Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.

单调包含问题对于解决工程问题以及在科学的不同分支中出现的问题至关重要。在本文中，我们提出了一种新颖的两步惯性Douglas-Rachford算法，以基于常规S迭代方法来解决两个最大单调算子之和的单调包含问题（Sahu，DR：S迭代过程的应用约束最小化问题和分裂可行性问题定点理论12（1），187-204（2011）。我们已经研究了所提出算法的收敛性。基于提出的方法，我们开发了一种惯性原始对偶算法来求解包含线性组合和并行和类型算子的混合物的高度结构化的单调包含。最后，我们应用提出的惯性原始对偶算法来解决高度结构化的最小化问题。我们还进行了数值实验来解决广义Heron问题，并将拟议的惯性对偶算法与已知算法的性能进行了比较。