Stochastic Programming - Index of

RECOURSE PROBLEMS 179
φ
φ(ξ)
ξ)
1
2
ξ)
ξ
Figure 13
Two possible lower bounding functions.
oil from the second country gives 6 and 3 units of the same products. Company
1 wants at least 180 + ξ 1 units of Product 1 and Company 2 at least 162 + ξ 2
units of Product 2. The goal now is to find the expected value of φ(ξ); in other
words, we seek the expected value of the “wait-and-see” solution. Note that
this interpretation is not the one we adopted in Section 1.3.
3.4.1 The Jensen Lower Bound
Assume that q(ξ) ≡ q 0 , so that randomness affects only the right-hand side.
The purpose of this section is to find a lower bound on Q(ˆx, ξ), for fixed ˆx,
and for that purpose we shall, as just mentioned, use φ(ξ) ≡ Q(ˆx, ξ) fora
fixed ˆx.
Since φ(ξ) is a convex function, we can bound it from below by a linear
function L(ξ) =cξ + d. Since the goal will always be to find a lower bound
that is as large as possible, we shall require that the linear lower bound be
tangent to φ(ξ) at some point ˆξ. Figure 13 shows two examples of such lowerbounding
functions. But the question is which one should we pick. Is L 1 (ξ)
or L 2 (ξ) the better
If we let the lower bounding function L(ξ) be tangent to φ(ξ) atˆξ, theslope
must be φ ′ (ˆξ), and we must have
φ(ˆξ) =φ ′ (ˆξ)ˆξ + b,
since φ(ˆξ) =L(ˆξ). Hence, in total, the lower-bounding function is given by
L(ξ) =φ(ˆξ)+φ ′ (ˆξ)(ξ − ˆξ).
Since this is a linear function, we easily calculate the expected value of the