An Analysis of the Barzilai and Borwein Gradient Method for Unsymmetric Linear Equations

The Barzilai and Borwein gradient method does not ensure descent in the objective function at each iteration, but performs better than the classical steepest descent method in practical computations. Combined with the technique of nonmonotone line search etc., such a method has found successful applications in unconstrained optimization, convex constrained optimization and stochastic optimization. In this paper, we give an analysis of the Barzilai and Borwein gradient method for two unsymmetric linear equations with only two variables. Under mild conditions, we prove that the convergence rate of the Barzilai and Borwein gradient method is

Q

-superlinear if the coefficient matrix

A

has the same eigenvalue; if the eigenvalues of

A

are different, then the convergence rate is

R

-superlinear.