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Subject: Re: Approaches to solve constrained mixed-norm optmization problema
Date: Tue, 18 Aug 2009 17:05:21 +0000 (UTC)
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Prime Mover <emilsonpl@gmail.com> wrote in message <26c8a015-8817-4b87-a0b3-1d22e6464c2d@z31g2000yqd.googlegroups.com>...
> Dear friends,
> 
> What are the approaches available in MATLAB to solve a problem to find
> a vector of parameters r such that the sum
> 
> || W*r - s ||^2 + lambda1*| r | + lambda2*|| H*r - p ||^2
> 
> is minimized?
> 
> W and H are matrices with known values; s and p are vector with known
> values; and lambda1 and lambda2 are a set of given weights.
==================


The title of your post says that this is a constrained problem, yet you haven't mentioned any constraints on r. If there are no constraints, then I would be interested to know how the following Majorize-Minimize approach performs. It can easily be modified for box constraints on r:

1. First, reformulate the objective function as suggested by others to be in the form

f(x) = 1/2 *x'*Q*x+b*x


2. Proceed according to the following algorithm


MajCurvs=sum(abs(Q)<2);
ImportantQuantity=lambda1./MajCurvs;

r=InitialValue;

for ii=1:numIterations

 QuadGradient=Q*x+b;

  Center=r-QuadGradient./MajCurvs;
  Candidate1=Center-ImportantQuantity;
  Candidate2=Center+ImportantQuantity;

  r=Candidate1.*(Candidate1>0)+ Candidate2.*(Candidate2<0);

end