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
where u = (x, λ, w) ∈ Rn × Rm ⋆+ × Rm ⋆+ and ζ > m. This reformulation of the potential
function emphasizes the three components u = (x, λ, w). For the line-search, the gradient ∇p
is given by
⎛
⎞
∇p(x, λ, w) =
⎜
⎝
2ζ
||x|| 2 2 +||λ||2 2 +||w||2 2
2ζ
||x|| 2 2 +||λ||2 2 +||w||2 2
2ζ
||x|| 2 2 +||λ||2 2 +||w||2 2
x
λ − λ−1 w − w−1 ⎟
⎠ ,
where λ and w have positive components and terms λ −1 and w −1 correspond to the componentwise
inverse vector. Compared to the semismooth reformulation, the root function H is now
C1 . The Jacobian is given by
⎛
Jac xD(x, λ)
JacH(x, λ, w) = ⎝
diag ∇xigi (x)
Jac xg(x)
i
0
0
I
⎞
⎠ .
0 diag[w] diag[λ]
As reported in Dreves et al. (2011), the computation of the direction dk = (dx,k, dλ,k, dw,k) in
Equation (3.30) can be simplified due to the special structure of the above Jacobian matrix.
The system reduces to a linear system of n equations to find dx,k and the 2m components
dλ,k, dw,k are simple linear algebra. Using the classic chain rule, the gradient of the merit
function is given by
∇ψ(x, λ, w) = JacH(x, λ, w) T ∇p(H(x, λ, w)).
Again the computation of this gradient can be simplified due to the sparse structure of JacH.
Theorem 4.3 of Dreves et al. (2011) is the direct application of the previous theorem in the
GNEP context. We do not restate here their theorem, but present their nonsingularity result
given in Theorem 4.6. The Jacobian matrix is nonsingular, if the matrix Jac xD(x, λ) is
nonsingular and
M = Jac xg(x)Jac xD(x, λ) −1 diag ∇xigi (x)
(3.31)
is a P0-matrix. This is exactly Equation (3.28) given in the semismooth setting.
3.3 Numerical results
In this section, we perform a numerical illustration to compare the different methods
presented in this paper. The implementation is done in the R statistical software and the
package GNE, freely available on internet.
Our test problem is a simple two-player polynomial-objective game for which there are
four generalized Nash equilibria. The objective functions (to be minimized) are given by
θ1(x) = (x1 − 2) 2 (x2 − 4) 4 and θ2(x) = (x2 − 3) 2 (x1) 4 ,
for x ∈ R 2 , while the constraint functions are
g1(x) = x1 + x2 − 1 ≤ 0 and g2(x) = 2x1 + x2 − 2 ≤ 0.
Objective functions are player strictly convave. This problem is simple but not simplistic,
since second-order partial derivatives of objective functions are not constant, as for other
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