Abstract
It is generally acknowledged that the conjugate gradient (CG) method achieves global convergence—with at most a linear convergence rate—because CG formulas are generated by linear approximations of the objective functions. The quadratically convergent results are very limited. We introduce a new PRP method in which the restart strategy is also used. Moreover, the method we developed includes not only n-step quadratic convergence but also both the function value information and gradient value information. In this paper, we will show that the new PRP method (with either the Armijo line search or the Wolfe line search) is both linearly and quadratically convergent. The numerical experiments demonstrate that the new PRP algorithm is competitive with the normal CG method.
Highlights
Consider min f ðxÞ; x2
Li and Tian [34] proved that the modifying the PRP (MPRP) method with a restart strategy exhibited quadratic convergence under certain inexact line searches and suitable assumptions
By applying the conclusion of [34], we prove that the new conjugate gradient (CG) method obtains global convergence for general functions and n-step quadratic convergence for uniformly convex functions
Summary
Consider min f ðxÞ; x2
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