Synopses & Reviews
• The book introduces the key ideas behind practical nonlinear optimization. • Computational finance--an increasingly popular area of mathematics degree programmes--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 behaviour 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. Audience The book is aimed at lecturers and students (undergraduate and postgraduate) in mathematics, computational finance and related subjects. It is also useful for researchers and practitioners who need a good introduction to nonlinear optimization.
Review
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.)
Synopsis
This instructive book introduces the key ideas behind practical nonlinear optimization, accompanied by computational examples and supporting software. It combines computational finance with an important class of numerical techniques.
Table of Contents
List of Figures List of Tables Preface 1: PORTFOLIO OPTIMIZATION 1. Nonlinear optimization 2. Portfolio return and risk 3. Optimizing two-asset portfolios 4. Minimimum risk for three-asset portfolios 5. Two- and three-asset minimum-risk solutions 6. A derivation of the minimum risk problem 7. Maximum return problems 2: ONE-VARIABLE OPTIMIZATION 1. Optimality conditions 2. The bisection method 3. The secant method 4. The Newton method 5. Methods using quadratic or cubic interpolation 6. Solving maximum-return problems 3: OPTIMAL PORTFOLIOS WITH N ASSETS 1. Introduction 2. The basic minimum-risk problem 3. Minimum risk for specified return 4. The maximum return problem 4: UNCONSTRAINED OPTIMIZATION IN N VARIABLES 1. Optimality conditions 2. Visualising problems in several variables 3. Direct search methods 4. Optimization software and examples 5: THE STEEPEST DESCENT METHOD 1. Introduction 2. Line searches 3. Convergence of the steepest descent method 4. Numerical results with steepest descent 5. Wolfe's convergence theorem 6. Further results with steepest descent 6: THE NEWTON METHOD 1. Quadratic models and the Newton step 2. Positive definiteness and Cholesky factors 3. Advantages and drawbacks of Newton's method 4. Search directions from indefinite Hessians 5. Numerical results with the Newton method 7: QUASINEWTON METHODS 1. Approximate second derivative information 2. Rauk-two updates for the inverse Hessian 3. Convergence of quasi-Newton methods 4. Numerical results with quasi-Newton methods 5. The rank-one update for the inverse Hessian 6. Updating estimates of the Hessian 8: CONJUGATE GRADIENT METHODS 1. Conjugate gradients and quadratic functions 2. Conjugate gradients and general functions 3. Convergence of conjugate gradient methods 4. Numerical results with conjugate gradients 5. The truncated Newton method 9: OPTIMAL PORTFOLIOS WITH RESTRICTIONS 1. Introduction 2. Transformations to exclude short-selling 3. Results from Minrisk2u and Maxret2u 4. Upper and lower limits on invested fractions 10: LARGER-SCALE PORTFOLIOS 1. Introduction 2. Portfolios with increasing numbers of assets 3. Time-variation of optimal portfolios 4. Performance of optimized portfolios 11: DATA-FITTING AND THE GAUSS-NEWTON METHOD 1. Data fitting problems 2. The Gauss-Newton method 3. Least-squares in time series analysis 4. Gauss-Newton applied to time series 5. Least-squares forms of minimum-risk problems 6. Gauss-Newton applied to Minrisk1 and Minrisk2 12: EQUALITY CONSTRAINED OPTIMIZATION 1. Portfolio problems with equality constraints 2. Optimality conditions 3. A worked example 4. Interpretation of Lagrange multipliers 5. Some example problems 13: LINEAR EQUALITY CONSTRAINTS 1. Equality constrained quadratic programming 2. Solving minimum-risk problems as EQPs 3. Reduced-gradient methods 4. Projected gradient methods 5. Results with methods for linear constraints 14: PENALTY FUNCTION METHODS 1. Introduction 2. Penalty functions 3. The Augmented Lagrangian 4. Results with P-SUMT and AL-SUMT 5. Exact penalty functions 15: SEQUENTIAL QUADRATIC PROGRAMMING 1. Introduction 2. Quadratic/linear models 3. SQP methods based on penalty functions 4. Results with AL-SQP 5. SQP line searches and the Maratos effect 16: FURTHER PORTFOLIO PROBLEMS 1. Including transaction costs 2. A re-balancing problem 3. A sensitivity problem 17: INEQUALITY CONSTRAINED OPTIMIZATION 1. Portfolio problems with inequality constraints 2. Optimality conditions 3. Transforming inequalities to equalities 4. Transforming inequalities to simple bounds 5. Example problems 18: EXTENDING EQUALITY-CONSTRAINT METHODS 1. Inequality constrained quadratic programming 2. Reduced gradients for inequality constraints 3. Penalty functions for inequality constraints 4. AL-SUMT for inequality constraints 5. SQP for inequality constraints 6. Results with P-SUMT, AL-SUMT and AL-SQP 19: BARRIER FUNCTION METHODS 1. Introduction 2. Barrier functions 3. Numerical results with B-SUMT 20: INTERIOR POINT METHODS 1. Introduction 2. Approximate solutions of problem B-NLP 3. An interior point algorithm 4. Numerical results with IPM 21: DATA FITTING USING INEQUALITY CONSTRAINTS 1. Minimax approximation 2. Trend channels for time series data 22: PORTFOLIO RE-BALANCING AND OTHER PROBLEMS 1. Re-balancing allowing for transaction costs 2. Downside risk 3. Worst-case analysis 23: GLOBAL UNCONSTRAINED OPTIMIZATION 1. Introduction 2. Multi-start methods 3. DIRECT 4. Numerical examples 5. Global optimization in portfolio selection Appendix References Index