In this paper, we propose a generalized shift-splitting (GSS) preconditioner, along with its two relaxed variants to solve the double saddle point problem (DSPP). The convergence of the associated GSS iterative method is analyzed, and sufficient conditions for its convergence are established. Spectral analyses are performed to derive sharp bounds for the eigenvalues of the preconditioned matrices. Numerical experiments based on examples arising from the PDE-constrained optimization problems demonstrate the effectiveness and robustness of the proposed preconditioners compared with existing state-of-the-art preconditioners.

翻译：本文针对双鞍点问题提出了一种广义移位分裂预条件子及其两种松弛变体。我们分析了相应广义移位分裂迭代法的收敛性，并建立了其收敛的充分条件。通过谱分析，推导了预条件矩阵特征值的尖锐界。基于偏微分方程约束优化问题实例的数值实验表明，与现有先进预条件子相比，所提预条件子具有优越的有效性和鲁棒性。