1#. Show, using the deﬁnition of convex set, that the set S = {(x, y)|4x − 5y ≥ 20} is

convex.

2#. Let C ⊂ R^ n be convex, and suppose f : C → R and g : C → R are convex. Deﬁne

h : C → R by h(x) = max {f(x), g(x)}.

(a) Prove that epi h = epi f ∩ epi g.

(b) Use part (a) to show that h is convex.

3# (a) Let g : R

n → R be convex with g(x) ≥ 3 for all x ∈ R^n. Show that the function f

deﬁned by

f(x) = (g(x) − 3)2

is convex on R^n.

(b) If we no longer require g(x) ≥ 3, give an example of a convex function g : R → R such

that (g(x) − 3)2 is not convex.

4#. Let f : R^n → R have continuous ﬁrst-order partial derivatives at all points in R^n. Show

that if f is strictly convex and ∇f(x) = ∇f(y), then x = y.

5# Suppose that f is a convex function with continuous ﬁrst partials deﬁned on a convex set C in R^n. Prove that a point x∗ is a global minimizer of f on C if and only if ∇f(x∗) · (x − x∗) ≥ 0 for all x in C.

Subject | Mathematics |

Due By (Pacific Time) | 02/28/2014 12:00 am |

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