[R] minimization a quadratic form with some coef fixed and some constrained

[Rolf Turner]

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