equilibrium problems with equilibrium constraints - Convex ...

6 Chapter 2 Mathematical Program with Equilibrium Constraints
where I GH (¯x) is known as the degenerate set. If I GH (¯x) = ∅, then the feasible
vector ¯x is said to fulfill the strict complementarity conditions.
Associated with any given feasible vector ¯x of MPEC (2.1), there is a nonlinear
program, called the tightened NLP (TNLP(¯x)) [46, 58]:
minimize
f(x)
subject to g(x) ≤ 0, h(x) = 0,
G i (x) = 0, i ∈ I G (¯x),
G i (x) ≥ 0, i ∈ IG c (¯x),
(2.3)
H i (x) = 0, i ∈ I H (¯x),
H i (x) ≥ 0, i ∈ IH c (¯x).
Similarly, there is a relaxed NLP (RNLP(¯x)) [46, 58] defined as follows:
minimize
f(x)
subject to g(x) ≤ 0, h(x) = 0,
G i (x) = 0, i ∈ I c H (¯x),
G i (x) ≥ 0, i ∈ I H (¯x),
(2.4)
H i (x) = 0, i ∈ I c G (¯x),
H i (x) ≥ 0, i ∈ I G (¯x).
It is well known that an MPEC cannot satisfy the standard constraint qualifications,
such as linear independence constraint qualification (LICQ) or Mangasarian-
Fromovitz constraint qualification (MFCQ), at any feasible point [6, 58]. This
implies that the classical KKT theorem on necessary optimality conditions (with
the assumption that LICQ or MFCQ is satisfied at local minimizers) is not appropriate
in the context of MPECs. One then needs to develop suitable variants
of CQs and concepts of stationarity for MPECs. Specifically, the MPEC-CQs are
closely related to those of the RNLP (2.4).
Definition 2.1. The MPEC (2.1) is said to satisfy the MPEC-LICQ (MPEC-
MFCQ) at a feasible point ¯x if the corresponding RNLP(¯x) (2.4) satisfies the