# Let a vector z Rn be given. For X > 0 consider the problem (i) Show...

###### Question:

Let a vector z Rn be given. For X > 0 consider the problem (i) Show that for any λ 0 this problem has a unique solution「. (ii) Determine the unique solution「(as a function of λ and 2) Hint: Note that Λ is not differentiable everywhere. Remark: The solution of (ii) is really interesting and beautiful, since you will see that the solutions x\ are so-called sparse vectors, i.e. vector having many zero components. Indeed, χλ 0 whenever λ > llzlla . The reason for this behavior lies in the use of the 1-norm (instead of the Euclidean norm) in the second term

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