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I am aware of how to implement the Newton's method for minimization for a smooth analytic function.

I am also aware of log-barrier for constraint minimization.

Now, I am looking to solve the the minimization of L-2 norm with L-1 regularization. Thus, I want to solve,

$$ \text{minimize}_\beta \ ||\textbf{y}-\textbf{X}\beta||_2 $$ $$ \text{subject to} \ ||\beta||_1 \le T $$

$\beta \in R^p$, $X \in R^{n \times p}$. T is a known constant. How do I incoperate the L-1 norm in constraint minimization.

mkuse
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  • What's your purpose here? There are lots of existing tools to do this. If you just want to solve a particular problem, you might as well use one of those. – Michael Grant Oct 04 '14 at 21:00
  • Actually, I am wanting to learn about newton's method for minimization adapted for constraint-optimization (log-barrier). – mkuse Oct 06 '14 at 03:26
  • Then you should probably be consulting textbooks, like Boyd & Vandenberghe, Bertsekas, Renegar... – Michael Grant Oct 07 '14 at 13:09
  • actually, I did finally solve this using 2^p linear equations constraints as mentioned in sec 2.6 of http://www.cs.ubc.ca/~schmidtm/Documents/2005_Notes_Lasso.pdf. Although it also mentions other better solutions, but I got the point now....and thanks for all the help – mkuse Oct 07 '14 at 13:27
  • That looks like a nice report indeed... – Michael Grant Oct 07 '14 at 15:39

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