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I am trying to find the minimum of the following problem $$\frac{\theta}{2}\lVert\beta-x\rVert_2^2+\lambda\lVert x\rVert_2-\frac{1}{2\tau}\lVert x\rVert_2^2+\alpha^Tx$$

by taking the derivative with respect to $x$ and setting it to zero, I have $$-\theta(\beta-x)+\lambda\frac{x}{\lVert x\rVert_2}-\frac{1}{\tau}x+\alpha=0$$

I have trouble solving the problem because I have $\lVert x\rVert_2$ in my solution.

Simple
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  • Welcome to Cross Validated! What is the connection to statistics? You’ve tagged the question as involving the statistics topics of [tag:regression] and [tag:machine-learning] but have not mentioned either in your question. Is this about, for instance, regularized regression? – Dave Jun 09 '24 at 14:26
  • https://math.stackexchange.com/questions/1681658/closed-form-solution-of-arg-min-x-left-x-y-right-22-lamb?noredirect=1&lq=1 I found what i need to. – Simple Jun 10 '24 at 00:55

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