Foundations of Optimization

ISBN-13: 9780387344317 / Angielski / Twarda / 2010 / 442 str.

Osman Guler

Opis

Optimization is everywhere. It is human nature to seek the best option among all that are available. Nature, too, seems to be guided by optimizationmany laws of nature have a variational character. Among geometric gures in the plane with a xed perimeter, the circle has the greatest area. Such isoperim- ric problems involving geometric gures date back to ancient Greece. Fermat's principle, discovered in 1629, stating that the tangent line is horizontal at a minimum point, seems to have in uenced the development of calculus. The proofs of Rolle's theorem and the mean value theorem in calculus use the Weierstrass theorem on the existence of maximizers and minimizers. The - troduction of the brachistochrone problem in 1696 by Johann Bernoulli had a tremendous impact on the development of the calculus of variations and in uenced the development of functional analysis. The variational character of laws of mechanics and optics were discovered in the seventeenth and ei- teenth centuries. Euler and Lagrange forged the foundations of the calculus of variations in the eighteenth century. In the nineteenth century, Riemann used Dirichlet's principle, which has a variational character, in his investigations in complex analysis. The simplex method for linear programming was disc- ered shortly after the advent of computers in the 1940s, and in uenced the subsequent development of mathematical programming. The emergence of the theory of optimal control in the 1950s was in response to the need for contr- ling space vehicles and various industrial processes.

Recenzje

From the reviews:

"This book is an advanced graduate level text on the mathematical theory of optimization. ... filled with many useful examples and counterexamples that provide intuition to support the more formal theorems and proofs. ... There are also numerous exercises for the student. ... The book will also be useful as a reference for researchers working in various areas of optimization." (Brian Borchers, The Mathematical Association of America, October, 2010)

"Güler's book ... is intended for postgraduates or researchers in optimization theory; however, it is also suitable as a textbook in a first-year graduate level course. The book covers a wide range of mathematical tools and results concerning the fundamental principles of optimization in finite-dimensional spaces. ... this book can be a solid reference textbook, useful for graduate students in applied mathematics, economics, engineering, operations research, etc., and, more generally, for anyone wishing to learn the essential mathematical principles of optimization theory." (Giorgio Giorgi, Mathematical Reviews, Issue 2011 e)

"...this textbook presents the state of the art in the theory of continuous optimization in a very transparent and accessible way. Several results are proved in two or more independent ways to gain further insight into the problem structure, and to provide instructors with alternative ways of exposition. The book may be warmly recommended to graduate students and researchers in optimization, operations research, and other fields which apply optimization methods." (Zentralblatt Math, 2010)

Spis treści

Differential Calculus.- Unconstrained Optimization.- Variational Principles.- Convex Analysis.- Structure of Convex Sets and Functions.- Separation of Convex Sets.- Convex Polyhedra.- Linear Programming.- Nonlinear Programming.- Structured Optimization Problems.- Duality Theory and Convex Programming.- Semi-infinite Programming.- Topics in Convexity.- Three Basic Optimization Algorithms.

Blurb

The book gives a detailed and rigorous treatment of the theory of optimization (unconstrained optimization, nonlinear programming, semi-infinite programming, etc.) in finite-dimensional spaces. The fundamental results of convexity theory and the theory of duality in nonlinear programming and the theories of linear inequalities, convex polyhedra, and linear programming are covered in detail. Over two hundred, carefully selected exercises should help the students master the material of the book and give further insight. Some of the most basic results are proved in several independent ways in order to give flexibility to the instructor. A separate chapter gives extensive treatments of three of the most basic optimization algorithms (the steepest-descent method, Newton’s method, the conjugate-gradient method). The first chapter of the book introduces the necessary differential calculus tools used in the book. Several chapters contain more advanced topics in optimization such as Ekeland’s epsilon-variational principle, a deep and detailed study of separation properties of two or more convex sets in general vector spaces, Helly’s theorem and its applications to optimization, etc. The book is suitable as a textbook for a first or second course in optimization at the graduate level. It is also suitable for self-study or as a reference book for advanced readers. The book grew out of author’s experience in teaching a graduate level one-semester course a dozen times since 1993. Osman Guler is a Professor in the Department of Mathematics and Statistics at University of Maryland, Baltimore County. His research interests include mathematical programming, convex analysis, complexity of optimization problems, and operations research.