MATLAB FINANCIAL DERIVATIVES TOOLBOX Manuel d'utilisateur télécharger pdf (Page 108)

107

Specifically, )x(''f for a function with n variables is:













∂

∂∂

∂

∂∂

∂

∂∂

∂

∂∂

∂

)x(f

)x(''f

MMMM

• Step #3: Increase: k

←

k+1 and if the current iteration index k is

larger than the maximum number of iterations or if

1−

)]x(''f)[x('f

<e then stop and return

1+k

x , otherwise go to Step

#2 and perform one more iteration of the algorithm (the e is the desire

accuracy, usually set to a small quantity such as 1e-6, whereas the

maximum number of iterations depends solely by the experience of the

researcher).

Both the gradient descent and the Newton descent algorithms have inherent

problems with their implementation. For example, the user should

“magically” define the correct value of the learning parameter a in order to

succeed a robust convergence of the algorithm when it comes to implement

the gradient descent algorithm, or concerning the Newton descent one, when

the Hessian matrix is singular and its inverse does not exist then it is not

applicable. Moreover, there is no way to secure that the initial point to set

up the algorithm will eventually help in reaching the minimum point. Most

of the times and when the user does not have a detail overview of the faced

problem, the initial value of x might be too far from the minimum point,

preventing in this way the algorithm to converge.

There are many suggestions to suppress these problems (i.e [4], [5]). In here,

we will implement these two algorithms just for getting the feeling on the

way that a minimization algorithm can be used. The user can later improve

the functions that will be illustrated according to his/her needs.

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