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
Fct. call Jac. call Time x ⋆ 1 x ⋆ 2 λ ⋆ 1 λ ⋆ 1 ||F (z ⋆ )|| Code
Newton - GLS 96 24 0.008 2 -2 -2.8e-12 160 2.8e-12 1
Newton - QLS 67 20 0.007 11 3
Newton - PTR 322 217 0.102 7e-04 1 324 -2.3e-17 8.6e-09 1
Newton - DTR 317 217 0.056 0.00066 1 324 -4.6e-17 6.9e-09 1
Broyden - GLS 78 4 0.005 -2 3 8 2.4e-09 3e-09 1
Broyden - QLS 52 3 0.005 -2 3 8 4.8e-13 1.2e-09 1
Broyden - PTR 91 3 0.006 -2 3 8 8.5e-09 1.2e-08 1
Broyden - DTR 127 3 0.008 -2 3 8 6.6e-13 1.1e-09 1
LM min - GLS 29 29 0.02 -2 3 8 -1.4e-09 3.7e-09 1
LM adaptive 368 184 0.111 0.00019 6
Mod. CE Newton 1782 158 0.295 2500 6
Table 3.2: Results with starting point (5, 5, 0, 0) and φF B
z ⋆1 z ⋆2 z ⋆3 z ⋆4 ∞
min Newton GLS 58 213 280 394 55
FB Newton GLS 183 198 211 238 170
min Broyden PTR 106 362 45 385 102
FB Broyden PTR 104 381 35 248 232
Table 3.3: Number of GNEs found for 1000 random initial points
To further compare these methods and the complementarity function, we draw uniformly
1000 random initial points such that z0 ∈ [−10, 10] × [−10, 10] × {1} × {1} and run algorithms
on each of them. For simplicity, we restrict out comparison to Newton GLS and Broyden
PTR methods test the Newton GLS method both with the minimum and Fischer-Burmeister
complementarity functions. Results are summarized in Table 3.3, the first four columns store
the number of sequences converging to a particular GNE, while the last column contains
the number of diverging sequences (termination criteria remain the same as in the previous
example.). With this comparison, the best method seems to be the Newton GLS method
combined with the minimum function. We observe that using the minimum function tends
to get only two GNEs, namely z ⋆2 and z ⋆4 . The method finding almost equally all GNEs is
the Newton GLS method with the Fischer-Burmeister function. Finally, the Broyden PTR
method with the Fischer-Burmeister function seems very poor on this example.
3.4 Conclusion
The generalized Nash equilibrium problem (GNEP) is a useful tool for modelling many
concrete applications in economics, computer science and biology, just to name a few. The
demand for computational methods of the GNEP in general form is increasing. This survey
paper aims to present and to compare the current optimization methods available for the
GNEP. Our numerical experiments show an advantage for the KKT reformulation of the
GNEP compared to the constrained equation reformulation. But, in Dreves et al. (2011), the
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