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Appendix 2

Nonlinear Least-squares Minimisation

A2.1 INTRODUCTION AND OBJECTIVES

In this appendix we give an introduction to multi-variable optimisation and in particular to nonlinear least-squares problems. We use the Levenberg-Marquardt method and we also integrate its C# implementation from the ALGLIB library into our applications (www.alglib.net). This appendix complements the discussions in Chapters 16 and 17 and is included for the sake of completeness. There are many nonlinear solvers in use; we use Levenberg-Marquardt because it satisfied our needs.

A2.2 NONLINEAR PROGRAMMING AND MULTI-VARIABLE OPTIMISATION

In this section we introduce the mathematical formulation of unconstrained multi-variable optimisation problems (minimise or maximise a given objective function). This will prepare the way for a discussion of nonlinear least squares problems in computational finance.

Consider the scalar function that maps n-dimensional Euclidean space into the real line. In general, we wish to find a point x in Euclidean space that minimises or maximises this function. The general optimisation problem is to find a real number z such that:

Here F is called the objective function and the problem is unconstrained because there are no restrictions placed on the solution. We note that ...

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