This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-and-pepper noise removal. Numerical results show the good performance of the proposed algorithms.

中文翻译：

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预处理的Douglas-Rachford型原始对偶方法在解决复合单调包含问题中的应用

本文涉及单调包含，其中包含有限个最大单调算子的和与两个最大单调算子与有界线性算子的并行和。为了解决这种单调包含，我们首先将其转换为在适当乘积空间中三个最大单调算子之和的公式。然后，我们导出了两种有效的迭代算法，它们将部分逆方法与预处理的Douglas-Rachford分裂算法和预处理的近端点算法相结合。此外，我们开发了一种迭代算法，该算法依靠预处理的Douglas-Rachford分裂算法，而不使用部分逆方法。我们仔细分析了所提出算法的理论收敛性。最后，为了证明这些算法的有效性和效率，我们在用于椒盐噪声去除的新型图像去噪模型上进行了数值实验。数值结果表明了所提算法的良好性能。