Suppose that the constraint correspondence G(y) in the constrained optimization problem (example 2.30)

is defined by a set of inequalities (example 2.40)
g1(x, θ) ‰¤ 0, g2(x, θ) ‰¤ 0, ... , gm (x, θ) ‰¤ 0
If each functional gj (x; θ) ˆŠ F(X × Θ) is convex jointly in x and y, then the correspondence
G(θ) = {x ˆŠ X: gj(x, θ) ‰¤ 0, j = 1,2, ... , m}
is convex.

max fx, 0)