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Re: LeastSquares using LinearProgramming?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89032] Re: [mg88986] LeastSquares using LinearProgramming?
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Sat, 24 May 2008 03:53:28 -0400 (EDT)
  • References: <200805230705.DAA25655@smc.vnet.net>

Gareth Russell wrote:
> Hi,
> 
> Is it possible to specify a least-squares minimization through the 
> LinearProgramming function? In other words, exactly the same as 
> LeastSquares, with the extra constraint that all x>=0?
> 
> Presumably it comes down to specifying the input c correctly in the 
> LinearProgramming function. But I can't see how to do that such that 
> what is being minimized is the standard least-squares function 
> ||m.x-b||^2
> 
> Thanks,
> 
> Gareth

That objective function would be quadratic, so no, LinearProgramming 
will not like that.

You could instead try

FindMinimum[{objective,constraints}, vars,
   Method->"QuadraticProgramming"]

This method is, alas, not documented, and I'd imagine it could disappear 
(which would be a shame, because it works really well, bordering on 
magic, for some problems). A documented alternative that might work well 
is "InteriorPoint";, as it also can handle e.g. nonnegativity constraints.

If you can settle for an L_1 norm (so it's no longer least squares), you 
can minimize a new variable abs, with new constraints -abs<=m.x-b<=abs.

Daniel Lichtblau
Wolfram Research


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