The quadprog package

G'day Thomas,

On Mon, 3 Sep 2007 10:08:08 +0200

What's wrong with my code?

Require(quadprog)

library(quadprog) ? :)

Dmat<-diag(1,7,7)
# muss als quadratische Matrix eingegeben werden
Dmat
dvec<-matrix(0,7,1) # muss als Spaltenvektor eingegeben werden
dvec
mu<-0 # (in Mio. ?)
bvec<-c(1,mu,matrix(0,7,1)) # muss als Spaltenvektor eingegeben werden
bvec
mu_r<-c(19.7,33.0,0.0,49.7, 82.5, 39.0,11.8)	
Amat<-matrix(c(matrix(1,1,7),7*mu_r,diag(1,7,7)),9,7,byrow=T) 
# muss als Matrix angegeben werden, wie sie wirklich ist
Amat
meq<-2
loesung<-solve.QP(Dmat,dvec,Amat=t(Amat),bvec=bvec,meq=2)

With the commands above, I get on my system:

loesung<-solve.QP(Dmat,dvec,Amat=t(Amat),bvec=bvec,meq=2)

Error in solve.QP(Dmat, dvec, Amat = t(Amat), bvec = bvec, meq = 2) : 
	constraints are inconsistent, no solution!

Which is a bit strange, since Dmat is the identity matrix, so there
should be little room for numerical problems.

OTOH, it is known that the Goldfarb-Idnani algorithm can have problem
in rare occasions if the problem is "ill-scaled"; see the work of
Powell. 

Changing the definition of Amat to

Amat<-matrix(c(matrix(1,1,7),mu_r,diag(1,7,7)),9,7,byrow=T) 

leads to a successful call to solve.QP.  And since mu=0, I really
wonder why you scaled up the vector mu_r by a factor of 7 when putting
it into Amat.... :)

loesung<-solve.QP(Dmat,dvec,Amat=t(Amat),bvec=bvec,meq=2)

Now, with the solution I obtained the rest of your commands show:

loesung$solution %*% mu_r

[,1]
[1,] 6.068172e-15

sum(loesung$solution)

[1] 1

for (i in 1:7){

a<-loesung$solution[i]>=0
	print(a)
}
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] FALSE
[1] FALSE
[1] TRUE
 
But:

 for (i in 1:7){

a<-loesung$solution[i]>=- .Machine$double.eps*1000
	print(a)
}

[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE
[1] TRUE

 for (i in 1:7) print(loesung$solution[i])

[1] 0
[1] 0
[1] 1
[1] 0
[1] -4.669169e-17
[1] -1.436586e-17
[1] 8.881784e-16

This is a consequence of finite precision arithmetic.  Just as you
should not compare to numeric numbers directly for equality but rather
that the absolute value of their difference is smaller than an
appropriate chosen threshold, you should not check whether a number is
bigger or equal to zero, but whether it is bigger or equal to an
appropriately chosen negative threshold.  More information are given in
FAQ 7.31.

HTH.

Cheers,

	Berwin

=========================== Full address =============================
Berwin A Turlach                            Tel.: +65 6515 4416 (secr)
Dept of Statistics and Applied Probability        +65 6515 6650 (self)
Faculty of Science                          FAX : +65 6872 3919       
National University of Singapore     
6 Science Drive 2, Blk S16, Level 7          e-mail: statba at nus.edu.sg
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Hi Thomas,

On my computer the solution is (0,0,1,0,0,0,0) (within
machine accuracy) and it satisfies the constraints.

Hi everybody,

I'm using Windows XP Prof, R 2.5.1 and a Pentium 4
Processor.
Now, I want to solve a quadratic optimization
program (Portfolio Selection) with the quadprog
package

I want to minimize (\omega'%*%\Sigma%*%\omega)
Subject to
	(1) \iota' %*% \omega = 1 (full investment)
	(2) R'%*%\omega = \mu (predefined expectation
value)
	(3) \omega \ge 0 (no short sales).

Where 
	\omega is a Nx1 vector of the weights invested in
each asset
	\Sigma is the NxN variance-covariance matrix
	\iota is a 1xN vector of 1's
	R' is a Nx1 vector of the expected returns
	and
	\mu is a number, the postualted return of the
investor

I've done the following code but it doesn't make
what I want it to do. The weights aren't all
positive and the \mu isn't reached. What's wrong
with my code?

Require(quadprog)
Dmat<-diag(1,7,7)
# muss als quadratische Matrix eingegeben werden
Dmat
dvec<-matrix(0,7,1) # muss als Spaltenvektor
eingegeben werden
dvec
mu<-0 # (in Mio. ???)
bvec<-c(1,mu,matrix(0,7,1)) # muss als Spaltenvektor
eingegeben werden
bvec
mu_r<-c(19.7,33.0,0.0,49.7, 82.5, 39.0,11.8)	

Amat<-matrix(c(matrix(1,1,7),7*mu_r,diag(1,7,7)),9,7,byrow=T)

# muss als Matrix angegeben werden, wie sie wirklich
ist
Amat
meq<-2

loesung<-solve.QP(Dmat,dvec,Amat=t(Amat),bvec=bvec,meq=2)

loesung

# ??berpr??fen, ob System richtig gel??st wurde
loesung$solution %*% mu_r
sum(loesung$solution)
for (i in 1:7){
	a<-loesung$solution[i]>=0
	print(a)
}

Thanks in advance for your answers.

______________________________

Thomas Schwander

MVV Energie
Konzern-Risikocontrolling

Telefon 0621 - 290-3115
Telefax 0621 - 290-3664

E-Mail: Thomas.Schwander at mvv.de .  Internet:
www.mvv.de
MVV Energie AG . Luisenring 49 . 68159 Mannheim
Handelsregister-Nr. HRB 1780, Amtsgericht Mannheim
Vorsitzender des Aufsichtsrates: Herr Dr. Peter Kurz
Vorstand: Dr. Rudolf Schulten (Vorsitzender) . 
Matthias Br??ckmann . Dr. Werner Dub . Hans-J??rgen
Farrenkopf 




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