Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
Chapitre 3. Calcul d’équilibre de Nash généralisé
The constrained equation reformulation
Dreves et al. (2011) also propose a constrained equation of the KKT system. Let ◦ denote
the component-wise product, i.e. x ◦ y = (x1y1, . . . , xNyN). Let w ∈ R m be slack variables
(to transform inequality into equality). The KKT conditions are equivalent to
H(x, λ, w) = 0, (x, λ, w) ∈ Z,
⎛ ⎞
D(x, λ)
and H(x, λ, w) = ⎝g(x)
+ w⎠
,
λ ◦ w
(3.4)
where Z is the set Z = {(x, λ, w) ∈ R n+2m , λ ≥ 0, w ≥ 0}.
3.1.2 The QVI reformulation
Let us first define the Variational Inequality (VI) and Quasi-Variational Inequality (QVI)
problems. Variational inequality problems VI(K, F (x)) consist in finding x ∈ K such that
∀y ∈ K, (y − x) T F (x) ≥ 0,
where F : K ↦→ R n . VI problems typically arise in minimization problems. Quasi-variational
inequality problems are an extension of VI problems where the constraint set K depends on
x. The QVI problem is defined as
∀y ∈ K(x), (y − x) T F (x) ≥ 0,
which is denoted by QVI(K(x), F (x)). Note that a QVI has very complex structure since y
must satisfy y ∈ K(x) for a vector x we are looking for.
The GNEP in Equation (3.1) can be reformulated as a QVI problem
∀y ∈ X(x), (y − x) T F (x) ≥ 0,
⎛
∇x1
⎜
with F (x) = ⎝
θ1(x)
.
∇xN θN(x)
⎞
⎟
⎠ , (3.5)
and a constrained set X(x) = {y ∈ R n , ∀i, g i (yi, x−i) ≤ 0}. We have the following theorem
precising the equivalence between the GNEP and the QVI.
Theorem. If objective functions are C 1 and player-convex, the action sets Xν(x−ν) are closed
convex, then we have x ⋆ solves the GNEP if and only if x ⋆ solves the QVI(X(x), F (x)) defined
in Equation (3.5).
This is Theorem 3.3 of Facchinei and Kanzow (2009) or Equation (3) Kubota and Fukushima
(2010).
Penalized sequence of VI
We can express the KKT conditions for the QVI problem, but we naturally get back to
Equation (3.2). According to Facchinei and Kanzow (2009), methods to solve general QVI
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