equilibrium problems with equilibrium constraints - Convex ...

12 Chapter 2 Mathematical Program with Equilibrium Constraints
This further implies that for every such x the gradient vectors of the active constraints
in (2.14) are linearly independent. Hence, NLP-LICQ holds at every
feasible point x ∈ N(¯x).
Given a feasible point ¯x of Reg(t) (2.13), we define the following index sets of
active constraints:
I g (¯x, t) := {i | g i (¯x; a g t) = 0},
I h (¯x, t) := {i | h i (¯x; a h t ) = 0},
I G (¯x, t) := {i | G i (¯x; a G t ) = 0},
(2.16)
I H (¯x, t) := {i | H i (¯x; a H t ) = 0},
I GH (¯x, t) := {i | G i (¯x; a H t )H i (¯x; a H t ) = t}.
Lemma 2.10. If the MPEC-LICQ holds at the feasible point ¯x of the MPEC
(2.10), then there exists a neighborhood N(¯x) and a scalar ¯t > 0 such that for
every t ∈ (0, ¯t ), the NLP-LICQ holds at every feasible point x ∈ N(¯x) of Reg(t)
(2.13).
Proof. This follows from Lemma 2.9 and the following relations on the index sets
of active constraints:
I g (x, t) ⊆
I h (x, t) ⊆
I g (¯x),
I h (¯x),
I G (x, t) ∪ I H (x, t) ∪ I GH (x, t) ⊆
I G (x, t) ∩ I GH (x, t) = ∅,
I G (¯x) ∪ I H (¯x),
(2.17)
I H (x, t) ∩ I GH (x, t) = ∅,
which hold for all x in a sufficiently small neighborhood N(¯x) and all t ∈ (0, ¯t)
for sufficiently small ¯t > 0.