Inexact Barzilai-Borwein method for saddle point problems

This paper considers the inexact Barzilai-Borwein algorithm applied to saddle point problems. To this aim, we study the convergence properties of the inexact Barzilai-Borwein algorithm for symmetric positive definite linear systems. Suppose that gk and g̃k are the exact residual and its approximation of the linear system at the k-th iteration, respectively. We prove the R-linear convergence of the algorithm if ‖g̃k − gk‖ ≤ η‖g̃k‖ for some small η > 0 and all k. To adapt the algorithm for solving saddle point problems, we also extend the Rlinear convergence result to the case when the right hand term ‖g̃k‖ is replaced by ‖g̃k−1‖. Although our theoretical analyses cannot provide a good estimate to the parameter η, in practice we find that η can be as large as the one in the inexact Uzawa algorithm. Further numerical experiments show that the inexact Barzilai-Borwein algorithm performs well for the tested saddle point problems.