Stochastic Programming - Index of

DYNAMIC SYSTEMS 137
optimization is to move constraints that are seen as complicated into the
objective function, and penalize deviations. We outlined a number of different
approaches. For scenario aggregation, the appropriate one is the augmented
Lagrangian method. Its properties, when used with equality constraints such
as (6.2), were given in Propositions 1.27 and 1.28. Note that if we move the
implementability constraints into the objective, the remaining constraints are
separable in the scenarios (meaning that there are no constraints containing
information from more than one scenario). Our objective then becomes
{
∑ ∑ T
p(s) α t[ r t (zt s ,x s t,ξt s )
s∈S t=0
⎫
+wt s (x s t − ∑ p s′ x
]
⎬
s′
t
p({s} t ) ) + α T +1 Q(zT s +1)
⎭
s ′ ∈{s} t
(6.3)
where wt s is the multiplier for implementability for scenario s in period t.
If we add an augmented Lagrangian term, this problem can, in principle,
be solved by an approach where we first fix w, then solve the overall problem,
then update w and so on until convergence, as outlined in Section 1.8.2.4.
However, a practical problem (and a severe one as well) results from the fact
that the augmented Lagrangian term will change the objective function from
one where the different variables are separate, to one where products between
variables occur. Hence, although this approach is acceptable in principle, it
does not work well numerically, since we have one large problem instead of
many scenario problems that can be solved separately. What we then do is to
replace
∑ p s′ x s′
t
p({s} t )
s ′ ∈{s} t
with
x({s} t )=
∑ p s′ x s′
t
p({s} t )
s ′ ∈{s} t
from the previous iteration. Hence, we get
{
∑ ∑ T
p(s)
s∈S
t=0
α t[ ]
r t (zt s ,x s t,ξt s )+wt s [x s t − x({s} t )] + α T +1 Q(zT s +1)
But since, for a fixed w, thetermswt sx({s} t) are fixed, we can as well drop
them. If we then add an augmented Lagrangian term, we are left with
∑
{
∑ T
p(s) α t[ r t (zt s ,xs t ,ξs t )+ws t xs t + 1 2 ρ[xs t − x({s} t)] 2] + α T +1 Q(zT s +1
}. )
s∈S
t=0
}
.