equilibrium problems with equilibrium constraints - Convex ...

10 Chapter 2 Mathematical Program with Equilibrium Constraints
2.2 A Generalization of Scholtes’s Regularization
In this section, we present a generalization of Scholtes’s regularization scheme [62].
Our approach suggests relaxing the complementarity constraints and perturbing
the coefficients in the objective function and constraints simultaneously. Hence,
Scholtes’s scheme is a special case of our approach if the objective function and
constraints are not perturbed. We show that the convergence analysis studied
in [62] can be extended to our method without any difficulty. The convergence
results of our method will be applied to establish the convergence of the sequential
nonlinear complementarity algorithm in the next section.
For any mapping F : R n × A F → R m , where R n is the space of variables and
A F is the space of (fixed) parameters, we denote the mapping as F(x; ā F ) with
x ∈ R n and ā F ∈ A F . The order of elements in ā F is mapping specific. For any
positive sequence {t} tending to 0, we perturb the parameters in F and denote
the new parameter vector as a F t with a F t → ā F as t → 0, and a F t = ā F when t = 0.
Note that the perturbation on ā F does not require the perturbed vector a F t to be
parameterized by t.
To facilitate the presentation, we let Ω := {f, g, h, G, H} denote the collection
of all the functions in the MPEC (2.1). With the notation defined above, the
MPEC (2.1) is presented as
minimize f(x; ā f )
subject to g(x; ā g ) ≤ 0, h(x; ā h ) = 0,
(2.10)
0 ≤ G(x; ā G ) ⊥ H(x; ā H ) ≥ 0,
where ā ω ∈ A ω , for all ω ∈ Ω.
For any positive sequence {t} tending to 0, we perturb every parameter vector
ā ω and denote the perturbed parameter vector as a ω t for all ω ∈ Ω. The perturbed
vector a ω t should satisfy the following two conditions for all ω ∈ Ω:
a ω t → ā ω , as t → 0 + . (2.11)
a ω t = ā ω , when t = 0. (2.12)