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I have $$g(W)=\lambda ||W||_1 + \gamma \sum_{i=1}^{k} ||w_i||_2$$ where $W \in R^{m \times T}, ||W||_1=max_j \sum_{j=1}^{T}|a_{ij}|$ and $w_i \in R^T$. I know that, $$\partial (\gamma \sum_{i=1}^{k} ||w_i||_2)=\gamma \sum_{i=1}^{k} \frac{w_i}{||w_i||_2}, w_i \neq0$$ $$\partial (\gamma \sum_{i=1}^{k} ||w_i||_2)=?, w_i =0$$

I would like drive the sub-differential of this term. I was wondering whether someone can help me with this ?

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    This should help: http://math.stackexchange.com/questions/2044993/how-to-compute-sub-differential-of-matrix-l1-norm – 2017-01-02
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    Yes, but it's get bite tricky by second term. – 2017-01-02
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    Well, subgradients do add... – 2017-01-02
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    I would like to compute the proximal operator of whole term, so I was trying to write down the optimality condition for that. Now I'm stuck with driving sub-differential part $g(W)$. would you please help me to write down the proximal operator of this term? – 2017-01-02

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