Stochastic Programming - Index of

RECOURSE PROBLEMS 235
Exercises
1. The second-stage constraints of a two-stage problem look as follows:
( ) ( ) ( )
1 3 −1 0 −6 5 −1 0
y = ξ +
x
2 −1 2 1 −4 0 2 4
y ≥ 0
where ˜ξ is a random variable with support Ξ = [0, 1]. Write down the LP
(both primal and dual formulation) needed to check if a given x produces
a feasible second-stage problem. Do it in such a way that if the problem
is not feasible, you obtain an inequality in x that cuts off the given x. If
you have access to an LP code, perform the computations, and find the
inequality explicitly for ˆx =(1, 1, 1) T .
2. Look back at problem (4.1) we used to illustrate the bounds. Add one extra
constraint, namely
x raw1 ≤ 40.
(a) Find the Jensen lower bound after this constraint has been added.
(b) Find the Edmundson–Madansky upper bound.
(c) Find the piecewise linear upper bound.
(d) Try to find a good variable for partitioning.
3. Assume that you are facing a decision problem where randomness is
involved. You have no idea about the distribution of the random variables
involved. However, you can obtain samples from the distribution by running
an expensive experiment. You have decided to use stochastic decomposition
to solve the problem, but are concerned that you may not be able to
perform enough experiments for convergence to take place. The cost of a
single experiment is much higher than the costs involved in the arithmetic
operations of the algorithm.
(a) Argue why (or why not) it is reasonable to use stochastic decomposition
under the assumptions given. (You can assume that all necessary
convexity is there.)
(b) What changes could you suggest in stochastic decomposition in order
to (at least partially) overcome the fact that samples are so expensive
4. Let ϕ be a convex function. Show that
(See the definition following (9.7).
x ⋆ ∈ arg min ϕ iff 0 ∈ ∂ϕ(x ⋆ ).