Nonlinear Optimization with Financial Applications

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 problems2: 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 problems3: OPTIMAL PORTFOLIOS WITH N ASSETS 1. Introduction 2. The basic minimum-risk problem 3. Minimum risk for specified return 4. 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. Introduction 2. 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 step 2. Positive definiteness and Cholesky factors3. Advantages and drawbacks of Newton's method4. Search directions from indefinite Hessians5. Numerical results with the Newton method7: QUASINEWTON METHODS 1. Approximate second derivative information 2. 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 METHODS 1. 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. Introduction 2. Transformations to exclude short-selling3. Results from Minrisk2u and Maxret2u 4. Upper and lower limits on invested fractions10: LARGER-SCALE PORTFOLIOS1. Introduction 2. 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 example 4. 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-SQP 5. SQP line searches and the Maratos effect16: FURTHER PORTFOLIO PROBLEMS1. Including transaction costs 2. A re-balancing problem3. A sensitivity problem17: INEQUALITY CONSTRAINED OPTIMIZATION1. Portfolio problems with inequality constraints2. Optimality conditions3. Transforming inequalities to equalities 4. Transforming inequalities to simple bounds5. Example