Optimization Problems

ISBN-13: 9780387901435 / Angielski / Miękka / 1975 / 356 str.

L. Collatz

Opis

The German edition of this book, first published in 1966, has been quite popular; we did not, however, consider publishing an English edition because a number of excellent textbooks in this field already exist. In recent years, how- ever, the wish was frequently expressed that, especially, the description of the relationships between optimization and other subfields of mathematics, which is not to be found in this form in other texts, might be made available to a wider readership; so it was with this in mind that, be- latedly, a translation was undertaken after all. Since the appearance of the German edition, the field of optimization has continued to develop at an unabated rate. A completely current presentation would have required a total reworking of the book; unfortunately, this was not possible. For example, we had to ignore the extensive progress which has been made in the development of numerical methods which do not require convexity assumptions to find local maxima and minima of non-linear optimization problems. These methods are also applicable to boundary value, and other, problems. Many new results, both of a numerical and a theoretical na- ture, which are especially relevant to applications, are to be found in the areas of optimal contol and integer optimiza- tion.

Spis treści

I. Linear Optimization.- §1. Introduction.- §2. Linear Optimization and Polyhedra.- §3. Vertex Exchange and the Simplex Method.- §4. Algorithmic Implementation of the Simplex Method.- §5. Dual Linear Optimization Problems.- II. Convex Optimization.- §6. Introduction.- §7. A Characterization of Minimal Solutions for Convex Optimization.- §8. Convex Optimization for Differentiable Functions.- §9. Convex Optimization with Affine Linear Constraints.- §10. The Numerical Treatment of Convex Optimization Problems.- III. Quadratic Optimization.- §11. Introduction.- §12. The Kuhn-Tucker Theorem and Applications..- §13. Duality for Quadratic Optimization.- §14. The Numerical Treatment of Quadratic Optimization Problems.- IV. Tchebychev Approximation and Optimization.- § 15. Introduction.- §16. Discrete Linear Tchebychev Approximation.- §17. Further Types of Approximation Problems.- V. Elements of Game Theory.- §18. Matrix Games (Two Person Zero Sum Games).- §19. n-Person Games.- Problems.