Stochastic Programming - Index of

BASIC CONCEPTS 91
where the bounded convex set B is assumed to contain an interior point ŷ.
Under the assumptions mentioned, it is possible to include the feasible set
B of problem (8.11) in a convex polyhedron P, which—after our discussions
in Section 1.7—we may expect to be able to represent by linear constraints.
Observe that the inclusion P⊃Bimplies the inequality
min
x∈P cT x ≤ min
x∈B cT x.
The cutting-plane method for problem (8.11) proceeds as follows.
Step 1 Determine a ŷ ∈ int B and a convex polyhedron P 0 ⊃B;letk := 0.
Step 2 Solve the linear program
min{c T x | x ∈P k },
yielding the solution ˆx k .
If ˆx k ∈Bthen stop (ˆx k solves problem (8.11)); otherwise, i.e. if
ˆx k ∉B, determine
λ k := min{λ | λŷ +(1− λ)ˆx k ∈B}
and let
z k := λ k ŷ +(1− λ k )ˆx k .
(Obviously we have z k ∈Band moreover z k is a boundary point of B
on the line segment between the interior point ŷ of B and the point
ˆx k , which is “external” to B.)
Step 3 Determine a “supporting hyperplane” of B in z k (i.e. a hyperplane
being tangent to B at the boundary point z k ). Let this hyperplane be
given as
H k := {x | (a k ) T x = α k }
such that the inequalities
hold. Then define
(a k ) Tˆx k >α k ≥ (a k ) T x ∀x ∈B
P k+1 := P k
⋂
{x | (a k ) T x ≤ α k },
let k := k + 1, and return to step 2.
In Figure 30 we illustrate one step of the cutting-plane method.
Remark 1.11 By construction—see steps 1 and 3 of the above method—we
have
P k ⊃B, k =0, 1, 2, ···,