Stochastic Programming - Index of

RECOURSE PROBLEMS 193
φ
φ
α
β
α
β
ξ
ξ
Figure 19 An illustration of a situation where both α and β give good
information about curvature.
The parameters α and β contain information about the curvature of φ(ξ).
In particular, note that if, for example, α =0thenπ B is an optimal dual
solution corresponding to ξ = A. Ifπ A ≠ π B in such a case, we are simply
facing dual degeneracy. In line with this argument, a small α (or β) should
mean little curvature. But we may, for example, have α large and β small. So
what is going on
Figure 19 shows two different cases (both in one dimension), where both α
and β are good indicators of how important a random variable is. Intuitively, it
seems reasonable to say that the left part of the figure indicates an important
random variable, and the right part an unimportant one. And, indeed, in the
left part both α and β will be large, whereas in the right part both will be
small.
But then consider Figure 20. Intuitively, the random variable is
unimportant, but, in fact, the slopes at the end points are the same as in
the left part of Figure 19, and the slopes describe how the objective changes
as a function of the random variable. However, in this case α is very small,
whereas β is large. What is happening is that α and β pick up two properties
of the recourse function. If the function is very flat (as in the right part of
Figure 19) then both parameters will be small. If the function is very nonlinear
(as in the left part of Figure 19), both parameters will be large. But if