Special Topics in Operations Research 16:711:611
Convex Analysis and Optimization
Spring 2009 Rutgers University Prof. Eckstein
Homework 2
Due Wednesday, February 11
1. Prove the following fact, used in the proof of Proposition 1.5.1(b) in the text: let
x, y ∈Rn, and let {zk} ⊂ Rnbe a sequence with kzkk→∞; then
lim
k→∞
kzk−xk
kzk−yk= 1.
Hints: consider the square of the ratio above, instead of the ratio. Expand the numer-
ator and denominator and use the Cauchy-Schwarz inequalities −kzkkwk≤hz, wi ≤
kzkkwk. Finally, you may use without proof the following standard calculus fact about
limits of ratios of polynomials: given a1, . . . , am, b1...,bm∈Rwith am, bm6= 0, one
has
lim
t→∞ Pm
i=1 aiti
Pm
i=1 biti=am
bm
.
2. Suppose that f:Rn→(−∞,+∞] is a convex function and x∈dom f. Show that for
any d∈Rnthe function gd: (0,∞)→(−∞,+∞] defined by
gd(α) = f(x+αd)−f(x)
α
is nondecreasing.
3. Nonconvex Projections (similar to exercise 2.11 in the text): Let C⊂Rnbe a closed
but possibly nonconvex set, and consider any point x∈Rn.
(a) Show that the function g(w) = kw−xkmust have a nonempty, compact set of
minima over C. Denote this set by PC(x).
(b) Show that dC(x) = infw∈Ckw−xkis an everywhere finite-valued and continuous
function of x∈Rn.
(c) Give an example showing that if Cis not convex, dCneed not be convex.
4. Given a set X⊆Rn, its indicator function is the function δX:Rn→(∞,+∞] given
by
δX(x) = 0,if x∈X
+∞,if x6∈ X.
(a) Show that if Xis a closed set, δXis a closed function.
(b) Show that if Xis a convex set, δXis a convex function.
1
