Stochastic Programming - Index of

40 STOCHASTIC PROGRAMMING
Figure 14 Linear mapping violating assumption (5.1).
Since the B l are convex polyhedral cones in IR m1
(see Section 1.7) with
nonempty interiors, they may be represented by inequality systems
C l z ≤ 0,
where C l ≠ 0 is an appropriate matrix with no row equal to zero. Fix l and
let ξ ∈ D l (x) such that, by (5.1), h(ξ) − T (ξ)x ∈ intB l .Then
C l [h(ξ) − T (ξ)x] < 0,
i.e. for any fixed j there exists a ˆτ lj > 0 such that
or, equivalently,
C l [h(ξ) − T (ξ)(x ± τ lj e j )] ≤ 0
C l [h(ξ) − T (ξ)x] ≤∓τ lj C l T (ξ)e j ∀τ lj ∈ [0, ˆτ lj ].
Hence for γ(ξ) =max i
∣ ∣(C l T (ξ)e j ) i
∣ ∣ there is a
t l > 0: C l [h(ξ) − T (ξ)x] ≤−|t|γ(ξ)e ∀|t| 0such
that
C l [h(ξ) − T (ξ)x] ≤−|t|γe ∀|t| 0such
that
D l (x; t) :={ξ | C l [h(ξ) − T (ξ)x] ≤−|t|γe} ≠ ∅∀|t|