Re: help needed in an optimization problem


In article <dmkr3f$7j8$1@xxxxxxxxxxxxxxxx>,
"yigido" <mygurdal@xxxxxxxxx> writes:
>Dear All,
>I encountered the following optmization problem during my research. I
>wanted to ask for help.
>How to choose q to maximize:
>-c( q )+ exp( q + s^2 )[ 1 - Phi( (lnA - q - s^2) / s ) ]
>where c( ) is a convex function and Phi( ) is the CDF of standard
>normal distribution.
>all help would be appreciated a lot.
>thanks
>

I think this can be done numerically only. but since this is maximization in R^1
with a smooth function you could use e.g. a combination of golden section
search and local polynomial models, e.g. Brent's fmin which is quite
easy to use. you may also try a 1D global zero finder for the derivative, e.g.
Brents "fzero"
you get fmin and fzero from
http://www.netlib.org/fmm
(also available in MATLAB)
Be careful regarding the numerical evaluation of your expression!
it may be useful to use the identity
1-Phi(x) = 0.5*erfc(x/sqrt(2))
for erfc there exist good evaluation codes. the continued fraction
erfc(x) =(1/sqrt(pi))*exp(-x^2)*(1/(x+0.5/(x+1/(x+.5/x+(.....))))))
shows that the dangerous looking term exp(q+s^2) cancels out essentialy
for large q.
hth
peter



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