Re: Approaches to solve constrained mixed-norm optmization problema

Prime Mover <emilsonpl@xxxxxxxxx> wrote in message <26c8a015-8817-4b87-a0b3-1d22e6464c2d@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>...
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.
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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
.

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