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
Therefore, the semismoothness of function H at x requires the Hadamard directional
derivative at x to exists along any direction converging to h and not only along h.
Examples of semismooth functions are smooth, convex and piecewise linear functions. An
important property is that composite, scalar products, sums, minimum, maximum preserve
semismoothness. Indeed for the minimum function, the breakpoint is at (0, 0). By standard
calculations, the directional derivate of φ∧ at this point is φ∧((0, 0); (a, b)) = min(a, b).
Futhermore, for all nonzero vector (a, b), φ∧ is differentiable and is uniquely given by
We deduce
∂Bφ∧(0, 0) =
1 0
,
0 1
1a≤b
∇φ∧(a, b) = .
1a>b
and ∂φ∧(0, 0) =
Furthermore, for all V ∈ ∂φ∧(c, d), such that (c, d) → (a, b), we have
V
λ
, λ ∈ [0, 1] . (3.17)
1 − λ
a
− φ∧((0, 0); (a, b)) = a 1c≤d + b 1c>d − min(a, b) = o((a, b)) . (3.18)
b
By Appendix 3.5.1, we conclude that φ∧ is semismooth at (0,0).
Finally, we introduce the strong semismoothness, also called 1-order semismoothness, that
will be used in most convergence theorems.
Definition (strongly semismooth). A locally Lipschitzian function G is strongly semismooth
at x if for all d → 0, ∀V ∈ ∂G(x + d), we have
V d − G ′ (x, d) = O ||d|| 2 .
Based on Equation (3.18), we conclude that the minimum function φ∧ is not strongly
semismooth at (0, 0). But, the Fischer-Burmeister function is strongly semismooth at (0, 0).
In fact, by standard calculations, for all nonzero vector (a, b), we have
We deduce
∇φF B(a, b) =
√ a − 1
a2 +b2 √ b − 1
a2 +b2 ∂BφF B(0, 0) = {∇φF B(a, b), (a, b) = (0, 0)} and ∂φF B(0, 0) = ¯ B((−1, −1), 1), (3.19)
where ¯ B denotes the closed ball.
Furthermore, for all V ∈ ∂φF B(c, d), such that (c, d) → (a, b), we have
Hence, we have
V
a
= φF
b
B(a, b) and φF B((0, 0); (a, b)) = φF B(a, b).
V
a
− φF
b
B((0, 0); (a, b)) = 0.
.
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