This undergraduate textbook introduces students of science and engineering to the fascinating field of optimization. It is a unique book that brings together the subfields of mathematical programming, variational calculus, and optimal control, thus giving students an overall view of all aspects of optimization in a single reference. As a primer on optimization, its main goal is to provide a succinct and accessible introduction to linear programming, nonlinear programming, numerical optimization algorithms, variational problems, dynamic programming, and optimal control. Prerequisites have been kept to a minimum, although a basic knowledge of calculus, linear algebra, and differential equations is assumed. There are numerous examples, illustrations, and exercises throughout the text, making it an ideal book for self-study. Applied mathematicians, physicists, engineers, and scientists will all find this introduction to optimization extremely useful.
Author(s): Pablo Pedregal
Series: Texts in Applied Mathematics
Edition: 1
Publisher: Springer
Year: 2003
Language: English
Pages: 256
Contents......Page 10
Series Preface......Page 7
Preface......Page 8
1. Some examples......Page 12
2. The Mathematical Setting......Page 17
3. The Variety of optimization problems......Page 25
4. Exercises......Page 26
1. Introduction......Page 34
2. The simplex method......Page 41
3. Duality......Page 54
4. Some practical issues......Page 60
5. Integer programming......Page 70
6. Exercises......Page 74
1. Model problem......Page 78
2. Lagrange multipliers......Page 80
3. Karush–Kuhn–Tucker optimality conditions......Page 90
4. Convexity......Page 97
5. Sufficiency of the KKT conditions......Page 106
6. Duality and convexity......Page 113
7. Exercises......Page 118
1. Introduction......Page 122
2. Line search methods......Page 124
3. Gradient methods......Page 127
4. Conjugate gradient methods......Page 131
5. Approximation under constraints......Page 135
6. Final remarks......Page 142
7. Exercises......Page 143
1. Introduction......Page 148
2. The Euler–Lagrange Equation: examples......Page 151
3. The Euler–Lagrange Equation: justification......Page 164
4. Natural boundary conditions......Page 168
5. Variational problems under integral and pointwise restrictions......Page 171
6. Summary of restrictions for variational problems......Page 179
7. Variational problems of different order......Page 184
8. Dynamic programming: Bellman’s equation......Page 189
9. Some basic ideas on the numerical approximation......Page 196
10. Exercises......Page 201
1. Introduction......Page 206
2. Multipliers and the hamiltonian......Page 208
3. Pontryagin’s principle......Page 215
4. Another format......Page 235
5. Some comments on the numerical approximation......Page 237
6. Exercises......Page 243
References......Page 248
C......Page 252
G......Page 253
N......Page 254
S......Page 255
Z......Page 256
