Stochastic Programming - Index of

208 STOCHASTIC PROGRAMMING
Figure 25 Simple recourse: supporting Q i(χ i)by ˆQ i(χ i, ξ 1 i , ξ2 i ).
P (ξ i ∈ I iν ), ξ iν = E˜ξ(˜ξ i | ξ i ∈ I iν ):
min x,χ
[c T x +
k∑
i=1
s.t. Ax = b,
Tx− χ =0,
x ≥ 0,
N∑
i−1
ν=0
]
p iν ˆQi (χ i , ξ iν )
yielding the solution ˆx and ˆχ = T ˆx. Obviously relation (6.5) holds for
conditional expectations Q iν (ˆχ i ) (with respect to I iν ) as well. Then for each
component of ˆχ there are three possibilities.
(a) If ˆχ i ≤ α i ,then
ˆQ i (ˆχ i , ξ iν )=Q iν (ˆχ i )=E˜ξ( ˆQ i (ˆχ i , ˜ξ i ) | ξ i ∈ I iν ), ν =0, ···,N i − 1,
and hence
Q i (ˆχ i )=
N∑
i−1
ν=0
p iν ˆQi (ˆχ i , ξ iν ),
i.e. there is no error with respect to this component.
(b) If ˆχ i ≥ β i , then it again follows from (6.5) that
Q i (ˆχ i )=
N∑
i−1
ν=0
p iν ˆQi (ˆχ i , ξ iν ).