ISyE 6664 Stochastic Optimization
Fall 2007
Assignment 1 (Revision)
Issued: August 21, 2007
Due: August 28, 2007
Problem 1
Integration Preserves Convexity: Consider a function G:X×Ω→ R,whereXis a vec-
tor space. Suppose G(x, ω)isconvexinxfor each ω∈Ω. Let Pbe any probability
distribution on Ω. Assume that the necessary measurability conditions hold. Show that
g(x):=EP[G(x, ω)] is convex in x. (If you need additional assumptions for the result to
hold, then state those assumptions and motivate why the assumptions are needed.)
Problem 2
Consider a function G:X×Ω→ R,whereΩ=Rnfor some n. Suppose G(x, ω)isconvex
in ωfor each x∈X.LetPbe any probability distribution on Ω. Assume that the necessary
measurability conditions hold and that EP[ω] is finite. Show that G(x, EP[ω]) ≤EP[G(x, ω)]
for each x∈X. That is, the objective function G(x, EP[ω]) of the deterministic optimiza-
tion problem minx∈X G(x, EP[ω]) that replaces ωwith its mean is a biased estimate of the
objective function EP[G(x, ω)] of the stochastic optimization problem minx∈X EP[G(x, ω)].
(If you cannot prove the result for Ω = Rn, then prove the result for Ω = R.Forbonus
points, give a counterexample for the assertion above for a function G:X×Ω→ R∪{+∞},
where Ω is a vector space.)
Problem 3
Consider a function g:X1×X
2→ R, with decisions x1∈X
1and x2∈X
2.
1. Show that
sup
x1∈X1
sup
x2∈X2
g(x1,x
2) = sup
x2∈X2
sup
x1∈X1
g(x1,x
2) = sup
(x1,x2)∈X1×X2
g(x1,x
2)
2. Show that
sup
x1∈X1
inf
x2∈X2
g(x1,x
2)≤inf
x2∈X2
sup
x1∈X1
g(x1,x
2)
3. Give an example of a function g(x1,x
2) such that
sup
x1∈X1
inf
x2∈X2
g(x1,x
2)<inf
x2∈X2
sup
x1∈X1
g(x1,x
2)
