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I have an application in which I need to minimize the following cost function. I made myself familiar with optimization up until very recently. Could someone kindly let me know what kind of minimization $J$ is and how I can convert it to a SeDuMi format?

$J = \min_\boldsymbol{\rm x}\left\{\|A\boldsymbol{\rm x}_\lambda-b\|_2^2 + \lambda\|\boldsymbol{\rm x}_\lambda\|_2\right\} = \min_\boldsymbol{\rm x}\left\{\left(A\boldsymbol{\rm x}_\lambda-b\right)^T\left(A\boldsymbol{\rm x}_\lambda-b\right) + \lambda\sqrt{{\boldsymbol{\rm x}}_\lambda^T{\boldsymbol{\rm x}}_\lambda}\right\}$

I get very good results when I use CVX, which is an interface to SeDuMi. My goal is to understand how SeDuMi solves this minimization, which allows me to write my own solver.

shashashamti2008
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    Sedumi uses interior point methods, which are very effective for small or medium sized problems but also somewhat difficult to implement. It would be much easier to implement your own solver using a proximal algorithm such as the proximal gradient method or FISTA. Have you considered using a proximal algorithm? – littleO Feb 10 '18 at 09:36
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    For reference, I explained an approach using proximal algorithms here: https://math.stackexchange.com/questions/2643922/regularized-least-squares-with-non-squared-2-norm-penalty-term – littleO Feb 10 '18 at 10:00
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    I just read your answer in the other question. Sounds very interesting. Thank you for sharing. I am going to read the lectures you kindly posted on that and start the implementation. Do you happen to know if proximal algorithm are comparable to interior points in terms of accuracy and speed? – shashashamti2008 Feb 10 '18 at 10:02

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