Conjugate Gradient Methods

Neculai Andrei ()
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Neculai Andrei: Center for Advanced Modeling and Optimization

Chapter 5 in Modern Numerical Nonlinear Optimization, 2022, pp 169-260 from Springer

Abstract: Abstract These methods are characterized by very strong convergence properties and modest storage requirements. They are dedicated to solving large-scale unconstrained optimization problems and applications. The history of these methods starts with the researches of Cornelius Lanczos (1950, 1952), Magnus Hestenes (1951, 1955, 1956a, 1956b), Rosser (1953), Forsythe, Hestenes and Rosser (1951), (Stein), and others from the Institute for Numerical Analysis – National Bureau of Standards, Los Angeles, as well as with the researches of Eduard Stiefel (1958) from Eidgenössische Technische Hochschule Zürich. During over 70 years of researches in this area, an impressive number of developments, variants, and algorithms of these methods have appeared. A thorough and detailed presentation of these methods was given by Andrei (2020a). The search direction of these methods is a linear combination of the negative gradient and the previous search direction. The conjugate gradient methods require only the first order derivatives. As we will see, these methods may include the second order information given by an approximation of the Hessian of the minimizing function, thus increasing their convergence properties.

Date: 2022
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DOI: 10.1007/978-3-031-08720-2_5

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