Optim with two constraints

Hello,

I have a follow-up from Jens's question and Professor Ripley's response.

Jens wants to do quadratic optimization with 2 constraints:
# I need two constraints:
# 1. each element in par needs to be between 0 and 1
 # 2. sum(par)=1, i.e. the elements in par need to sum to 1
how does one set both constraints in quadprog, per Prof. Ripley's suggestion?

i know how to get quadprog to handle the second constraint, but not
BOTH, since quadprog only takes as inputs the constraint matrix "A"
and constraint vector "b"--
unlike in "ipop" (kernlab), there is no additional option for box constraints.

apologies if i am not seeing something obvious here.

thanks in advance,

alexis

-----Urspr??ngliche Nachricht-----
Von: Prof Brian Ripley [mailto:ripley at stats.ox.ac.uk]
Gesendet: Thursday, October 13, 2005 2:46 AM
An: Jens Hainmueller
Cc: r-help at stat.math.ethz.ch
Betreff: Re: [R] Optim with two constraints

This is actually quadratic programming, so why do you want to
use optim()?
There are packages specifically for QP, e.g. quadprog.

A more general approach is to eliminate one variable, which
gives you an inequality constrained problem in n-1 variables
to which you could apply contrOptim().  Other
re-parametrizations (e.g. of weights as a log-linear model)
will work provided none of the parameters are going to be
zero at the optimum (one cannot be one without all the others
being zero).

On Wed, 12 Oct 2005, Jens Hainmueller wrote:

Hi R-list,

I am new to optimization in R and would appreciate help on the
following question. I would like to minimize the following function
using two
constraints:

######
fn <- function(par,H,F){

    fval <- 0.5 * t(par) %*% H %*% par + F%*% par
    fval

 }

# matrix H is (n by k)
# matrix F is (n by 1)
# par is a (n by 1) set of weights

# I need two constraints:
# 1. each element in par needs to be between 0 and 1 # 2.
sum(par)=1
i.e. the elements in par need to sum to 1

## I try to use optim
res <- optim(c(runif(16),fn, method="L-BFGS-B", H=H, F=f
,control=list(fnscale=-1), lower=0, upper=1) ######

If I understand this correctly, using L-BFGS-B with lower=0 and
upper=1 should take care of constraint 1 (box constraints).
What I am
lacking is the skill to include constraint no 2.

I guess I could solve this by reparametrization but I am
not sure how
exactly. I could not find (i.e. wasn't able to infer) the answer to
this in the archives despite the many comments on optim and
constrained optimization (sorry if I missed it there). I am
using version 2.1.1 under windows XP.
Thank you very much.

Jens

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--
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595

"AD" == Alexis Diamond <alexisjdiamond at gmail.com> writes:
AD> I have a follow-up from Jens's question and Professor Ripley's
    AD> response.

    AD> Jens wants to do quadratic optimization with 2 constraints:

    >> > > # I need two constraints:
    >> > > # 1. each element in par needs to be between 0 and 1
    >> > > # 2. sum(par)=1, i.e. the elements in par need to sum to 1

    AD> how does one set both constraints in quadprog, per
    AD> Prof. Ripley's suggestion?  i know how to get quadprog to
    AD> handle the second constraint,
The first is actually not one constraint but 2*k constraints, where k
is the number of elements in "par".  But there is quite some
redundancy in this set of equation.  It suffices to constrain each
element to be bigger or equal to 0 and that they should sum to 1.  If
these constraints are fulfilled, then each element must be less or
equal to one.

    AD> but not BOTH, since quadprog only takes as inputs the
    AD> constraint matrix "A" and constraint vector "b"--
So what stops you from coding the k constraints from 1.) in the form
that quadprog requires them?  From memory, i.e. untested:

m <- length(par)
Amat <- cbind(rep(1,m), diag(m))
bvec <- c(1,rep(0,m))
meq <- 1

solve.QP(Dmat, dvec, Amat, bvec, meq)

    AD> unlike in "ipop" (kernlab), there is no additional option for
    AD> box constraints.
Well, the problems that I had (and still have) usually don't involve
box constraints, but I see that other people use them again and again.
So probably it would be a good idea to implement them...  But, more
importantly, would be to implement Powell's modifications of the
Goldfarb-Idnani algorithm to make it numerically more robust...  Oh,
yeah, and a warm start option from a feasible point would be nice
too....  Probably all in a future version which should be released
sometime before Xmas 20xx. :)

    AD> apologies if i am not seeing something obvious here.
Apologies accepted. :)

Cheers,

        Berwin