Nonlinear Optimization with Financial Applications

By: Michael Bartholomew-Biggs

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Hardcover | 4 January 2005

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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.

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From the reviews:

"The book is intended for readers who have an understanding of linear algebra, and the Taylor mean value theorems in several variables. It presents numerical approaches to nonlinear optimization which typically have been applied for 30-40 years to practical problems in science and engineering. ... Summing up, one could say that the author had an interesting idea to present the classical algorithmic methods while teaching contemporary portfolio theory." (Leszek S. Zaremba, Zentralblatt MATH, Vol. 1083 (9), 2006)

"This book is a timely and very useful addition to the literature on practical mathematical optimization. ... Indeed, the outstanding and almost unique aspect of the work is the thorough integration of the methods with application ... . The text is very readable and will be easy to teach from. The passion of the author for, and his fascination with, optimization are conveyed to the reader. His mathematically flavored poems, scattered throughout the text, entertain and amuse ... ." (Jan A. Snyman, SIAM Review, Vol. 48 (1), 2006)

"This book contains many computational examples demonstrating the practical behavior of the proposed methods and their application to practical financial problems." (Mathematical Reviews, Zimmermann, K.)

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You Can Find This Book In

Non-Fiction Mathematics Optimisation Linear Programming Computing & I.T.Computer Science Applied Mathematics Accounting & Finance Finance Algebra Computer Programming & Software Development Algorithms & Data Structures

Business & Management Operational Research Game Theory

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Published: 21st November 2014

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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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Nonlinear Optimization with Financial Applications

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