Stochastic Programming - Index of

PROBABILISTIC CONSTRAINTS 247
Case 3
Step 2 Solve (1.5). If still τ ∗ ≤ ε, go to step 1; otherwise, case 3
applies.
When τ ∗ >ε, w ∗T =(u ∗T ,v ∗T ) is accepted as search direction.
If a search direction w ∗T =(u ∗T ,v ∗T ) has been found, a line search follows
using bisection. Since the line search in this case amounts to determining the
intersection of the ray x + µw ∗ ,µ ≥ 0 with the boundary bdB(α) within the
tolerance ε, the evaluation of G(x) becomes important. For this purpose a
special Monte Carlo technique is used, which allows efficient computation of
upper and lower bounds of G(x) aswellasthegradient∇G(x).
If the next iterate ˇx, resulting from the line search, still satisfies strict
nondegeneracy, the whole step is repeated with the same partition of D into
basic and nonbasic parts; otherwise, a basis exchange is attempted to reinstall
strict nondegeneracy for a new basis.
4.2 Separate Chance Constraints
Let us now consider stochastic linear programs with separate (or single) chance
constraints as introduced at the end of Section 1.4. Using the formulation given
there we are dealing with the problem
}
min x∈X E˜ξc T (˜ξ)x
(2.1)
s.t. P ({ξ | T i (ξ)x ≥ h i (ξ)}) ≥ α i ,i=1, ···,m,
where T i (ξ) istheith row of T (ξ). The main question is whether or under
what assumptions the feasibility set defined by any one of the constraints
in (2.1),
{x | P ({ξ | T i (ξ)x ≥ h i (ξ)} ≥α i },
is convex. As we know from Section 1.6, this question is very simple to answer
for the special case where T i (ξ) ≡ T i , i.e. where only the right-hand side h i (˜ξ)
is random. That is, with F i the distribution function of h i (˜ξ),
{x | P ({ξ | T i x ≥ h i (ξ)}) ≥ α i } = {x | F i (T i x) ≥ α i }
= {x | T i x ≥ F −1
i
(α i )}.
It follows that the feasibility set for this particular chance constraint is just
the feasibility set of an ordinary linear constraint.
For the general case let us first simplify the notation as follows. Let
B i (α i ):={x | P ({(t T ,h) T | t T x ≥ h}) ≥ α i },
where (˜t T , ˜h) T is a random vector. Assume now that (˜t T , ˜h) T has a joint
normal distribution with expectation µ ∈ IR n+1 and (n +1)× (n +1)