An increasing-angle property of the conjugate gradient method and the implementation of large-scale minimization algorithms with line searches

Abstract

The search direction in unconstrained minimization algorithms for large-scale problems is usually computed as an iterate of the preconditioned) conjugate gradient method applied to the minimization of a local quadratic model. In line-search procedures this direction is required to satisfy an angle condition that says that the angle between the negative gradient at the current point and the direction is bounded away from π/2. In this paper, it is shown that the angle between conjugate gradient iterates and the negative gradient strictly increases as far as the conjugate gradient algorithm proceeds. Therefore, the interruption of the conjugate gradient sub-algorithm when the angle condition does not hold is theoretically justified. Copyright © 2002 John Wiley & Sons, Ltd.