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

tel-00703797, version 2 - 7 Jun 2012
3.1. Problem reformulations
problem arising from GNEP are still missing. However, Fukushima and Pang (2005) propose
to solve the QVI(X(x), F (x)) as a sequence of penalized variational inequality problems
VI( ˜ X, ˜ Fk), where ˜ Fk is defined as
with
⎛
∇x1
˜Fk(x)
⎜
= ⎝
θ1(x)
.
∇xN θN(x)
⎞ ⎛ ⎞
P1(x)
⎟ ⎜ ⎟
⎠ + ⎝ . ⎠ , (3.6)
PN(x)
mν
Pν(x) = u k i + ρkg ν
i (x)
+ ∇xν g ν i (x).
i=1
The set ˜ X is either Rn or a box constraint set [l, u] ⊂ Rn , (ρk) an increasing sequence of
penalty parameters and (uk) a bounded sequence. Theorem 3 of Fukushima and Pang (2005)
shows the convergence of the VIP solutions x⋆ k to a solution of the QVI under some smoothness
conditions. We will see later that the QVI reformulation for a certain class of generalized games
reduces to a standard VI problem. In that case, it makes sense to use that reformulation.
3.1.3 Nikaido-Isoda reformulation
We present a last reformulation of the GNEP, which was originally introduced in the
context of standard Nash equilibrium problem. We define the Nikaido-Isoda function as the
function ψ from R 2n to R by
ψ(x, y) =
N
[θ(xν, x−ν) − θ(yν, x−ν)]. (3.7)
ν=1
This function represents the unilateral player improvement of the objective function between
actions x and y. Let ˆ V be the gap function
ˆV (x) = sup
y∈X(x)
ψ(x, y).
Theorem 3.2 of Facchinei and Kanzow (2009) shows the relation between GNEPs and the
Nikaido-Isoda function.
Theorem. If objective functions θi are continuous, then x ⋆ solves the GNEP if and only if
x ⋆ solves the equation
ˆV (x) = 0 and x ∈ X(x), (3.8)
where the set X(x) = {y ∈ R n , ∀i, g i (yi, x−i) ≤ 0} and ˆ V defined in (3.7). Furthermore, the
function ˆ V is such that ∀x ∈ X(x), ˆ V (x) ≥ 0.
As for the QVI reformulation, Equation (3.8) has a very complex structure. There is no
particular algorithm able to solve this problem for a general constrained set X(x). But a
simplification will occur in a special case.
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