-EBB in terms of the KKT residual. Finally, we apply PADMM-EBB to handle the nonnegative dual graph regularized low-rank representation problem. Promising results on synthetic and real datasets corroborate the efficacy of PADMM-EBB.
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