Stochastic Programming - Index of

48 STOCHASTIC PROGRAMMING
Figure 18
Chance constraints: nonconvex feasible set.
It follows that the feasible set for the above constraints is nonconvex, as shown
in Figure 18.
✷
As above, define S(x) :={ξ | g(x, ξ) ≤ 0}. Ifg(·, ·) is jointly convex in (x, ξ)
then, with x i ∈ B(α),i = 1, 2, ξ i ∈ S(x i ) and λ ∈ [0, 1], for (¯x, ¯ξ) =
λ(x 1 ,ξ 1 )+(1− λ)(x 2 ,ξ 2 )itfollowsthat
g(¯x, ¯ξ) ≤ λg(x 1 ,ξ 1 )+(1− λ)g(x 2 ,ξ 2 ) ≤ 0,
i.e. ¯ξ = λξ 1 +(1− λ)ξ 2 ∈ S(¯x), and hence 5
S(¯x) ⊃ [λS(x 1 )+(1− λ)S(x 2 )]
implying
P (S(¯x)) ≥ P (λS(x 1 )+(1− λ)S(x 2 )).
By our assumption on g (joint convexity), any set S(x) isconvex.Nowwe
conclude immediately that B(α) isconvex∀α ∈ [0, 1], if
P (λS 1 +(1− λ)S 2 ) ≥ min[P (S 1 ),P(S 2 )] ∀λ ∈ [0, 1]
for all convex sets S i ∈F, i =1, 2, i.e. if P is quasi-concave. Hence we have
proved the following
5 The algebraic sum of sets ρS 1 + σS 2 := {ξ := ρξ 1 + σξ 2 | ξ 1 ∈ S 1 , ξ 2 ∈ S 2 }.