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

From: Rolf Turner <rolf_at_erdos.math.unb.ca>
Date: Thu 10 Aug 2006 - 02:08:25 EST


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

______________________________________________
R-help@stat.math.ethz.ch mailing list
https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Thu Aug 10 02:12:01 2006

Archive maintained by Robert King, hosted by the discipline of statistics at the University of Newcastle, Australia.
Archive generated by hypermail 2.1.8, at Thu 10 Aug 2006 - 10:19:39 EST.

Mailing list information is available at https://stat.ethz.ch/mailman/listinfo/r-help. Please read the posting guide before posting to the list.