equilibrium problems with equilibrium constraints - Convex ...

16 Chapter 2 Mathematical Program with Equilibrium Constraints
an index j ∈ I GH (¯x) such that
− lim
ν→∞
λ GH
j,ν H j(x ν ; a H t ν
) = ¯λ G j < 0,
− lim
ν→∞
λ GH
j,ν G j(x ν ; a G t ν
) = ¯λ H j ≤ 0.
From (i), this further implies that j ∈ I 0 and G j (x ν ; a G t ν
)H j (x ν ; a H t ν
) = t ν for every
sufficiently large ν.
For every ν, we construct a matrix B(x ν ) with rows being the transpose of the
vectors
∇g i (x ν ; a g t ν
), i ∈ I g (¯x),
∇h i (x ν ; a h t ν
), i ∈ I h (¯x),
∇G i (x ν ; a G t ν
), i ∈ I G (¯x),
∇H i (x ν ; a H t ν
), i ∈ I H (¯x).
The sequence of matrices {B(x ν )} converges to the matrix B(¯x) with linearly
independent rows
∇g i (¯x; ā g ), i ∈ I g (¯x),
∇h i (¯x; ā h ), i ∈ I h (¯x),
∇G i (¯x; ā G ), i ∈ I G (¯x),
∇H i (¯x; ā H ), i ∈ I H (¯x).
Since the MPEC-LICQ holds at ¯x, it follows that the rows in the matrix B(x ν )
are linearly independent for every sufficiently large ν. Consequently, the following
system has no solutions for ν large enough:
B(x ν ) T z ν = 0, z ν ≠ 0. (2.22)
By Gale’s theorem of alternatives [34, p. 34], the following system has a solution