An inexact augmented Lagrangian algorithm for unsymmetric saddle-point systems

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.