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From: "Bruno Luong" <b.luong@fogale.findmycountry>
Newsgroups: comp.soft-sys.matlab
Subject: Re: Approaches to solve constrained mixed-norm optmization problema
Date: Mon, 17 Aug 2009 19:21:01 +0000 (UTC)
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Emilson <emilsonpl@gmail.com> wrote in message <2a703ae3-f962-43d2-b3f0-6e226ca70490@k30g2000yqf.googlegroups.com>...

> 
> || Wr - s ||  and || Hr - p || = L2-norm; | r | = L1-norm
> 

Here are few though. First without restrict the generality, let's assume there is only one L2 term (put them together):

J2(r) := || A r - b ||^2 = alpha*|| Wr - s || ^2 + beta*|| Hr - p ||^2

A, b are matrices derived from the known parameters.

Our main objective function is

J0(r) := J2(r) + |r|

Let r2/r0 are respectively solutions minimizing J2/J0. I believe (!) we can show the *sign* of r0 and r2 (components) are identical.

Thus we can replace the pb minimizing J0(r) + |r| by

Minimizing:
J0(r) + <s,r>
<s,r>>=0

where s := sign(r2)

This problem is a box quadratic least-square minimization, thus it can be solved with an appropriate algorithm.

I hope my intuition is good and I don't make any false reasoning.

Bruno