[SOLVED] Ece490 homework 1 p0

Required reading: Wright and Recht, Ch. 2.
1 In this problem, you will prove the Cauchy–Schwarz inequality for vectors in Rn: for u = (u1,…,un)T and v = (v1,…,vn)T,
|uTv| ≤ kukkvk,
where k · k stands for the `2 norm k · k2.
(a) Start by proving the following inequality:
, for all γ > 0.
Hint: First prove it for n = 1, then go from there.
(b) Deduce the Cauchy–Schwarz inequality from the inequality proved in part (a).
2 (Exercise 2 from Wright and Recht, Ch. 2) Prove that the epigraph epif is a convex subset of Rn × R for any convex function f.
3 (Exercise 5 from Wright and Recht, Ch. 2) Prove that no function f : Rn → R can be simultaneously strongly convex and Lipschitz.
4 (Exercise 7 from Wright and Recht, Ch. 2) Suppose that f : Rn → R is a convex function with LLipschitz gradient and a minimizer x∗ with function value f∗ = f(x∗).
(a) Show that, for any x ∈ Rn, we have
.
(b) Prove the following co-coercivity property: For all x,y ∈ Rn,
.
Hint: Apply part (a) to the following two functions: hx(z) := f(z) − ∇f(x)Tz and hy(z) := f(z) − ∇f(y)Tz.
5 (Exercise 8 in Wright and Recht, Ch. 2) Let f : Rn → R be an m-strongly convex function with L-Lipschitz gradient.
(a) Show that the function is convex with (L − m)-Lipschitz gradient.
(b) By applying the co-coercivity property fron the previous problem to this function q, show that the following holds for all x,y ∈ Rn:
.
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