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
We conclude that φF B is strongly semismooth at (0, 0), as proved in Kanzow and Kleinmichel
(1998).
Now, we can express appropriately the generalized Jacobian of the GNEP. We denote by
JΦ(z) elements of the generalized Jacobian ∂Φ(z). Using chain rules and previous definitions,
we have
Jac
JΦ(z) =
xD(x, λ) diag ∇xigi (x)
i , (3.20)
−Da(z)Jac xg(x) Db(z)
where Jac x denotes the Jacobian with respect to x and diag[...] represents a block diagonal
matrix, see Appendix 3.5.1 for a extended representation of the generalized Jacobian JΦ.
The diagonal matrices Da and Db are given by
Da(z) = diag[a 1 (x, λ 1 ), . . . , a N (x, λ N )] and Db(z) = diag[b 1 (x, λ 1 ), . . . , b N (x, λ N )],
with ai (x, λi ), bi (x, λi ) ∈ Rmi defined as
(a i j(x, λ i j), b i j(x, λ i
j)) =
φ ′ a(−gi j (x), λij ), φ′ b (−gi j (x), λij )
if (−gi j (x), λi j ) = (0, 0),
(ξij, ζij) if (−gi j (x), λij ) = (0, 0),
where φ ′ a (resp. φ ′ b ) denotes the derivative of φ with respect to the first (second) argument a
(b) and (ξij, ζij) ∈ ¯ B(pφ, cφ), the closed ball at pφ of radius cφ.
Let us precise the top-left part.
⎛
Jac xD(x, λ) =
⎜
⎝
Jac x1 L1(x, λ 1 ) . . . Jac xN L1(x, λ 1 )
.
Jac x1 LN(x, λ N ) . . . Jac xN LN(x, λ N )
where Li(x, λi ) = ∇xiθi(x)+ mi j=1 ∇xigi j (x)λij . The top-right part is block diagonal and given
by
⎛
Jac x1
⎜
⎝
g1 (x) T 0
. ..
0 Jac xN gN (x) T
⎞
⎟
⎠ .
Let us specify the parameters pφ and cφ for the two considered complementarity functions.
For the minimum function, using Equation (3.17), we have pφ = (1/2, 1/2) and cφ = 1/2.
For the Fischer-Burmeister function, using Equation (3.19), we have pφ = (−1, −1) and cφ =
1. We refer to Kanzow and Kleinmichel (1998) and Facchinei and Pang (2003) for other
complementarity functions.
If functions θi and gi are Ck+1 , by the chain rule, the root function Φ defined in Equation
(3.20) is Ck except at points z such that gi j (x) = λij = 0. At these points, when the
complementarity function is φ∧, Φ is semismooth, while for φF B, Φ is strongly semismooth.
Extension and local convergence in the semismooth framework
As the Jacobian of the root function is not available, the direction computation of local
methods presented in Subsection 3.2.1 must be adapted. The solution consists in replacing
the Jacobian by an element of the generalized Jacobian.
Considering the Newton method (3.12), the direction solves
154
.
⎞
⎟
⎠ ,
Jkdk = −F (zk), (3.21)