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
points zk+1 = zk − V −1
k F (zk). If (zk)k converges to z ⋆ then (zk)k converges superlinearly to
z ⋆ and F (z ⋆ ) = 0 is equivalent to ∃Wk ∈ ∂bF (zk),
with sk = zk+1 − zk.
||(Vk − Wk)sk||
lim
= 0,
k→+∞ ||sk||
Local convergence of the generalized Levenberg-Marquardt method is studied by Facchinei
and Kanzow (1997), cf. Theorem 6.
Theorem. Let z ⋆ a solution of F (z ⋆ ) = 0, with F a locally Lipschitzian, semismooth at z ⋆
and ∀J ⋆ ∈ ∂BF (z ⋆ ) are nonsingular. If the Levenberg-Marquardt parameters (λk) converge to
0, then the (generalized) Levenberg-Marquardt converges superlinearly to z ⋆ . If in addition, F
is strongly semismooth and locally directionnally differentiable at z ⋆ , and λk = O(||Jkgk||) or
λk = O(||gk||), then the sequence converges quadratically.
We turn our attention to the assumption analysis of preceding theorems. All theorems
require the root function to be semismooth at a solution z ⋆ . This is verified for our function
Φ defined in Equation (3.3) as long as the complementarity function φ is semismooth. Furthermore,
strong semismoothness improves the convergence rate. In the previous subsection,
we have seen this requires for φ to be strongly semismooth, as e.g., the Fischer-Burmeister
function.
The nonsingularity condition is required only for elements J ⋆ of the limiting Jacobian at
a solution z ⋆ . As analyzed in Facchinei et al. (2009), the limiting Jacobian (3.20) might have
some identical rows at the bottom part. Let us investigate first this issue. We recall first the
expression of generalized Jacobian
Jac
JΦ(z) =
xD(x, λ) diag ∇xigi (x)
i .
−Da(z)Jac xg(x) Db(z)
As only terms (ξij, ζij) in diagonal matrices D a and D b change between the generalized and
the limiting Jacobian of Φ, we study the nonsingularity condition directly on the generalized
Jacobian. In a detailed form, the bottom part has the following structure
⎛
⎜
⎝
−D a 1 (x, λ1 )Jac x1 g1 (x) . . . −D a 1 (x, λ1 )Jac xN g1 (x)
.
.
D b 1 (x, λ1 ) 0
. ..
−Da N (x, λN )Jac x1gN (x) . . . −Da N (x, λN )Jac xN gN (x) 0 Db N (x, λN )
(3.23)
where Da i and Db i are mi × mi diagonal matrices. In the following, we denote by Da-part and
Db-part the left and right parts of Equation (3.23).
Assume the generalized Jacobian has two identical rows, say for players i and ĩ and components
ji and jĩ . The, the Db-part requires that the jith row of Db i and the jĩth row of Dbĩ
equals zero
⎞
⎟
⎠ ,
b i j(x, λ i j) = b ĩ jĩ (x, λĩj ) = 0, (3.24)
ĩ
with b i j (x, λi j ) = φ′ b (−gi j (x), λi j ). Identical rows in the Da -part is equivalent to the n dimen-
sional equation
156
a i j(x, λ i j)Jac xg i j(x) = a ĩ jĩ (x, λĩ jĩ )Jac xg ĩ j (x). (3.25)
ĩ