[SOLVED] Ece490 homework 5 p0

Required reading: Wright and Recht, Ch. 8.
1 (Exercise 1 in Wright and Recht, Ch. 8) Prove that, if f is convex and x ∈ dom(f), the subdifferential ∂f(x) is closed and convex.
Reminder: A set S ⊆ Rn is closed if the limit of any convergent sequence (vn)n≥1 of elements of S is also an element of S:
vn ∈ S for all n = 1,2,… and v = lim vn exists =⇒ v ∈ S. n→∞
2 (Exercise 5 in Wright and Recht, Ch. 8) For the following norm functions f over Rn, find the subdifferential ∂f(x) and the directional derivative f0(x,v) for all x,v ∈ Rn:
(a) The `1 norm.
(b) The `∞ norm f(x) = kxk∞ = max |xi|.
1≤i≤n
(c) The `2 (Euclidean) norm .
3 (Exercise 7 in Wright and Recht, Ch. 8) Find the subdifferential ∂f(x) of the piecewise-linear convex function f : Rn → R defined by
,
where ai ∈ Rn and bi ∈ R for i = 1,…,m.
4 Suppose that f is defined as a maximum of m convex, continuously differentiable functions; that is, f(x) = max1≤i≤m fi(x). Show that

X X
∂f(x) = λi∇fi(x) : λi ≥ 0, λi = 1 . i:fi(x)=f(x) i:fi(x)=f(x) 1