Preface (p. XVI)
This book has grown out of undergraduate and postgraduate lecture courses given at the University of Hertfordshire and the University of Bergamo. Its pri- mary focus is on numerical methods for nonlinear optimization. Such methods can be applied to many practical problems in science, engineering and manage- ment: but, to provide a coherent secondary theme, the applications considered here are all drawn from financial mathematics. (This puts the book in good company since many classical texts in mathematics also dealt with commer- cial arithmetic.) In particular, the examples and case studies are concerned with portfolio selection and with time-series problems such as fitting trend- lines and trend-channels to market data.
The content is intended to be suitable for final-year undergraduate students in mathematics (or other subjects with a high mathematical or computational content) and exercises are provided at the end of most sections. However the book should also be useful for postgraduate students and for other researchers and practitioners who need a foundation for work involving development or application of optimization algorithms.
It is assumed that readers have an un- derstanding of the algebra of matrices and vectors and of the Taylor and Mean Value Theorems in several variables. Prior experience of using computational methods for problems such as solving systems of linear equations is also de- sirable, as is familiarity with iterative algorithms (e.g., Newton's method for nonlinear equations in one variable).
The approach adopted in this book is a blend of the practical and theoretical. A description and derivation is given for most of the currently popular methods for continuous nonlinear optimization. For each method, important conver- gence results are outlined (and we provide proofs when it seems instructive to do so). This theoretical material is complemented by numerical illustrations which give a flavour of how the methods perform in practice.
It is not always obvious where to draw the line between general descriptions of algorithms and the more subtle and detailed considerations relating to re- search issues. The particular themes and emphases in this book have grown out of the author's experience at the Numerical Optimization Centre (NOC). This was established in 1968 at the Hatfield College of Technology (predeces- sor of the University of Hertfordshire) as a centre for research in optimization techniques.
Since its foundation, the NOC has been engaged in algorithm de- velopment and consultancy work (mostly in the aerospace industry). The NOC staff has included, at various times, Laurence Dixon, Ed Hersom, Joanna Go- mulka, Sean McKeown and Zohair Maany who have all made contributions to the state-of-the-art in fields as diverse as quasi-Newton methods, sequential quadratic programming, nonlinear least-squares, global optimization, optimal control and automatic differentiation.
The computational results quoted in this book have been obtained using a Fortran90 module called SAMPO. This is based on the NOC's OPTIMA library - a suite of subroutines for different types of minimization problem. The name SAMPO serves as an acronym for Software And Methods for Portfolio Optimization. (However, it is also worth mentioning that The Sampo appears in Finnish mythology as a magical machine which grinds out endless supplies of corn, salt and gold. |
