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Zbl 0616.65067
Grippo, L.; Lampariello, F.; Lucidi, S.
A nonmonotone line search technique for Newton's method. (English)
[J] SIAM J. Numer. Anal. 23, 707-716 (1986). ISSN 0036-1429; ISSN 1095-7170/e

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.
[R.Hettich]

MSC 2000:: *65K05 Mathematical programming (numerical methods)
90C30 Nonlinear programming

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

Cited in: Zbl 1220.65060 Zbl 1115.65067 Zbl 1068.65073 Zbl 1073.90024 Zbl 1060.90071 Zbl 0858.90122 Zbl 0822.90129 Zbl 0763.65042 Zbl 0732.65055

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