Stochastic Programming - Index of

246 STOCHASTIC PROGRAMMING
(f T ,g T ) and a descent direction w T =(u T ,v T ). Assume further that for some
tolerance ε>0,
y j >ε ∀j (strict nondegeneracy). (1.3)
Then the search direction w T =(u T ,v T ) is determined by the linear program
⎫
max τ
s.t. f T u + g T v ≤−τ,
⎪⎬
∇ y G(x) T u + ∇ z G(x) T v ≥ θτ if G(x) ≤ α + ε,
(1.4)
Bu + Nv =0,
v j ≥ 0 if z j ≤ ε, ⎪⎭
‖v‖ ∞ ≤ 1,
where θ>0 is a fixed parameter as a weight for the directional derivatives
of G and ‖v‖ ∞ =max j {|v j |}. Accordingtotheaboveassumption,wehave
from (1.4)
u = −B −1 Nv,
which renders (1.4) into the linear program
where obviously
max τ
s.t. r T v ≤−τ,
s T v ≥ θτ if G(x) ≤ α + ε,
v j ≥ 0ifz j ≤ ε,
‖v‖ ∞ ≤ 1,
⎫
⎪⎬
⎪⎭
(1.5)
r T = g T − f T B −1 N,
s T = ∇ z G(x) T −∇ y G(x) T B −1 N
are the reduced gradients of the objective and the probabilistic constraint
function. Problem (1.5)—and hence (1.4)—is always solvable owing to its
nonempty and bounded feasible set. Depending on the obtained solution
(τ ∗ ,u ∗T ,v ∗T ) the method proceeds as follows.
Case 1 When τ ∗ = 0, ε is replaced by 0 and (1.5) is solved again. If
τ ∗ = 0 again, the feasible solution x T =(y T ,z T ) is obviously optimal.
Otherwise the steps of case 2 below are carried out, starting with
the original ε>0.
Case 2 When 0