Stochastic Programming - Index of

138 STOCHASTIC PROGRAMMING
procedure scenario(s, x,x s );
begin
Solve the problem
{ T
}
∑
min α t [r t (z t ,x t ,ξt s )+wt s x t + 1 ρ(x 2 t − x) 2 ]+α T +1 Q(z T +1 )
t=0
s.t.
z t+1 = G t (z t ,x t ,ξt s ) for t =0,...,T, with z 0 given,
A t (z t ) ≤ x t ≤ B t (z t )fort =0,...,T,
to obtain x s =(x s 0 ,...,xs T )andzs =(z s 0 ,...,zs T +1 );
end;
Figure 13
Procedure for solving individual scenario problems.
Our problem is now totally separable in the scenarios. That is what we need
to define the scenario aggregation method. See the algorithms in Figures 13
and 14 for details. A few comments are in place. First, to find an initial
x({s} t ), we can solve (6.1) using expected values for all random variables.
Finding the correct value of ρ, and knowing how to update it, is very hard.
We discussed that to some extent in Chapter 1: see in particular (8.17). This is
a general problem for augmented Lagrange methods, and will not be discussed
here. Also, we shall not go into the discussion of stopping criteria, since the
details are beyond the scope of the book. Roughly speaking, though, the goal
is to have the scenario problems produce implementable solutions, so that x s
equals x({s} t ).
Example 2.3 This small example concerns a very simple fisheries
management model. For each time period we have one state variable, one
decision variable, and one random variable. Let z t be the state variable,
representing the biomass of a fish stock in time period t, and assume that
z 0 is known. Furthermore, let x t be a decision variable, describing the portion
of the fish stock caught in a given year. The implicit assumption made here
is that it requires a fixed effort (measured, for example, in the number of
participating vessels) to catch a fixed portion of the stock. This seems to be
a fairly correct description of demersal fisheries, such as for example the cod
fisheries. The catch in a given year is hence z t x t .
During a year, fish grow, some die, and there is a certain recruitment. A
common model for the total effect of these factors is the so called Schaefer
model, where the total change in the stock, due to natural effects listed above,