1. Consider the following non-smooth function on the real line:

f(x) = max!

Describe the subdifferential f(x) at every point x R.

2. Prove or disprove: the subdifferential f(x) of a convex function is a convex set at every x R.
3. Recall the subdifferential for the nuclear norm for an n₁ n₂ matrix X.

$X$ = #UV ^T + W : UW = 0,WV = 0,$W$ 1$

In the expression above, X has rank r and its SVD is X = UV ^T, where U is n₁ r, is r r and V is n₂ r. Recall that

x⁺ = prox_t!c_dot!!(X) = argmin

Z

if and only if

XX⁺ tX⁺

Show that we can compute the prox operator above by singular value thresholding:

X max(_it,0).

1