equilibrium problems with equilibrium constraints - Convex ...

14 Chapter 2 Mathematical Program with Equilibrium Constraints
KKT system of Reg(t ν ) (2.13) for sufficiently small t ν :
∇f(x ν ;a f t ν
) + ∇g(x ν ;a g t ν
) T λ g ν + ∇h(x ν ;a h t ν
) T λ h ν
− ∇G(x ν ;a G t ν
) T λ G ν − ∇H(x ν ;a H t ν
) T λ H ν
+ ∇G(x ν ;a G t ν
) T [H(x ν ;a H t ν
) ◦ λ GH
ν ] + ∇H(x ν ;a H t ν
) T [G(x ν ;a G t ν
) ◦ λ GH
ν ] = 0,
h(x ν ;a h t ν
) = 0,
g(x ν ;a g t ν
) ≤ 0, λ g ν ≥ 0, g(x ν ;a g t ν
) T λ g ν = 0,
(2.19)
G(x ν ;a G t ν
) ≤ 0, λ G ν ≥ 0, G(x ν;a G t ν
) T λ G ν = 0,
H(x ν ;a H t ν
) ≤ 0, λ H ν ≥ 0, H(x ν ;a H t ν
) T λ H ν = 0,
G(x ν ;a G t ν
) ◦ H(x ν ;a H t ν
) − t ν e ≤ 0, λ GH
ν ≥ 0,
[G(x ν ;a G t ν
) ◦ H(x ν ;a H t ν
) − t ν e] T λ GH
ν = 0.
(i) Define ˜λ G i,ν = −λGH i,ν H i(x ν ; a H t ν
) and ˜λ H i,ν = −λGH i,ν G i(x ν ; a G t ν
) for i ∈ I GH (x ν , t ν )
and rewrite the first equation in (2.19) as
−∇f(x ν ; a f t ν
)
= ∑
λ g i,ν ∇g i(x ν ; a g t ν
) + ∑
i∈I g(x ν)
−
∑
i∈I G (x ν,t ν)
i∈I h (x ν)
λ G i,ν∇G i (x ν ; a G t ν
) −
λ h i,ν ∇h i(x ν ; a h t ν
)
∑
i∈I H (x ν,t ν)
λ H i,ν∇H i (x ν ; a H t ν
)
∑
[
−
˜λH i,ν ∇H i (x ν ; a H t ν
) + G i(x ν ; a G ]
t ν
)
H
i∈I GH (x ν,t ν)∩IG c i (x ν ; a H (¯x) t ν
) ∇G i(x ν ; a G t ν
)
∑
[
−
˜λG i,ν ∇G i (x ν ; a G t ν
) + H i(x ν ; a H ]
t ν
)
G
i∈I GH (x ν,t ν)∩IH c i (x ν ; a G (¯x) t ν
) ∇H i(x ν ; a H t ν
)
∑
−
[˜λG i,ν ∇G i (x ν ; a G t ν
) + ˜λ H i,ν ∇H i(x ν ; a H t ν
)]
.
i∈I GH (x ν,t ν)∩I G (¯x)∩I H (¯x)
(2.20)
For every sufficient large ν, we construct a matrix A(x ν ) with rows being the