Ebook: Convexity and Well-Posed Problems
Author: Roberto Lucchetti (auth.)
- Tags: Calculus of Variations and Optimal Control, Optimization, Operations Research Mathematical Programming, Functional Analysis
- Series: Canadian Mathematical Society
- Year: 2006
- Publisher: Springer-Verlag New York
- Edition: 1
- Language: English
- pdf
Intended for graduate students especially in mathematics, physics, and
economics, this book deals with the study of convex functions and of
their behavior from the point of view of stability with respect to
perturbations. The primary goal is the study of the problems of
stability and well-posedness, in the convex case. Stability means the
basic parameters of a minimum problem do not vary much if we slightly
change the initial data. Well-posedness means that points with values
close to the value of the problem must be close to actual solutions.
In studying this, one is naturally led to consider perturbations of
both functions and of sets.
The book includes a discussion of numerous topics, including:
* hypertopologies, ie, topologies on the closed subsets of a metric space;
* duality in linear programming problems, via cooperative game theory;
* the Hahn-Banach theorem, which is a fundamental tool for the study of convex functions;
* questions related to convergence of sets of nets;
* genericity and porosity results;
* algorithms for minimizing a convex function.
In order to facilitate use as a textbook, the author has included a
selection of examples and exercises, varying in degree of difficulty.
Robert Lucchetti is Professor of Mathematics at Politecnico di Milano. He has taught this material to graduate students at his own university, as well as the Catholic University of Brescia, and the University of Pavia.
Intended for graduate students especially in mathematics, physics, and
economics, this book deals with the study of convex functions and of
their behavior from the point of view of stability with respect to
perturbations. The primary goal is the study of the problems of
stability and well-posedness, in the convex case. Stability means the
basic parameters of a minimum problem do not vary much if we slightly
change the initial data. Well-posedness means that points with values
close to the value of the problem must be close to actual solutions.
In studying this, one is naturally led to consider perturbations of
both functions and of sets.
The book includes a discussion of numerous topics, including:
* hypertopologies, ie, topologies on the closed subsets of a metric space;
* duality in linear programming problems, via cooperative game theory;
* the Hahn-Banach theorem, which is a fundamental tool for the study of convex functions;
* questions related to convergence of sets of nets;
* genericity and porosity results;
* algorithms for minimizing a convex function.
In order to facilitate use as a textbook, the author has included a
selection of examples and exercises, varying in degree of difficulty.
Robert Lucchetti is Professor of Mathematics at Politecnico di Milano. He has taught this material to graduate students at his own university, as well as the Catholic University of Brescia, and the University of Pavia.
Intended for graduate students especially in mathematics, physics, and
economics, this book deals with the study of convex functions and of
their behavior from the point of view of stability with respect to
perturbations. The primary goal is the study of the problems of
stability and well-posedness, in the convex case. Stability means the
basic parameters of a minimum problem do not vary much if we slightly
change the initial data. Well-posedness means that points with values
close to the value of the problem must be close to actual solutions.
In studying this, one is naturally led to consider perturbations of
both functions and of sets.
The book includes a discussion of numerous topics, including:
* hypertopologies, ie, topologies on the closed subsets of a metric space;
* duality in linear programming problems, via cooperative game theory;
* the Hahn-Banach theorem, which is a fundamental tool for the study of convex functions;
* questions related to convergence of sets of nets;
* genericity and porosity results;
* algorithms for minimizing a convex function.
In order to facilitate use as a textbook, the author has included a
selection of examples and exercises, varying in degree of difficulty.
Robert Lucchetti is Professor of Mathematics at Politecnico di Milano. He has taught this material to graduate students at his own university, as well as the Catholic University of Brescia, and the University of Pavia.
Content:
Front Matter....Pages i-xiv
Convex sets and convex functions: the fundamentals....Pages 1-19
Continuity and ?(X)....Pages 21-30
The derivatives and the subdifferential....Pages 31-54
Minima and quasi minima....Pages 55-77
The Fenchel conjugate....Pages 79-97
Duality....Pages 99-116
Linear programming and game theory....Pages 117-137
Hypertopologies, hyperconvergences....Pages 139-167
Continuity of some operations between functions....Pages 169-183
Well-posed problems....Pages 185-217
Generic well-posedness....Pages 219-248
More exercises....Pages 249-256
Back Matter....Pages 257-305
Intended for graduate students especially in mathematics, physics, and
economics, this book deals with the study of convex functions and of
their behavior from the point of view of stability with respect to
perturbations. The primary goal is the study of the problems of
stability and well-posedness, in the convex case. Stability means the
basic parameters of a minimum problem do not vary much if we slightly
change the initial data. Well-posedness means that points with values
close to the value of the problem must be close to actual solutions.
In studying this, one is naturally led to consider perturbations of
both functions and of sets.
The book includes a discussion of numerous topics, including:
* hypertopologies, ie, topologies on the closed subsets of a metric space;
* duality in linear programming problems, via cooperative game theory;
* the Hahn-Banach theorem, which is a fundamental tool for the study of convex functions;
* questions related to convergence of sets of nets;
* genericity and porosity results;
* algorithms for minimizing a convex function.
In order to facilitate use as a textbook, the author has included a
selection of examples and exercises, varying in degree of difficulty.
Robert Lucchetti is Professor of Mathematics at Politecnico di Milano. He has taught this material to graduate students at his own university, as well as the Catholic University of Brescia, and the University of Pavia.
Content:
Front Matter....Pages i-xiv
Convex sets and convex functions: the fundamentals....Pages 1-19
Continuity and ?(X)....Pages 21-30
The derivatives and the subdifferential....Pages 31-54
Minima and quasi minima....Pages 55-77
The Fenchel conjugate....Pages 79-97
Duality....Pages 99-116
Linear programming and game theory....Pages 117-137
Hypertopologies, hyperconvergences....Pages 139-167
Continuity of some operations between functions....Pages 169-183
Well-posed problems....Pages 185-217
Generic well-posedness....Pages 219-248
More exercises....Pages 249-256
Back Matter....Pages 257-305
....