## Nonlinear optimization with financial applications

Author: Bartholomew-Biggs, Michael Publisher: Springer, 2005.Language: EnglishDescription: 261 p. ; 24 cm.ISBN: 1402081103Type of document: BookBibliography/Index: Includes bibliographical references and indexItem type | Current location | Collection | Call number | Status | Date due | Barcode | Item holds |
---|---|---|---|---|---|---|---|

Europe Campus Main Collection |
QA402.5 .B37 2005
(Browse shelf) 001212152 |
Available | 001212152 |

Includes bibliographical references and index

Digitized

Nonlinear Optimization with Financial Applications 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 xi xiii xv 1 1 3 6 11 12 16 17 19 19 20 23 24 28 29 33 33 34 36 38 vi 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. QUASI-NEWTON METHODS 1 Approximate second derivative information 2 Rank-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 41 41 43 45 47 51 51 52 54 57 58 63 65 65 67 70 71 73 77 77 78 81 82 84 85 87 87 90 91 92 94 Contents vii 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 97 97 100 101 104 107 107 108 111 113 117 117 119 121 122 125 127 131 131 132 135 135 137 139 139 140 142 146 147 151 151 153 viii 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-SU/4T 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 157 160 163 165 165 165 167 173 176 179 179 182 184 187 187 190 192 193 194 197 197 201 204 206 206 207 211 211 211 215 219 219 220 Contents ix 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 223 225 227 227 228 22. PORTFOLIO RE-BALANCING AND OTHER PROBLEMS 233 1 Re-balancing allowing for transaction costs 233 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 237 240 243 243 244 245 247 249 253 255 259

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