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Conjugate Gradient Methods

In: Modern Numerical Nonlinear Optimization

Author

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  • Neculai Andrei

    (Center for Advanced Modeling and Optimization)

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.

Suggested Citation

  • Neculai Andrei, 2022. "Conjugate Gradient Methods," Springer Optimization and Its Applications, in: Modern Numerical Nonlinear Optimization, chapter 5, pages 169-260, Springer.
    • Handle: RePEc:spr:spochp:978-3-031-08720-2_5
    DOI: 10.1007/978-3-031-08720-2_5
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