Stochastic Programming - Index of

RECOURSE PROBLEMS 229
that
0 ≥ E˜ξF
(x ∗ , ˜ξ) − E˜ξF
(x ν , ˜ξ) ≥ E(v ν | x 0 , ···,x ν ) T (x ∗ − x ν )+γ ν , (9.10)
where
γ ν = −b νT (x ∗ − x ν ). (9.11)
Intuitively, if we assume that {x ν } converges to x ⋆ and all v ν are uniformly
bounded, i.e. |v ν |≤α for some constant α, we should require that ‖b ν ‖ ν→∞
−→ 0,
ν→∞
implying γ ν −→ 0 as well. Observe that the particular choice of a stochastic
subgradient
v ν ∈ ∂ x F (x ν ,ξ ν ), (9.12)
or more generally
v ν = 1
N ν
∑N ν
w µ , w µ ∈ ∂ x F (x ν ,ξ νµ ), (9.13)
µ=1
where the ξ ν or ξ νµ are independent samples of ˜ξ, would yield b ν = 0,
γ ν =0∀ν, provided that the operations of integration and differentiation may
be exchanged, as asserted for example by Proposition 1.2 for the differentiable
case.
Finally, assume that for the step size ρ ν together with v ν and γ ν we have
ρ ν ≥ 0,
∞∑
ρ ν = ∞,
ν=0
∞∑
E˜ξ(ρ ν |γ ν | + ρ 2 ν‖v ν ‖ 2 ) < ∞. (9.14)
ν=0
With the choices (9.12) or (9.13), for uniformly bounded v ν this assumption
could obviously be replaced by the step size assumption
ρ ν ≥ 0,
∞∑
ρ ν = ∞,
ν=0
∞∑
ρ 2 ν < ∞. (9.15)
ν=0
With these prerequisites, it can be shown that, under the assumptions (9.3),
(9.8) and (9.14) (or (9.3), (9.12) or (9.13), and (9.15)) the iterative
method (9.4) converges almost surely (a.s.) to a solution of (9.2).
3.10 Solving Many Similar Linear Programs
In both the L-shaped (continuous and integer) and stochastic decomposition
methods we are faced with the problem of solving many similar LPs. This is
most obvious in the L-shaped method: cut formation requires the solution of
many LPs that differ only in the right-hand side and objective. This amount of