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16 - Convex Optimization

from Part III - Nonlinear Equations and Optimization

Published online by Cambridge University Press:  29 September 2022

Abner J. Salgado
Affiliation:
University of Tennessee, Knoxville
Steven M. Wise
Affiliation:
University of Tennessee, Knoxville
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Summary

This chapter serves two purposes: it introduces several essential concepts of linear and nonlinear functional analysis that will be used in subsequent chapters and, as an illustration of them, studies the problem of unconstrained minimization of a convex functional. All the necessary notions of existence, uniqueness, and optimality conditions are presented and analyzed. Preconditioned gradient descent methods for strongly convex, locally Lipschitz smooth objectives in infinite dimensions are then presented and analyzed. A general framework to show linear convergence in this setting is then presented. The preconditioned steepest descent with exact and approximate line searches are then analyzed using the same framework. Finally, the application of Newton’s method to the Euler equations is discussed. The local convergence is shown, and how to achieve global convergence is briefly discussed.

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Chapter
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Classical Numerical Analysis
A Comprehensive Course
, pp. 451 - 506
Publisher: Cambridge University Press
Print publication year: 2022

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  • Convex Optimization
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book: Classical Numerical Analysis
  • Online publication: 29 September 2022
  • Chapter DOI: https://doi.org/10.1017/9781108942607.021
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  • Convex Optimization
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book: Classical Numerical Analysis
  • Online publication: 29 September 2022
  • Chapter DOI: https://doi.org/10.1017/9781108942607.021
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Convex Optimization
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book: Classical Numerical Analysis
  • Online publication: 29 September 2022
  • Chapter DOI: https://doi.org/10.1017/9781108942607.021
Available formats
×