Nonlinear Optimization with Financial Applications

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.

Contains state-of-the-art research findings

Includes supplementary material: sn.pub/extras

Contenu
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