Inexact Barzilai‐Borwein method for saddle point problems

Abstract

AbstractThis paper considers the inexact Barzilai–Borwein (BB) algorithm applied to saddle point problems. To this aim, we study the convergence properties of the inexact BB 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 kth 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 R‐linear 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 BB algorithm performs well for the tested saddle point problems. Copyright © 2007 John Wiley & Sons, Ltd.