Numerical experience with the truncated Newton method for unconstrained optimization
Journal of Optimization Theory and Applications
Choosing the forcing terms in an inexact Newton method
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
Theoretical efficiency of an inexact Newton method
Journal of Optimization Theory and Applications
Lancelot: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A)
Lancelot: A FORTRAN Package for Large-Scale Nonlinear Optimization (Release A)
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The Newton-PCG (preconditioned conjugate gradient) like algorithms are usually very efficient. However, their efficiency is mainly supported by the numerical experiments. Recently, a new kind of Newton-PCG-like algorithms is derived in (J. Optim. Theory Appl. 105 (2000) 97; Superiority analysis on truncated Newton method with preconditioned conjugate gradient technique for optimization, in preparation) by the efficiency analysis. It is proved from the theoretical point of view that their efficiency is superior to that of Newton's method for the special cases where Newton's method converges with precise Q-order 2 and α(≥ 2), respectively. In the process of extending such kind of algorithms to the more general case where Newton's method has no fixed convergence order, the first is to get the solutions to the one-dimensional optimization problems with many different parameter values of α. If these problems were solved by numerical method one by one, the computation cost would reduce the efficiency of the Newton-PCG algorithm, and therefore is unacceptable. In this paper, we overcome the difficulty by deriving an analytic expression of the solution to the one-dimensional optimization problem with respect to the parameter α.