Издательство Springer, 1972, -240 pp.
The course for which these notes were originally prepared was a one-semester graduate level course at Purdue University, dealing with optimization in general and best approximation in particular. The prerequisites were modest: a semester's worth of functional analysis together with the usual background required for such a course. A few prerequisite results of special importance have been gathered together for ease of reference in Part I.
My general aim was to present an interesting field of application of functional analysis. Although the tenor of the course is consequently rather theoretical, I made some effort to include a few fairly concrete examples, and to bring under consideration problems of genuine practical interest. Examples of such problems are convex programs (§'s 11-13), calculus of variations (§17), minimum effort control (§21), quadrature formulas (§24), construction of "good" approximations to functions (§'s 26 and 29), optimal estimation from inadequate data (§33), solution of various ill-posed linear systems (§'s 34-3S). Indeed, the bulk of the notes is devoted to a presentation of the theoretical background needed for the study of such problems.
No attempt has been made to provide encyclopedic coverage of the various topics. Rather I tried only to show some highlights, techniques, and examples in each of the several areas studied. Should a reader be stimulated to pursue a particular topic further, he will hopefully find an adequate sample of the pertinent literature included in the bibliographies. (Note that in addition to the main bibliography between Parts IV and V, each section in Part V has its own special set of references appended.)
Preliminaries.
Notation.
The Hahn-Banach Theorem.
S. The Separation Theorems.
The Alaoglu-Bourbaki Theorem.
The Krein-Milman Theorem.
Theory of Optimization.
Convex Functions.
Directional Derivatives.
Subgradients.
Normal Cones.
Subdifferential Formulas.
Convex Programs.
Kuhn-Tucker Theory.
Lagrange Multipliers.
Conjugate Functions.
Polarity.
Dubovitskii-Milyutin Theory.
An Application.
Conjugate Functions and Subdifferentials.
Distance Functions.
The Fenchel Duality Theorem.
Some Applications.
Theory of Best Approximation.
Characterization of Best Approximations.
Extremal Representations.
Application to Gaussian Quadrature.
Haar Subspaces.
Chebyshev Polynomials.
Rotundity.
Chebyshev Subspaces.
Algorithms for Best Approximation.
Proximinal Sets.
Comments on the Problems.
Selected Special Topics.
E-spaces.
Metric Projections.
Optimal Estimation.
Quasi-Solutions.
Generalized Inverses.
The course for which these notes were originally prepared was a one-semester graduate level course at Purdue University, dealing with optimization in general and best approximation in particular. The prerequisites were modest: a semester's worth of functional analysis together with the usual background required for such a course. A few prerequisite results of special importance have been gathered together for ease of reference in Part I.
My general aim was to present an interesting field of application of functional analysis. Although the tenor of the course is consequently rather theoretical, I made some effort to include a few fairly concrete examples, and to bring under consideration problems of genuine practical interest. Examples of such problems are convex programs (§'s 11-13), calculus of variations (§17), minimum effort control (§21), quadrature formulas (§24), construction of "good" approximations to functions (§'s 26 and 29), optimal estimation from inadequate data (§33), solution of various ill-posed linear systems (§'s 34-3S). Indeed, the bulk of the notes is devoted to a presentation of the theoretical background needed for the study of such problems.
No attempt has been made to provide encyclopedic coverage of the various topics. Rather I tried only to show some highlights, techniques, and examples in each of the several areas studied. Should a reader be stimulated to pursue a particular topic further, he will hopefully find an adequate sample of the pertinent literature included in the bibliographies. (Note that in addition to the main bibliography between Parts IV and V, each section in Part V has its own special set of references appended.)
Preliminaries.
Notation.
The Hahn-Banach Theorem.
S. The Separation Theorems.
The Alaoglu-Bourbaki Theorem.
The Krein-Milman Theorem.
Theory of Optimization.
Convex Functions.
Directional Derivatives.
Subgradients.
Normal Cones.
Subdifferential Formulas.
Convex Programs.
Kuhn-Tucker Theory.
Lagrange Multipliers.
Conjugate Functions.
Polarity.
Dubovitskii-Milyutin Theory.
An Application.
Conjugate Functions and Subdifferentials.
Distance Functions.
The Fenchel Duality Theorem.
Some Applications.
Theory of Best Approximation.
Characterization of Best Approximations.
Extremal Representations.
Application to Gaussian Quadrature.
Haar Subspaces.
Chebyshev Polynomials.
Rotundity.
Chebyshev Subspaces.
Algorithms for Best Approximation.
Proximinal Sets.
Comments on the Problems.
Selected Special Topics.
E-spaces.
Metric Projections.
Optimal Estimation.
Quasi-Solutions.
Generalized Inverses.