Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- 15 Solution of Nonlinear Equations
- 16 Convex Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
16 - Convex Optimization
from Part III - Nonlinear Equations and Optimization
Published online by Cambridge University Press: 29 September 2022
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- 15 Solution of Nonlinear Equations
- 16 Convex Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
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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- Information
- Classical Numerical AnalysisA Comprehensive Course, pp. 451 - 506Publisher: Cambridge University PressPrint publication year: 2022