Augmented Lagrangian (AL) methods are a well known class of algorithms for solving constrained optimization problems. They have been extended to the solution of saddle-point systems of linear equations. We study an AL (SPAL) algorithm for unsymmetric saddle-point systems and derive convergence and semi-convergence properties, even when the system is singular. At each step, our SPAL requires the exact solution of a linear system of the same size but with an SPD (2,2) block. To improve efficiency, we introduce an inexact SPAL algorithm. We establish its convergence properties under reasonable assumptions. Specifically, we use a gradient method, known as the Barzilai-Borwein (BB) method, to solve the linear system at each iteration. We call the result the augmented Lagrangian BB (SPALBB) algorithm and study its convergence. Numerical experiments on test problems from Navier-Stokes equations and coupled Stokes-Darcy flow show that SPALBB is more robust and efficient than BICGSTAB and GMRES. SPALBB often requires the least CPU time, especially on large systems.

翻译：增广拉格朗日（AL）方法是求解约束优化问题的一类经典算法。该方法已被推广用于求解线性方程组的鞍点系统。我们研究了一种面向非对称鞍点系统的增广拉格朗日（SPAL）算法，并推导了其收敛性与半收敛性性质，即使系统为奇异系统也适用。在每一步中，SPAL算法需要精确求解一个规模相同但具有SPD（2,2）块的线性系统。为提升效率，我们引入了一种非精确SPAL算法，并在合理假设下建立了其收敛性。具体而言，我们采用梯度方法——即Barzilai-Borwein（BB）方法——来求解每次迭代中的线性系统，并将所得算法称为增广拉格朗日BB（SPALBB）算法，进而研究其收敛性。基于Navier-Stokes方程和耦合Stokes-Darcy流测试问题的数值实验表明，SPALBB比BICGSTAB和GMRES具有更强的鲁棒性和更高的效率。SPALBB通常所需CPU时间最少，尤其在大规模系统上表现突出。