R help - Aug 2006 - minimization a quadratic form with some coef fixed and some constrained

Hello, all,

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!

Regards,
Yingfu
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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 at math.unb.ca