. The book introduces the key ideas behind practical nonlinear optimization.

"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. TOC: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.- The conjugate gradient method.- Optimal portfolios with restrictions.- Larger-scale portfolio problems.- Data-fitting and the Gauss-Newton method.- Equality constrained optimization.- Methods for 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.- Global unconstrained optimization."

List of FiguresList of TablesPreface 1: PORTFOLIO OPTIMIZATION1. Nonlinear optimization2. Portfolio return and risk3. Optimizing two-asset portfolios4. Minimimum risk for three-asset portfolios5. Two- and three-asset minimum-risk solutions6. A derivation of the minimum risk problem7. Maximum return problems2: ONE-VARIABLE OPTIMIZATION1. Optimality conditions2. The bisection method3. The secant method4. The Newton method5. Methods using quadratic or cubic interpolation6. Solving maximum-return problems3: OPTIMAL PORTFOLIOS WITH N ASSETS1. Introduction2. The basic minimum-risk problem3. Minimum risk for specified return4. The maximum return problem4: UNCONSTRAINED OPTIMIZATION IN N VARIABLES1. Optimality conditions2. Visualising problems in several variables3. Direct search methods4. Optimization software and examples5: THE STEEPEST DESCENT METHOD1. Introduction2. Line searches3. Convergence of the steepest descent method4. Numerical results with steepest descent5. Wolfe's convergence theorem6. Further results with steepest descent6: THE NEWTON METHOD1. Quadratic models and the Newton step2. Positive definiteness and Cholesky factors3. Advantages and drawbacks of Newton's method4. Search directions from indefinite Hessians5. Numerical results with the Newton method7: QUASINEWTON METHODS1. Approximate second derivative information2. Rauk-two updates for the inverse Hessian3. Convergence of quasi-Newton methods4. Numerical results with quasi-Newton methods5. The rank-one update for the inverse Hessian6. Updating estimates of the Hessian8: CONJUGATE GRADIENT METHODS1. Conjugate gradients and quadratic functions2. Conjugate gradients and general functions3. Convergence of conjugate gradient methods4.Numerical results with conjugate gradients5. The truncated Newton method9: OPTIMAL PORTFOLIOS WITH RESTRICTIONS1. Introduction2. Transformations to exclude short-selling3. Results from Minrisk2u and Maxret2u4. Upper and lower limits on invested fractions10: LARGER-SCALE PORTFOLIOS1. Introduction2. Portfolios with increasing numbers of assets3. Time-variation of optimal portfolios4. Performance of optimized portfolios11: DATA-FITTING AND THE GAUSS-NEWTON METHOD1. Data fitting problems2. The Gauss-Newton method3. Least-squares in time series analysis4. Gauss-Newton applied to time series5. Least-squares forms of minimum-risk problems6. Gauss-Newton applied to Minrisk1 and Minrisk212: EQUALITY CONSTRAINED OPTIMIZATION1. Portfolio problems with equality constraints2. Optimality conditions3. A worked example4. Interpretation of Lagrange multipliers5. Some example problems13: LINEAR EQUALITY CONSTRAINTS1. Equality constrained quadratic programming2. Solving minimum-risk problems as EQPs3. Reduced-gradient methods4. Projected gradient methods5. Results with methods for linear constraints14: PENALTY FUNCTION METHODS1. Introduction2. Penalty functions3. The Augmented Lagrangian4. Results with P-SUMT and AL-SUMT5. Exact penalty functions15: SEQUENTIAL QUADRATIC PROGRAMMING1. Introduction2. Quadratic/linear models3. SQP methods based on penalty functions4. Results with AL-SQP5. SQP line searches and the Maratos effect16: FURTHER PORTFOLIO PROBLEMS1. Including transaction costs2. A re-balancing problem3. A sensitivity problem17: INEQUALITY CONSTRAINED OPTIMIZATION1. Portfolio problems with inequality constraints2. Optimality conditions3. Transforming inequalities to equalities4. Transforming inequalities to simple bounds5. Example