Hello,
 
Thanks for reply! But I think that solution is right without the constrain b'b=1. With this constrain, the solution is not so simple. :(
 
Regards,
Yingfu

________________________________

Från: Rolf Turner [mailto:rolf@erdos.math.unb.ca]
Skickat: on 2006-08-09 18:08
Till: Yingfu Xie; r-help@stat.math.ethz.ch
Ämne: Re: [R] minimization a quadratic form with some coef fixed and some constrained



Yingfu Xie wrote:

> I had problems with an extension to a classic optimization problem.
>
> The target is to minimize a quadratic form a'Ma with respect to vector
> b, where vector a=(b',-1)', i.e., a is the expand of b, and M is a
> symmetric matrix (positive definite if needed). One more constrain on b
> is b'b=1. I want to solve b given M.
>
> I tried but it seems impossible to find an analytic solution for b. Any
> objection?
>
> Now, come to the numerical.  Does anybody have any idea on how to
> parameterize this to use, e.g. optim() or constrOptim()?
>
> Any help are appreciated very much!

        The analytic solution is trivial.  Write M as

                | M_11 c |
                | c'   m |

        Then given that M_11 is positive definite, the
        minimizer is

                b = (M_11)^{-1}c

                                cheers,

                                        Rolf Turner
                                        rolf@math.unb.ca


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