Download Stochastic Optimization Assignment 2 for ISyE 6664, Fall 2007 and more Assignments Systems Engineering in PDF only on Docsity!
ISyE 6664 Stochastic Optimization
Fall 2007
Assignment 2
Issued: August 28, 2007
Due: September 11, 2007
Problem 1 Consider the newsvendor problem in which the demand D has a known probability distri- bution with cumulative distribution function F. No other assumptions are made regarding F (whether F has a density, or whether F is a discrete cdf, etc.). Show that, in general, the set of optimal order quantities is given by
F −^1
r − c r − s
where F −^1 (p) denotes the set of p-quantiles of F , that is
F −^1 (p) ≡ {x ∈ R : F (y) ≤ p ∀ y < x and F (y) ≥ p ∀ y ≥ x}
Problem 2 Consider the optimization problem
min x∈ |x − ω| + 1
where ω ∈ Ω ⊂ is unknown to the decision maker at the time the decision x has to be made.
- Assume that Ω = [a, b]. Determine the worst-case solutions x 1 , x 2 , and x 3.
- Suppose ω has an exponential probability distribution with rate λ, but this fact is unknown to the decision maker. The decision maker has an iid exponential(λ) sample ω 1 ,... , ωk of ω, and uses the empirical distribution to choose a decision ˆx. Compute the value of perfect information, the value of the stochastic solution, and the expected optimality gap γek(λ) of solutions based on the empirical distribution, as a function of λ (it may also depend on the other parameters). Also compute the expected optimality gaps γi(λ) of the worst-case solutions xi, i = 1, 2 , 3, and compare these with γke(λ).
Problem 3 Randomization does not help for static optimization problems: Suppose we want to maximize g(x) over all x ∈ X. That is, the optimization problem is
sup x∈X
g(x)
(The following also holds if g(x) ≡ E[G(x, ω)].) A deterministic decision chooses one par- ticular decision x ∈ X. A randomized decision chooses a probability distribution P on X (we can choose a sufficiently rich σ-field on X ), and then a decision x ∈ X is generated according to probability distribution P. The objective value of such a randomized decision P is EP [g(X)] (where g(X) is a random variable if the σ-field on X is sufficiently rich). Show that randomization does not help for such a static optimization problem, unless supx∈X g(x) = ∞. (There are dynamic optimization problems for which randomization does help. Furthermore, there are worst-case type problems for which randomization may appear to help.) That is, show the following:
- For any probability distribution P on X ,
EP [g(X)] ≤ sup x∈X
g(x)
That is, sup P ∈P
EP [g(X)] = sup x∈X
g(x)
where P denotes the set of probability distributions on X.
- If g∗^ ≡ supx∈X g(x) < ∞ and there exists no x ∈ X such that g(x) = g∗, that is, there exists no deterministic decision x ∈ X that attains the supremum, then there exists no P ∈ P such that EP [g(X)] = g∗, that is, there exists no randomized decision P ∈ P that attains the supremum.
- Suppose we have a static optimization problem where g(x) < ∞ for all x ∈ X , but supx∈X g(x) = ∞. Show how to choose a probability distribution P such that EP [g(X)] = ∞. That is, there exists no deterministic decision x ∈ X that attains the supremum, but there exists a randomized decision P ∈ P that attains the supremum.
Problem 4 Randomization may appear to help for worst-case problems: Consider a function G(x, ω), with decision x ∈ X , and ω ∈ Ω unknown at the time the decision has to be made. Suppose we want to maximize G(x, ω) for the worst possible outcome ω ∈ Ω. That is, the optimization problem is sup x∈X
inf ω∈Ω G(x, ω)
A deterministic decision chooses one particular decision x ∈ X. A randomized decision chooses a probability distribution P on X , and then a decision x ∈ X is generated according
and assume that g∗^ < ∞. (If g∗^ = ∞ then we can choose a probability distribution P on X such that EP [infω∈Ω G(X, ω)] = ∞, and thus the expected value of the outcome is ∞ for both the case with the omniscient adversary and the case with the oblivious adversary.) For ease of comparison, suppose that x∗^ is an optimal deterministic decision, that is, G(x∗, ω) ≥ g∗^ for all ω ∈ Ω. For many problems, a decision x∗^ is good for some values of ω and not good for other values of ω. Suppose we have a set {x 1 ,... , xn} ⊂ X of n optimal deterministic decisions, each of which is good on some subset of Ω and reasonable on the rest of Ω. (Obviously, there are many examples of problems for which this holds—think about this a little.) Specifically, suppose Ω can be partitioned into n subsets, Ω = ∪ni=1Ωi, with Ωi ∩ Ωj = ∅ for all i = j, such that G(xi, ω) ≥ g∗^ for all ω ∈ Ω and
g∗ i ≡ inf ω∈Ωi
G(xi, ω) > g∗
(that is, xi is good on subset Ωi) for each i ∈ { 1 ,... , n}. (Note that, in general, for any x,
inf ω∈Ωi
G(x, ω) ≥ inf ω∈Ω
G(x, ω)
because Ωi ⊂ Ω.) Now show how to construct a randomized decision P˜ such that
inf ω∈Ω E (^) P˜ [G(X, ω)] > inf ω∈Ω G(x∗, ω) = g∗
that is, it seems as if randomization helps. (This approach is popular in the computer science community for constructing “randomized algorithms”.) Do you think the randomized decision is really better than optimal deterministic decision x∗? Why or why not?
