Stochastic Programming - Index of

80 STOCHASTIC PROGRAMMING
solutions, {u {1} , ···,u {k} , ···,u {r} },of{u | W T u ≤ q}, it follows that
ˆθ =max 1≤i≤k (h − T ˆx) T u {i}
≤ max 1≤i≤r (h − T ˆx) T u {i}
= f(ˆx)
≤ ˆθ
and hence ˆθ = f(ˆx), which implies the optimality of ˆx.
✷
Summarizing the above remarks we have the following.
Proposition 1.20 Provided that the program (7.17) is solvable and {x |
Ax = b, x ≥ 0} is bounded, the dual decomposition method yields an optimal
solution after finitely many steps.
We have described this method for the data structure of the linear program
(7.17) that would result if a stochastic linear program with recourse had just
one realization of the random data. To this end, we introduced the feasibility
and optimality cuts for the recourse function f(x) := min{q T y | Wy =
h − Tx, y ≥ 0}. The modification for a finite discrete distribution with K
realizations is immediate. From the discussion in Section 1.5, our problem is
of the form
{ K∑
min c T x + q iT y i}
i=1
s.t. Ax = b
T i x + Wy i = h i , i =1, ···,K
x ≥ 0,
y i ≥ 0, i =1, ···,K.
Thus we may simply introduce feasibility and optimality cuts for all the recourse
functions f i (x) :=min{q iT y i | Wy i = h i − T i x, y i ≥ 0}, i=1, ···,K,
yielding the so-called multicut version of the dual decomposition method. Alternatively,
combining the single cuts corresponding to the particular blocks
i =1, ···,K with their respective probabilities leads to the so-called L-shaped
method.
1.8 Nonlinear Programming
In this section we summarize some basic facts about nonlinear programming
problems written in the standard form
}
min f(x)
(8.1)
s.t. g i (x) ≤ 0, i =1, ···,m.