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The Newton and Cauchy perspectives on computational nonlinear optimization. (English) Zbl 0809.65062
An overview of model-based and metric-based methods for unconstrained nonlinear optimization is given as they cover the main methods of the subject. Firstly, the two methods are discussed for themselves, but also connections and relationships are drawn between them. The role of Newton’s and of Cauchy’s method is emphasized as some kind of complementary roots in the two classes.
Secondly, the Cauchy-Newton framework as a unifying point of view is established showing their relations with other methods such as quasi- Newton, conjugate gradients, affine reduced methods, etc. Also space for hinting at recent and future research directions is reserved.
This survey article addresses a broader readership such that technical details are dropped. They can be found in the “accompanying” monograph of the author [The Newton-Cauchy framework: A unified approach to unconstrained nonlinear minimization. Lecture Notes in Computer Science, 769, Berlin: Springer-Verlag (ISBN 3-540-57671-1). xii, 101 p. (1994)].

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
L-BFGS; minpack
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