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

tel-00703797, version 2 - 7 Jun 2012
In the jointly convex case, we get
3.2. Methods to solve nonlinear equations
ˆV (x) = 0 and x ∈ X(x), (3.11)
where the set X(x) = {y ∈ R n , g(yi, x−i) ≤ 0}. Still the computation of ˆ V is a complex optimization
over a constrained set X(x). As in the previous subsection, the class of GNE called
variational equilibrium can be characterized by the NI formulation. We have the folllowing
theorem.
Theorem. Assuming θi and g are C 1 functions and g is convex and θi player-convex. x ⋆ is
a variational equilibrium if and only if x ⋆ ∈ X and V (x ⋆ ) = 0 with V defined as
V (x) = sup ψ(x, y).
y∈X
In the rest of the paper, we do not study all algorithms but rather focus on the most
promising ones. We restrict our attention to general GNEPs and algorithms to solve the KKT
system presented in Subsection 3.1.1. So, we do not study jointly convex GNEPs for which
special methods have been proposed in the literature. These two situations differs widely,
since in the general GNEP, we have to solve a nonlinear equation, while for the jointly convex
case, we solve a fixed point equation or a minimization problem.
3.2 Methods to solve nonlinear equations
As introduced in many optimization books, see, e.g., Dennis and Schnabel (1996); Nocedal
and Wright (2006); Bonnans et al. (2006), an optimization method to solve a nonlinear
equation or more generally to find the minimum of a function is made of two components:
a local method and a globalization scheme. Assuming the initial point is not “far” from the
root or the optimal point, local methods use a local approximation of the function, generally
linear or quadratic approximation based on the Taylor expansion, that is easier to solve. The
globalization studies adjustments to be carried out, so that the iterate sequence still converges
when algorithms are badly initialized.
To emphasize the prominent role of the globalization, we first look at a simple example of
a nonlinear equation. Let F : R2 ↦→ R2 be defined as
x2 F (x) = 1 + x2 2
− 2
e x1−1 + x 3 2 − 2
This function only has two roots x ⋆ = (1, 1) and ¯x = (−0.7137474, 1.2208868). We notice that
the second component of F explodes as x1 tends to infinity.
On Figure 3.2, we plot the contour level of the norm ||F (x)||2, as well as two iterate
sequences (xn), (yn) (see numbers 0, 1, 2,. . . ) starting from the point (x0, y0) = (−1, −3/2).
The first sequence (xn) corresponds to a “pure” Newton method, which we will present after,
whereas the second sequence (yn) combine the Newton method with a line search (LS). We
can observe the sequence (yn) converges less abruptly to the solution x ⋆ than the sequence
(xn).
On Figure 3.3, we plot the contour level of the norm ||F (x)||2 with two iterate sequences
(xn), (yn), for pure and line-search Newton, respectively. But this time, sequences are initiated
at (x0, y0) = (2, 1/2). Despite being close the solution ¯x, the pure sequence (xn) wanders in
.
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