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The book introduces the key ideas behind practical nonlinear optimization. Computational finance - an increasingly popular area of mathematics degree programs - is combined here with the study of an important class of numerical techniques. The financial content of the book is designed to be relevant and interesting to specialists. However, this material - which occupies about one-third of the text - is also sufficiently accessible to allow the book to be used on optimization courses of a more general nature. The essentials of most currently popular algorithms are described, and their performance is demonstrated on a range of optimization problems arising in financial mathematics. Theoretical convergence properties of methods are stated, and formal proofs are provided in enough cases to be instructive rather than overwhelming. Practical behavior of methods is illustrated by computational examples and discussions of efficiency, accuracy and computational costs. Supporting software for the examples and exercises is available (but the text does not require the reader to use or understand these particular codes). The author has been active in optimization for over thirty years in algorithm development and application and in teaching and research supervision.
Table des matières
Portfolio Optimization.- One-Variable Optimization.- Optimal Portfolios with N Assets.- Unconstrained Optimization in N Variables.- The Steepest Descent Method.- The Newton Method.- Quasi-Newton Methods.- Conjugate Gradient Methods.- Optimal Portfolios with Restrictions.- Larger-Scale Portfolios.- Data-Fitting & The Gauss-Newton Method.- Equality Constrained Optimization.- Linear Equality Constraints.- Penalty Function Methods.- Sequential Quadratic Programming.- Further Portfolio Problems.- Inequality Constrained Optimization.- Extending Equality-Constraint Methods to Inequalities.- Barrier Function Methods.- Interior Point Methods.- Data Fitting Using Inequality Constraints.- Portfolio Re-Balancing and other Problems.- Global Unconstrained Optimization.
Résumé
The book introduces the key ideas behind practical nonlinear optimization. Computational finance – an increasingly popular area of mathematics degree programs – is combined here with the study of an important class of numerical techniques. The financial content of the book is designed to be relevant and interesting to specialists. However, this material – which occupies about one-third of the text – is also sufficiently accessible to allow the book to be used on optimization courses of a more general nature. The essentials of most currently popular algorithms are described, and their performance is demonstrated on a range of optimization problems arising in financial mathematics. Theoretical convergence properties of methods are stated, and formal proofs are provided in enough cases to be instructive rather than overwhelming. Practical behavior of methods is illustrated by computational examples and discussions of efficiency, accuracy and computational costs. Supporting software for the examples and exercises is available (but the text does not require the reader to use or understand these particular codes). The author has been active in optimization for over thirty years in algorithm development and application and in teaching and research supervision.
Texte suppl.
From the reviews:
"The book is intended for readers who have an understanding of linear algebra, and the Taylor mean value theorems in several variables. It presents numerical approaches to nonlinear optimization which typically have been applied for 30-40 years to practical problems in science and engineering. … Summing up, one could say that the author had an interesting idea to present the classical algorithmic methods while teaching contemporary portfolio theory." (Leszek S. Zaremba, Zentralblatt MATH, Vol. 1083 (9), 2006)
"This book is a timely and very useful addition to the literature on practical mathematical optimization. … Indeed, the outstanding and almost unique aspect of the work is the thorough integration of the methods with application … . The text is very readable and will be easy to teach from. The passion of the author for, and his fascination with, optimization are conveyed to the reader. His mathematically flavored poems, scattered throughout the text, entertain and amuse … ." (Jan A. Snyman, SIAM Review, Vol. 48 (1), 2006)
"This book contains many computational examples demonstrating the practical behavior of the proposed methods and their application to practical financial problems." (Mathematical Reviews, Zimmermann, K.)
Commentaire
From the reviews:
"The book is intended for readers who have an understanding of linear algebra, and the Taylor mean value theorems in several variables. It presents numerical approaches to nonlinear optimization which typically have been applied for 30-40 years to practical problems in science and engineering. ... Summing up, one could say that the author had an interesting idea to present the classical algorithmic methods while teaching contemporary portfolio theory." (Leszek S. Zaremba, Zentralblatt MATH, Vol. 1083 (9), 2006)
"This book is a timely and very useful addition to the literature on practical mathematical optimization. ... Indeed, the outstanding and almost unique aspect of the work is the thorough integration of the methods with application ... . The text is very readable and will be easy to teach from. The passion of the author for, and his fascination with, optimization are conveyed to the reader. His mathematically flavored poems, scattered throughout the text, entertain and amuse ... ." (Jan A. Snyman, SIAM Review, Vol. 48 (1), 2006)
"This book contains many computational examples demonstrating the practical behavior of the proposed methods and their application to practical financial problems." (Mathematical Reviews, Zimmermann, K.)
