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

tel-00703797, version 2 - 7 Jun 2012
3.2. Methods to solve nonlinear equations
– If ||p N || ≤ ∆k, then h ⋆ = p N .
– Else if ||p S || ≥ ∆k, then h ⋆ = ∆k/||p S || × p S .
– Otherwise, we choose a convex combination between the two points p S and p N . That is
we find a λ ∈ [0, 1] such that ||p S + λ(p N − p S )|| = ∆k. We get h ⋆ = λ ⋆ p N + (1 − λ ⋆ )p S
with
λ ⋆ = − < pS , pN − pS > + < pS , pN − pS > 2 −||pN − pS || 2 (||pS || 2 − ∆k)
||pN − pS || 2
,
where < ., . > denotes the scalar product and ||.|| denotes the Euclidean norm.
Nocedal and Wright (2006) also propose a “simple” dogleg method which remove the step
||p S || ≥ ∆k, see Algorithm 11.6.
The double dogleg method finds an approximate solution of the model function mk(h)
assuming the Hessian matrix is ˜ Hk = L T L. The double dogleg method is a variant Powell
dogleg method. Dennis and Schnabel (1996) propose the following procedure.
– If ||p N || ≤ ∆k, then h ⋆ = p N .
– Else if ηk||p N || ≤ ∆k, then h ⋆ = p N × ηk/∆k.
– Else if ||p S || ≥ ∆k then h ⋆ = p S × ∆k/||p S ||.
– Otherwise, we choose λ such that ||p S +λ(ηkp N −p S )|| = ∆k. We get back to the Powell
dogleg case with ηkp N instead of p N .
where the parameter ηk ≤ 1 is defined as
with α = ||gk|| 2 , β = ||Lgk|| 2 .
ηk = 0.2 + 0.8
α 2
β|g T k dN | ,
As for line-search techniques, we use the merit function f(z) = 1
the gradient is given by
g(z) = JacF (z) T F (z).
2 ||F (z)||2 2
. We recall that
Therefore, the approximated Hessian ˜ Hk is JacF (zk) T JacF (zk) as in the Gauss-Newton
model. Hence, the steepest descent point and the Newton point have the following expression
p S k = − gT k gk
gT k J T k Jkgk
gk and p N k = −J T k gk.
As in the previous subsection, when working a quasi-Newton method, the Jacobian is numerically
approximated by a forward difference.
The special case of the Levenberg-Marquardt method
Until now, all globalization methods are adapted for the Newton or the Broyden direction
defined in Equations (3.12) and (3.13). We need to precise how to globalize the Levenberg-
Marquardt direction. This method was introduced in the context of the least-square problem
min 1
2 ||F (z)||22 . In fact, there is a relation between the trust-region approach and the
Levenberg-Marquardt method. The solution to the quadratic problem
min f(zk) + g(xk)
||h||