Basic Properties and Theory

1
Basic Properties and Theory

• Two-Stage problem
• Part II

2
Two-Stage Program
Classical Two-Stage Stochastic Linear Program
with Fixed Recourse (Dantzig, Beale)
- vector of first-stage decisions, size (n ?
1) - first-stage data of sizes (n ? 1), (m ? n),
(m ? 1), respectively - random event -
second-stage data of sizes (k ? 1), (l ? 1), (l ?
k), respectively
3
Two-Stage Program (cont'd)
Reduction of classical two-stage stochastic
linear program with fixed recourse to
deterministic equivalent problem
Recourse function
Deterministic equivalent program (DEP)
4
Second Stage Value Function
Theorem 5
For a stochastic program with fixed recourse,
Q(x, ?) is
(a) a piecewise linear convex function in (h,
T) (b) a piecewise linear concave function in
q (c) a piecewise linear convex function in x
for all x in K K1 ? K2 .
is a convex function in b
Proof
Let
Note is a
feasible solution of
Consequently
Statements (b) and (c) can be proved similarly.
5
Second Stage Value Function (cont'd)
? has absolutely continuous distribution F(?)
if there exists function f? (x) such that In
such a case, f? (x) is a density function.
Definition
Theorem 6
For a stochastic program with fixed recourse
where ? has finite second moments
(a) Q(x) is a Lipschitzian convex function and
is finite on K2 . (b) When ? is finite, Q(x)
is piecewise linear. (c) If F(?) is an
absolutely continuous distribution, Q(x) is
differentiable on K2 .
Proof
Convexity can be proved similar to previous
theorem.
6
Complete Recourse
7
Simple Recourse
stochastic program has simple recourse if
Definition
Condition
Representation
8
Simple Recourse
Suppose the two-stage stochastic program is
feasible and has simple recourse and that ? has
finite second moments. Then Q(x) is finite if and
only if qi qi- ? 0 with probability one.
Theorem 7
Proof
Suppose
However, if then
. A similar argument applies if
. Hence, Q(x) is not finite.
Using proposition 2, the result is proven.
9
The Case with Finite Second Moments
Proposition 2
B is square submatrix of W, called basis, such
that
Note basis B may be different for different x
and ? .
Finite Second moment
10
Simple Recourse (cont'd)
Assuming ,
let
Let q and T are fixed. If hi have an
associated distribution Fi , mean value
, and then
where