ISEN 629 Engineering Optimization
Fall 2007
Homework # 2
Due Date: October 18, Thursday.
1. Prove Theorem 2.1.9.
2. Prove Theorem 2.1.11.
3. Use a one-dimensional example to illustrate Theorem 3.1.1 geometrically.
4. Given two convex functions f1and f2on Rn, define their convolution function as
f(x) = inf
y{f1(x−y) + f2(y)}.
Show that f(x) is convex.
5. A function f:X→[−∞,+∞], where S⊆Rnis called lower semi-continuous at a
point x∈Sif
f(x)≤lim
i→∞
f(xi)
for every sequence {xi:i≥0} ⊆ Ssuch that xi→x, i → ∞, and the limit of
{f(xi) : i≥0}exists in [−∞,+∞].
For an arbitrary function f:Rn→[−∞,+∞], prove that the following conditions
are equivalent:
(a) fis lower semi-continuous throughout Rn;
(b) Lf(α) = {x:f(x)≤α}is closed for every α∈R;
(c) epi(f) is a closed set in Rn+1.