nn-step quadratic convergence of the MPRP method with a restart strategy

It is well-known that the PRP conjugate gradient method with exact line search is globally and linearly convergent. If a restart strategy is used, the convergence rate of the method can be an nn-step superlinear/quadratic convergence. Recently, Zhang et al. [L. Zhang, W. Zhou, D.H. Li, A descent modified Polak–Ribière–Polyak conjugate gradient method and its global convergence, IMA J. Numer. Anal. 26 (2006) 629–640] developed a modified PRP (MPRP) method that is globally convergent if an inexact line search is used. In this paper, we investigate the convergence rate of the MPRP method with inexact line search. We first show that the MPRP method with Armijo line search or Wolfe line search is linearly convergent. We then show that the MPRP method with a restart strategy still retains nn-step superlinear/quadratic convergence if the initial steplength is appropriately chosen. We also do some numerical experiments. The results show that the restart MPRP method does converge quadratically. Moreover, it is more efficient than the non-restart method.