A course on optimization and best approximation
Holmes, Richard B.
A course on optimization and best approximation Holmes, Richard B. - Berlin Springer 1972 - viii, 233 p. - Lecture notes in mathematics; 257 .
Table of Contents:
Part I. Preliminaries . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . 1
The Hahn-Banach Theorem . . . . . . . . . . . . . 2
The Separation Theorems . . . . . . . . . . . . . . 4
The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . 7
The Krein-Milman Theorem . . . . . . . . . . . . . . 8
Part II. Theory of Optimization . . . . . . . . . . . . . . .
Convex Functions . . . . . . . . . . . . . . . . . . 14
Directional Derivatives . . . . . . . . . . . . . . 16
Subgradients . . . . . . . . . . . . . . . . . . . . 20
Normal Cones . . . . . . . . . . . . . . . . . . . . 23
Subdifferential Formulas . . . . . . . . . . . . 25
Convex Programs . . . . . . . . . . . . . . . . . . 29
Kuhn-Tucker Theory . . . . . . . . . . . . . . . . . 32
Lagrange Multipliers . . . . . . . . . . . . . . . . 36
Conjugate Functions . . . . . . . . . . . . . . . . 42
Polarity . . . . . . . . . . . . . . . . . . . . . . 48
Dubovitskii-Milyutin Theory . . . . . . . . . . . . 51
An Application . . . . . . . . . . . . . . . . . . . 56
Conjugate Functions and Subdifferentials ...... 58
Distance Functions . . . . . . . . . . . . . . . . . 61
The Fenchel Duality Theorem . . . . . . . . . . . . 65
Some Applications .
Part III. Theory of Best Approximation • . . . . . . . . . .
Characterization of Best Approximations ...... 76
Extremal Representations . . . . . . . . . . . . . . 81
Application to Gaussian Quadrature . . . . . . . . . 88
Haar Subspaces . . . . . . . . . . . . . . . . . . . 91
Chebyshev Polynomials . . . . . . . . . . . . . . . 98
Rotundity . . . . . . . . . . . . . . . . . . . . . 105
Chebyshev Subspaces . . . . . . . . . . . . . . . . 109
Algorithms for Best Approximation . . . . . . . . . 118
Proximinal Sets . . . . . . . . . . . . . . . . . . 123
Theory of Optimization . . . . . . . . . . . . . . .
Convex Functions . . . . . . . . . . . . . . . . . . 14
Directional Derivatives . . . . . . . . . . . . . . 16
Subgradients . . . . . . . . . . . . . . . . . . . . 20
Normal Cones . . . . . . . . . . . . . . . . . . . . 23
Subdifferential Formulas . . . . . . . . . . . . 25
Convex Programs . . . . . . . . . . . . . . . . . . 29
Kuhn-Tucker Theory . . . . . . . . . . . . . . . . . 32
Lagrange Multipliers . . . . . . . . . . . . . . . . 36
Conjugate Functions . . . . . . . . . . . . . . . . 42
Polarity . . . . . . . . . . . . . . . . . . . . . . 48
Dubovitskii-Milyutin Theory . . . . . . . . . . . . 51
An Application . . . . . . . . . . . . . . . . . . . 56
Conjugate Functions and Subdifferentials ...... 58
Distance Functions . . . . . . . . . . . . . . . . . 61
The Fenchel Duality Theorem . . . . . . . . . . . . 65
Some Applications . . . . . . . . . . . . . . . . . 7O
Part IV. Comments on the Problems . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Part V" Selected Special Topics s . . . . . . . . . . . . . . . .
E-spaces . . . . . . . . . . . . . . . . . . . . . . . 145
Metric Projections . . . . . . . . . . . . . . . . . . 157
Optimal Estimation . . . . . . . . . . . . . . . . . . 177
Quasi-Solutions . . . . . . . . . . . . . . . . . . . 203
Generalized Inverses . . . . . . . . . . . . . . . . . 214
This is a paperback book published by Spring-Verlag in 1972 as Volume 257 of Lecture Notes in Mathematics. Chapters include: Preliminaries; Theory of optimization; Theory of best approximation; Comments on the problems, and Selected special topics. There are numerous examples and equations. Holmes published this book while at Purdue University, Indiana.
