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How can I solve efficiently the problem:

$$\min_x \| A x \|_1 \quad \text{subject to} \; {b}^{T} x = c$$

for $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{n}, c \in \mathbb{R}$. The matrix $A$ has independent columns.

Is there a closed form solution? Or a trick to quickly calculate $x$?

I am aware it can be solved iteratively by projected gradient descent and by other alternating methods. Yet they converge pretty slowsly.

D.W.
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Mark
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    Welcome to [math.se] SE. Take a [tour]. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Another User Jan 24 '24 at 20:29
  • @AnotherUser, great guidance. I added notes about how I can solve it now. The motivation is for simpler method then iterative gradient descent. – Mark Jan 24 '24 at 21:00
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    Have you seen this https://math.stackexchange.com/questions/1639716/how-can-l-1-norm-minimization-with-linear-equality-constraints-basis-pu ? – CroCo Jan 24 '24 at 22:05
  • @CroCo, I am aware of the linear programming method to solve this. looking for something independent of specific solver. – Mark Jan 26 '24 at 07:56

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