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

tel-00703797, version 2 - 7 Jun 2012
x ⋆ 1 x⋆ 2 λ ⋆ 1 λ ⋆ 1
z ⋆1 2 -2 0 5 × 2 5
z ⋆2 -2 3 8 0
z ⋆3 0 1 4 × 3 4 0
z ⋆4 1 0 2 9 6
Table 3.1: Four GNEs
3.3. Numerical results
test problem such as the river basin pollution game of Krawczyk and Uryasev (2000) or the
example of Rosen (1965).
Due to the simple form of the objective function, we can solve the KKT system for this
GNEP, manually. The solutions are listed in Table 3.1.
Before discussing the results, we detail the stopping criteria used in our optimization
procedure. They are based on Chapter 7 of Dennis and Schnabel (1996). Algorithms always
stop after a finite number of iterations with an exit code specifying whether the sequence
converges or not: (1) convergence is achieved ||F (z)||∞ < ftol with ftol = 10 −8 , (2) algorithm
is stuck because two consecutive iterates are too close ||(zk − zk−1)/zk||∞ < xtol with xtol =
10 −8 , (3) stepsize is too small tk < xtol or radius is too small ∆k < xtol, (4) the iteration
limit is exceeded k > kmax with kmax = 300 or generalized Jacobian is too ill-conditionned
(5) or singular (6).
On this example, we compare the following methods: the Newton and Broyden methods
with a globalization scheme (line search or trust-region), the Levenberg-Marquardt (LM)
method with line search, the Levenberg-Marquardt with adaptive parameter, the constrainedequation
modified Newton. In the following tables, the results are labelled as follows: GLS
stands for geometric line search, QLS quadratic line search, PTR Powell trust-region, DTR
double dogleg trust-region. We report the number of calls to the root function and the generalized
Jacobian, the time (sec), the final iterate z ⋆ (when successful convergence), the value
of Euclidean norm at z ⋆ and the exit code. Meanings of exit code are 1 for successful convergence,
2 or 3 consecutive iterates too small, 4 iteration limit exceeded, 5 or 6 ill-conditionned
Jacobian.
In Table 3.2, we report the result with the complementarity function φF B and the starting
point z0 = (5, 5, 0, 0), while for the constrained equation method, the starting point is
z0 = (5, 5, 2, 2, 2, 2). Most methods converge to different equilibria. Surprisingly, the Newton
method with geometric line search converges to z ⋆1 , whereas all Broyden methods converge to
z ⋆2 and Newton trust-region methods converge to z ⋆3 , despite using the same initial points.
There is only one method diverging to a local minimum of the merit function 1/2||F (z ⋆ )|| 2 2
which is not a root of F : the Newton method with quadratic line search. The LM method
with adaptive parameter and modified Newton of constrained equation are stuck on singular
matrices. In overall, there is a clear advantage for classic semismooth methods solving the
extended KKT system on this example.
With the minimum complementarity function φ∧, we get similar results, see Table 3.4
in Appendix 3.5.2. Newton methods converges to a different GNE than convergent Broyden
methods. This time, Levenberg-Marquardt method with adaptive parameter is convergent
in relative few iterations. But again, the constrained equation Newton method is divergent
because of a singular Jacobian.
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