9783540057642
Functional analysis
Approximation theory
Mathematical optimization
510.8 / H6C6
A course on optimization and best approximation Holmes, Richard B. - Berlin Springer 1972 - viii, 233 p. - Lecture notes in mathematics; 257 .
Table of Contents:
Part I. Preliminaries . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . 1
The Hahn-Banach Theorem . . . . . . . . . . . . . 2
The Separation Theorems . . . . . . . . . . . . . . 4
The Alaoglu-Bourbaki Theorem . . . . . . . . . . . . 7
The Krein-Milman Theorem . . . . . . . . . . . . . . 8
Part II. Theory of Optimization . . . . . . . . . . . . . . .
Convex Functions . . . . . . . . . . . . . . . . . . 14
Directional Derivatives . . . . . . . . . . . . . . 16
Subgradients . . . . . . . . . . . . . . . . . . . . 20
Normal Cones . . . . . . . . . . . . . . . . . . . . 23
Subdifferential Formulas . . . . . . . . . . . . 25
Convex Programs . . . . . . . . . . . . . . . . . . 29
Kuhn-Tucker Theory . . . . . . . . . . . . . . . . . 32
Lagrange Multipliers . . . . . . . . . . . . . . . . 36
Conjugate Functions . . . . . . . . . . . . . . . . 42
Polarity . . . . . . . . . . . . . . . . . . . . . . 48
Dubovitskii-Milyutin Theory . . . . . . . . . . . . 51
An Application . . . . . . . . . . . . . . . . . . . 56
Conjugate Functions and Subdifferentials ...... 58
Distance Functions . . . . . . . . . . . . . . . . . 61
The Fenchel Duality Theorem . . . . . . . . . . . . 65
Some Applications .
Part III. Theory of Best Approximation • . . . . . . . . . .
Characterization of Best Approximations ...... 76
Extremal Representations . . . . . . . . . . . . . . 81
Application to Gaussian Quadrature . . . . . . . . . 88
Haar Subspaces . . . . . . . . . . . . . . . . . . . 91
Chebyshev Polynomials . . . . . . . . . . . . . . . 98
Rotundity . . . . . . . . . . . . . . . . . . . . . 105
Chebyshev Subspaces . . . . . . . . . . . . . . . . 109
Algorithms for Best Approximation . . . . . . . . . 118
Proximinal Sets . . . . . . . . . . . . . . . . . . 123
Theory of Optimization . . . . . . . . . . . . . . .
Convex Functions . . . . . . . . . . . . . . . . . . 14
Directional Derivatives . . . . . . . . . . . . . . 16
Subgradients . . . . . . . . . . . . . . . . . . . . 20
Normal Cones . . . . . . . . . . . . . . . . . . . . 23
Subdifferential Formulas . . . . . . . . . . . . 25
Convex Programs . . . . . . . . . . . . . . . . . . 29
Kuhn-Tucker Theory . . . . . . . . . . . . . . . . . 32
Lagrange Multipliers . . . . . . . . . . . . . . . . 36
Conjugate Functions . . . . . . . . . . . . . . . . 42
Polarity . . . . . . . . . . . . . . . . . . . . . . 48
Dubovitskii-Milyutin Theory . . . . . . . . . . . . 51
An Application . . . . . . . . . . . . . . . . . . . 56
Conjugate Functions and Subdifferentials ...... 58
Distance Functions . . . . . . . . . . . . . . . . . 61
The Fenchel Duality Theorem . . . . . . . . . . . . 65
Some Applications . . . . . . . . . . . . . . . . . 7O
Part IV. Comments on the Problems . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Part V" Selected Special Topics s . . . . . . . . . . . . . . . .
E-spaces . . . . . . . . . . . . . . . . . . . . . . . 145
Metric Projections . . . . . . . . . . . . . . . . . . 157
Optimal Estimation . . . . . . . . . . . . . . . . . . 177
Quasi-Solutions . . . . . . . . . . . . . . . . . . . 203
Generalized Inverses . . . . . . . . . . . . . . . . . 214
This is a paperback book published by Spring-Verlag in 1972 as Volume 257 of Lecture Notes in Mathematics. Chapters include: Preliminaries; Theory of optimization; Theory of best approximation; Comments on the problems, and Selected special topics. There are numerous examples and equations. Holmes published this book while at Purdue University, Indiana.
9783540057642
Functional analysis
Approximation theory
Mathematical optimization
510.8 / H6C6