Stochastic Programming - Index of

BASIC CONCEPTS 43
However—for finite discrete as well as for continuous distributions—we are
faced with a further problem, which we might discuss for the linear case (i.e.
for stochastic linear programs with fixed recourse (4.16)). By suppP we denote
the support of the probability measure P , i.e. the smallest closed set Ξ ⊂ IR k
such that P˜ξ(Ξ) = 1. With the practical interpretation of the second-stage
problem as given, for example, in Section 1.3, and assuming that Ξ = suppP˜ξ,
we should expect that for any first-stage decision x ∈ X the compensation
of deficiencies in the stochastic constraints is possible whatever ξ ∈ Ξ will be
realized for ˜ξ. In other words, we expect the program
Q(x, ξ) =minq T y
s.t. Wy = h(ξ) − T (ξ)x,
y ≥ 0
⎫
⎬
⎭
(5.3)
to be feasible ∀ξ ∈ Ξ. Depending on the defined recourse matrix W and the
given support Ξ, this need not be true for all first-stage decisions x ∈ X.
Hence it may become necessary to impose—in addition to x ∈ X—further
restrictions on our first-stage decisions called induced constraints. Tobemore
specific, let us assume that Ξ is a (bounded) convex polyhedron, i.e. the convex
hull of finitely many points ξ j ∈ Ξ ⊂ IR k :
Ξ=conv{ξ ⎧
1 , ···,ξ r }
⎫
⎨ ∣ ∣∣
r∑ r∑
⎬
=
⎩ ξ ξ = λ j ξ j , λ j =1,λ j ≥ 0 ∀j
⎭ .
j=1
j=1
From the definition of a support, it follows that x ∈ IR n allows for a feasible
solution of the second-stage program for all ξ ∈ Ξ if and only if this is true
for all ξ j , j =1, ···,r. In other words, the induced first-stage feasibility set
K is given as
K = {x | T (ξ j )x + Wy j = h(ξ j ), y j ≥ 0, j=1, ···,r}.
From this formulation of K (which obviously also holds if ˜ξ has a finite discrete
distribution, i.e. Ξ = {ξ 1 , ···,ξ r }), we evidently get the following.
Proposition 1.3 If the support Ξ of the distribution of ˜ξ is either a finite
set or a (bounded) convex polyhedron then the induced first-stage feasibility
set K is a convex polyhedral set. The first-stage decisions are restricted to
x ∈ X ⋂ K.
Example 1.3 Consider the following first-stage feasible set:
X = {x ∈ IR 2 + | x 1 − 2x 2 ≥−4,x 1 +2x 2 ≤ 8, 2x 1 − x 2 ≤ 6}.
For the second-stage constraints choose
( −1 3 5
W =
2 2 2
)
, T(ξ) ≡ T =
( ) 2 3
3 1