Stochastic Programming - Index of

PROBABILISTIC CONSTRAINTS 245
be treated computationally. In particular, we shall verify that, under our
assumptions, a program with joint chance constraints becomes a convex
program and that programs with separate chance contraints may be
reformulated to become a deterministic convex program amenable to standard
nonlinear programming algorithms.
4.1 Joint Chance Constrained Problems
Let us concentrate on the particular stochastic linear program
⎫
min c T x
⎪⎬
s.t. P ({ξ | Tx≥ ξ}) ≥ α
Dx = d, ⎪⎭
x ≥ 0.
(1.1)
For this problem we know from Propositions 1.5–1.7 in Section 1.6 that if the
distribution function F is quasi-concave then the feasible set B(α) is a closed
convex set.
Under the assumption that ˜ξ has a (multivariate) normal distribution,
we know that F is even log-concave. We therefore have a smooth convex
program. For this particular case there have been attempts to adapt penalty
and cutting-plane methods to solve (1.1). Further, variants of the reduced
gradient method as sketched in Section 1.8.2 have been designed.
These approaches all attempt to avoid the “exact” numerical integration
associated with the evaluation of F (Tx)=P ({ξ | Tx ≥ ξ}) and its gradient
∇ x F (Tx) by relaxing the probabilistic constraint
P ({ξ | Tx≥ ξ}) ≥ α.
To see how this may be realized, let us briefly sketch one iteration of the
reduced gradient method’s variant implemented in PROCON, a computer
program for minimizing a function under PRObabilistic CONstraints.
With the notation
let x be feasible in
G(x) :=P ({ξ | Tx≥ ξ}),
min c T x
s.t. G(x) ≥ α,
Dx = d,
x ≥ 0,
⎫
⎪⎬
⎪⎭
(1.2)
and—assuming D to have full row rank—let D be partitioned as D =(B,N)
into basic and nonbasic parts and accordingly partition x T =(y T ,z T ), c T =