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
compute the direction using the central vector a and (ii) find an appropriate stepsize with
a geometric line-search for which the merit function is ψ(u) = p(H(u)). Note that H(u) is
valued in R n. × R 2m. .
Init z0 ∈ ˚ Ω, 0 < ρ, α < 1 and choose σ0 ∈ [0, ¯σ[
Iterate until a termination criterion is satisfied,
– Solve the system ∗ to get dk
H(zk) + JacH(zk)d = σk
– Find the smallest integer mk such that
aT H(zk)
||a|| 2 a. (3.30)
2
ψ(zk + ρ mk dk) ≤ ψ(zk) + αρ mk ∇ψ(zk) T dk, and zk + ρ mk dk ∈ ˚ Ω.
– zk+1 = zk + ρ mkdk.
end Iterate
Due to the special structure H and a might have, the computation of dk in Equation (3.30), a
modified Newton direction because of the right-hand side term, may be further simplified by
decomposing into its components F and G. In this form, the algorithm is defined when the
Jacobian JacH is nonsingular at zk ∈ ˚ Ω. Lemma 2 of Monteiro and Pang (1999) shows that
the direction computed in the first step is a descent direction for the merit function ψ. So, the
algorithm is well-defined. Their Theorem 3 shows the convergence of the potential reduction
algorithm.
Theorem. Assume p is a potential function, the constrained Equation (3.29) satisfies the
constrained equation blanket assumptions, the Jacobian JacH(z) is nonsingular for all z ∈ ˚ Ω
and we have lim sup k σk < ¯σ. Let (zk) be a sequence generated by the potential reduction
Newton algorithm. We have (i) the sequence (H(zk)) is bounded and (ii) any accumulation
point, if there exists, solves the constrained Equation (3.29). In particular, if (zk) is bounded,
the constrained equation has a solution.
Application to GNEP
As already mentioned, Equation (3.4) of the GNEP can be reformulated as a constrained
equation. The root function H : R n × R 2m ↦→ R n × R 2m is defined as
⎛ ⎞
D(x, λ)
H(x, λ, w) = ⎝g(x)
+ w⎠
,
λ ◦ w
where the dimensions n, m correspond to the GNEP notation and (a, ¯σ) is given by ((0n, 1m), 1).
The potential function is given by
p (u) = ζ log ||x|| 2 2 + ||λ|| 2 2 + ||w|| 2 2 −
m
log(λk) −
k=1
m
log(wk),
∗. In Monteiro and Pang (1999), they use the directional derivative along d in the left-hand side of Equation
(3.30), which is equivalent to this formulation since H is C 1 under the blanket assumptions.
k=1
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