A nonmonotone line search technique for Newton’s method. (English) Zbl 0616.65067

Grippo, L.; Lampariello, F.; Lucidi, S.

SIAM J. Numer. Anal. 23, 707-716 (1986).

Newton’s method for finding the unrestricted minimum of a twice continuously differentiable function \(f\) is considered. To ensure convergence, a line search technique must be applied. Usually this is done such that a monotonic decrease in value of \(f\) is achieved. This – on the other hand – is known to eventually slow the rate of convergence. The authors propose a step-size rule, which may be considered as a generalization of Armijo’s rule in as much as a step is accepted if a certain improvement in function value is obtained not w.r.t. the last step but to one of the last \(M\) steps. By this a rule is obtained which is shown to preserve the global convergence properties without enforcing monotonic decrease in function values. By a number of examples it is demonstrated that there may be some gain in computational effort compared with the usual Armijo rule.

Reviewer: Rainer Hettich (Trier)

MSC:

65K05	Numerical mathematical programming methods
90C30	Nonlinear programming

Keywords:

unconstrained minimization; Newton’s method; line search technique; rate of convergence; global convergence; monotonic decrease; Armijo rule