| List of Figures | |
| List of Tables | |
| Preface | |
| Portfolio Optimization | |
| Nonlinear optimization | |
| Portfolio return and risk | |
| Optimizing two-asset portfolios | |
| Minimimum risk for three-asset portfolios | |
| Two- and three-asset minimum-risk solutions | |
| A derivation of the minimum risk problem | |
| Maximum return problems | |
| One-Variable Optimization | |
| Optimality conditions | |
| The bisection method | |
| The secant method | |
| The Newton method | |
| Methods using quadratic or cubic interpolation | |
| Solving maximum-return problems | |
| Optimal Portfolios With N Assets | |
| Introduction | |
| The basic minimum-risk problem | |
| Minimum risk for specified return | |
| The maximum return problem | |
| Unconstrained Optimization in N Variables | |
| Optimality conditions | |
| Visualising problems in several variables | |
| Direct search methods | |
| Optimization software and examples | |
| The Steepest Descent Method | |
| Introduction | |
| Line searches | |
| Convergence of the steepest descent method | |
| Numerical results with steepest descent | |
| Wolfe''s convergence theorem | |
| Further results with steepest descent | |
| The Newton Method | |
| Quadratic models and the Newton step | |
| Positive definiteness and Cholesky factors | |
| Advantages and drawbacks of Newton''s method | |
| Search directions from indefinite Hessians | |
| Numerical results with the Newton method | |
| Quasinewton Methods | |
| Approximate second derivative information | |
| Rauk-two updates for the inverse Hessian | |
| Convergence of quasi-Newton methods | |
| Numerical results with quasi-Newton methods | |
| The rank-one update for the inverse Hessian | |
| Updating estimates of the Hessian | |
| Conjugate Gradient Methods | |
| Conjugate gradients and quadratic functions | |
| Conjugate gradients and general functions | |
| Convergence of conjugate gradient methods | |
| Numerical results with conjugate gradients | |
| The truncated Newton method | |
| Optimal Portfolios With Restrictions | |
| Introduction | |
| Transformations to exclude short-selling | |
| Results from Minrisk2u and Maxret2u | |
| Upper and lower limits on invested fractions | |
| Larger-Scale Portfolios | |
| Introduction | |
| Portfolios with increasing numbers of assets | |
| Time-variation of optimal portfolios | |
| Performance of optimized portfolios | |
| Data-Fitting And The Gauss-Newton Method | |
| Data fitting problems | |
| The Gauss-Newton method | |
| Least-squares in time series analysis | |
| Gauss-Newton applied to time series | |
| Least-squares forms of minimum-risk problems | |
| Gauss-Newton applied to Minrisk1 and Minrisk 2 | |
| Equality Constrained Optimization | |
| Portfolio problems with equality constraints | |
| Optimality conditions | |
| A worked example | |
| Interpretation of Lagrange multipliers | |
| Some example problems | |
| Linear Equality Constraints | |
| Equality constrained quadratic programming | |
| Solving minimum-risk problems as EQPs | |
| Reduced-gradient methods | |
| Projected gradient methods | |
| Results with methods for linear constraints | |
| Penalty Function Methods | |
| Introduction | |
| Penalty functions | |
| The Augmented Lagrangian | |
| Results with P-SUMT and AL-SUMT | |
| Exact penalty functions | |
| Sequential Quadratic Programming | |
| Introduction | |
| Quadratic/linear models | |
| SQP methods based on penalty functions | |
| Results with AL-SQP | |
| SQP line searches and the Maratos effect | |
| Further Portfolio Problems | |
| Including transaction costs | |
| A re-balancing problem | |
| A sensitivity problem | |
| Inequality Constrained Optimization | |
| Portfolio problems with inequality constraints | |
| Optimality conditions | |
| Transforming inequalities to equalities | |
| Transforming inequalities to simple bounds | |
| Example problems | |
| Extending Equality-Constraint Methods | |
| Inequality constrained quadratic programming | |
| Reduced gradients for inequality constraints | |
| Penalty functions for inequality constraints | |
| AL-SUMT for inequality constraints | |
| SQP for inequality constraints | |
| Results with P-SUMT, AL-SUMT and AL-SQP | |
| Barrier Function Methods | |
| Introduction | |
| Barrier functions | |
| Numerical results with B-SUMT | |
| Interior Point Methods | |
| Introduction | |
| Approximate solutions of problem B-NLP | |
| An interior p | |
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