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I want to solve $$ \min \:\: \vert \vert x \vert \vert_1 \:\:\:\: s.t. \:\: Ax=y $$ using a Neural Network or Recurrent Neural Network.

I found only this paper, but there $x$ has to be greater or equal zero. Do you know any other good papers, tutorials or other projects about how to solve this problem using a network?

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    $\begingroup$ This may not be the best way, but if you substitute $x1 - x2$ for $x$, where $x1$ and $x2$ are constrained to be non-negative, then you get a problem equivalent to your original, and involving only variables which are constrained to be non-negative. In particular, $x$ is now unconstrained. $\endgroup$ Jun 26 '17 at 18:41
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    $\begingroup$ @whuber There's more than one way to skin a cat. In any event, a neural network is certainly not the most computationally efficient or robust way of solving a Linear Programming problem. So hopefully this whole exercise is just "for fun". $\endgroup$ Jun 26 '17 at 19:54
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    $\begingroup$ Use software designed for Linear Programming, which implements a Simplex or Interior Point solver. You can not just pick up a book, any book, and hope to reproduce anything anywhere near that quality or robustness. BTW, if your problem has special structure, such as network structure, that can be exploited to increase the efficiency. $\endgroup$ Jun 26 '17 at 20:27
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    $\begingroup$ @N8_Coder My point is that if you really want to solve a Linear programming problem, use software designed for it - don't use a neural network except for educational and experimental purposes. $\endgroup$ Jun 27 '17 at 19:22
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    $\begingroup$ @N8_Coder You would be able to comment if you merge your accounts so that the system understood you were the asker rather than some random person. Do NOT post comments as answers. $\endgroup$
    – Glen_b
    Jun 28 '17 at 6:56
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There is a recurrent neural network which solves the lasso problem. Here is the link to this paper: https://arxiv.org/pdf/1704.03443.pdf

I am also curious to know in what applications you need a neural network for the lasso problem.

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    $\begingroup$ Welcome to the site! Could you expand on your answer by briefly explaining the solution (if possible), and by providing a full citation (links tend to go dead)? $\endgroup$
    – mkt
    Mar 19 '18 at 8:06
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    $\begingroup$ Sure! In the paper, the lasso problem is firstly converted to a smooth equivalent minimization with some constraints. Then, a neural network is proposed to the smooth problem, which indeed solves the lasso problem. More details are in the paper and it is not lengthy at all $\endgroup$ Mar 23 '18 at 19:42

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