equilibrium problems with equilibrium constraints - Convex ...

20 Chapter 3 Equilibrium Problem with Equilibrium Constraints
shared decision variables y ∈ R n 0
:
minimize f k (x k , y; ¯x −k )
subject to g k (x k , y; ¯x −k ) ≤ 0, h k (x k , y; ¯x −k ) = 0,
(3.1)
0 ≤ G(x k , y; ¯x −k ) ⊥ H(x k , y; ¯x −k ) ≥ 0,
where f k : R n → R, g k : R n → R p k , h k : R n → R q k , G : R n → R m and
H : R n → R m are twice continuously differentiable functions in both x = (x k ) K k=1
and y, with n = ∑ K
k=0 n k. The notation ¯x −k means that x −k = (x j ) K j=1 \ xk
(∈ R n−n k−n 0
) is not a variable but a fixed vector. This implies that we can
view (3.1), denoted by MPEC(¯x −k ), as being parameterized by ¯x −k . Given ¯x −k ,
we assume the solution set of the k-th MPEC is nonempty and denote it by
SOL(MPEC(¯x −k )). Notice that in the above formulation, each MPEC shares the
same equilibrium constraints, represented by the complementarity system
0 ≤ G(x, y) ⊥ H(x, y) ≥ 0.
The EPEC, associated with K MPECs defined as above, is to find a Nash
equilibrium (x ∗ , y ∗ ) ∈ R n such that
(x k∗ , y ∗ ) ∈ SOL(MPEC(x −k∗ )) ∀ k = 1, . . .,K. (3.2)
Mordukhovich [39] studies the necessary optimality conditions of EPECs in the
context of multiobjective optimization with constraints governed by parametric
variational systems in infinite-dimensional space. His analysis is based on advanced
tools of variational analysis and generalized differential calculus [37, 38].
Since we only consider finite-dimensional optimization problems, following Hu [25],
we use the KKT approach and define stationary conditions for EPECs by applying
those for MPECs.
Definition 3.1. We call a vector (x ∗ , y ∗ ) a B-stationary (strongly stationary, M-
stationary, C-stationary, weakly stationary) point of the EPEC (3.2) if for each
k = 1, . . .,K, (x k∗ , y ∗ ) is a B-stationary (strongly stationary, M-stationary, C-
stationary, weakly stationary) point for the MPEC(x −k∗ ).