R help - Jun 2004 - optimize linear function

I am attempting to optimize a regression model's parameters to meet a
specific
target for the sum of positive errors over sum of the dependent variable 
(minErr below).  

I see two courses of action , 1) estimate a linear model then iteratively 
reduce the regressors to achieve the desired positive error threshold 
(naturally the regressors and predicted values are biased - but this is 
acceptable).  Were the problem only a single independent variable problem, 
this approach would be fairly trivial.  But the two variable model proves 
more troublesome from an efficiency point of view (efficiency being the mean 
negative error for a given minErr).

The second method, which seems more efficient, is to optimize the linear 
regressors until the criteria is met.  There are several optimizer packages 
out there, but none seem particularly well suited to this sort of task.  

Any suggestions out there?

===========  Sample Problem = first approach  ===============
# if we say... 
B1 <- .8
B2 <- .2

# then construct some data..
x1 <- rnorm(100,mean=100,sd=20)
x2 <- rnorm(100,mean=10,sd=2)
y <- B1*x1+B2*x2+rnorm(100,mean=5,sd=10)
# normally of course B1 and B2 are unknown to begin with

# linear model  (presumably yielding B1=.8 and B2=.2)
m<-lm(y ~ x1+ x2)
# test on summation of positive errors.
e <- resid(m)
minErr <- (sum(ifelse(e<0,0,e))/sum(y))-.03
while (minErr>.03){
	Make some adjustment to B1 and B2
	minErr <- (sum(ifelse(e<0,0,e))/sum(y))-.03
	}

======== Second Approach ==========
construct a function to minimize minErr with beginning values from 'm'.


Any suggestions for approaches greatly appreciated.

This is just a linear programming problem.  So the packages which do 
linear programming are `particularly well suited to this sort of task'
and theory tells you a lot about the solution.

Resarching how L_1 regression is done (that is that sum of the absolute 
values of the errors) should give you a lot of help.  This exact 
problem is a special case of one formulation of support vector machines.

On Sat, 12 Jun 2004, Michaell Taylor wrote:
> I am attempting to optimize a regression model's parameters to meet a
specific
> target for the sum of positive errors over sum of the dependent variable 
> (minErr below).  
The sum of the dependent variable is a constant so can be ignored.
> I see two courses of action , 1) estimate a linear model then iteratively 
> reduce the regressors to achieve the desired positive error threshold 
> (naturally the regressors and predicted values are biased - but this is 
> acceptable).  Were the problem only a single independent variable problem, 
> this approach would be fairly trivial.  But the two variable model proves 
> more troublesome from an efficiency point of view (efficiency being the
mean
> negative error for a given minErr).
> 
> The second method, which seems more efficient, is to optimize the linear 
> regressors until the criteria is met.  There are several optimizer packages
> out there, but none seem particularly well suited to this sort of task.  
> 
> Any suggestions out there?
> 
> ===========  Sample Problem = first approach  ===============> 
> # if we say... 
> B1 <- .8
> B2 <- .2
> 
> # then construct some data..
> x1 <- rnorm(100,mean=100,sd=20)
> x2 <- rnorm(100,mean=10,sd=2)
> y <- B1*x1+B2*x2+rnorm(100,mean=5,sd=10)
> # normally of course B1 and B2 are unknown to begin with
> 
> # linear model  (presumably yielding B1=.8 and B2=.2)
> m<-lm(y ~ x1+ x2)
> # test on summation of positive errors.
> e <- resid(m)
> minErr <- (sum(ifelse(e<0,0,e))/sum(y))-.03
> while (minErr>.03){
> 	Make some adjustment to B1 and B2
> 	minErr <- (sum(ifelse(e<0,0,e))/sum(y))-.03
> 	}
> 
> ======== Second Approach ==========> 
> construct a function to minimize minErr with beginning values from
'm'.
-- 
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