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
Definition (limiting Jacobian). The limiting Jacobian of G at x is defined as
∂BG(x) = {Vx, ∃(xk)k ∈ DG, xk → x, JacG(xk) → Vx},
where DG is the differentiability set of G.
For his desirable properties, the Clarke’s generalized Jacobian, based on the limiting Jacobian,
is commonly used.
Definition (generalized Jacobian). The generalized Jacobian ∂G(x) of G at x is the convex
hull of the limiting Jacobian ∂BG(x) ∗ .
Let us start with an example. Using the Example 7.1.2 of Facchinei and Pang (2003), we
define a function G : R 2 ↦→ R 2 with components G1(x) = min(x1, x2) and G2(x) = |x1| 3 − x2.
G is not differentiable at (0,0). By splitting R 2 into 4 parts, we can compute the limiting
Jacobian
∂BG((0, 0)) =
1 0 0 1
,
.
0 −1 0 −1
Then, the generalized Jacobian is ∂G((0, 0)) = co ∂BG((0, 0)). From a computational point
of view, using a component-wise version of the generalized Jacobian ∂G(x) ⊂ ∂G1(x) × · · · ×
∂Gm(x) is sometimes useful, see Appendix 3.5.1.
Now, we focus on the properties of the generalized Jacobian and more flexible criterion than
continuous differentiability. Before introducing the semismoothness, we present directional
derivative of a function.
Definition (directional derivative). For a locally Lipschitzian function G, the (classical) directional
derivative at x along h ∈ R n is defined as
G ′ (x; h) = lim
t↓0
The Hadamard directional derivative is
G(x + th) − G(x)
.
t
G ′ H(x; h) = lim
h ′ G(x + th
→h,t↓0
′ ) − G(x)
.
t
The difference between classical and Hadamard directional derivatives lies the fact that
we look at all directions h ′ → h and not only at h. Now, we can define the semismoothness.
Definition (semismooth). A locally Lipschitzian function G is semismooth at x, if for all
d ∈ R n , the following limit exists
lim
V ∈∂G(x+t ˜ V
d)
˜d→d,t↓0
˜ d.
In the case, this limit exists and equals to G ′ H (x; h).
∗. Note that if m = 1, the generalized Jacobian reduces to the generalized gradient, see Theorem 2.5.1 of
Clarke (1990) for this characterization of the generalized gradient.
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