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
λi j = gi j (x) = 0. Otherwise, F is differentiable for which the convergence is established. In
this paper, we test the Broyden method without proving the convergence. To our knowledge,
the Broyden method is not used in the GNEP context.
Global convergence
Let us study the global convergence starting by line-search techniques. The merit function
used is the same function as for the differentiable setting the residual norm.
f(z) = 1
2 || F (z)||2 .
The gradient is given by ∇f(z) = V T F (z), where V ∈ ∂F (z). As mentioned in Qi and Sun
(1998), the merit function may be C 1 , even if F is only semismooth.
Jiang and Ralph (1998) and Qi and Sun (1998) show the convergence of the Newton
method globalized with a backtracking (geometric) line search. When using the generalized
Jacobian, Jiang (1999) shows the convergence of the corresponding algorithm. All proofs rely
on the fact that after a large number of iteration, the full Newton step is accepted, i.e. we get
back to local convergence.
Theorem. Suppose that the root function F is a semismooth function and the merit function
f is C 1 . Then any accumulation points z ⋆ of the line-search generalized Newton method is a
stationary point of f, i.e. ∇f(z ⋆ ) = 0. If z ⋆ is a solution of F (z) = 0 and all matrices in
∂BF (z ⋆ ) are nonsingular, then the whole sequence converges superlinearly (resp. quadratically)
to z ⋆ (if F is strongly semismooth at z ⋆ ).
To our knowledge, the global convergence of general quasi-Newton methods for nonsmooth
equation is not established. However, for the Levenberg-Marquardt with a backtracking linesearch
technique, Jiang and Ralph (1998) and Jiang (1999) show the convergence. We present
below a light version of their theorem.
Theorem. Let (zk)k be a sequence of the generalized LM method globalized with a backtracking
line search to solve F (x) = 0 for a semismooth function F and a C 1 merit function f. Assuming
the direction step is solvable at each iteration, we denote by z ⋆ an accumulation point.
If λk = min(f(zk), ||∇f(zk)||) and elements in ∂BF (z ⋆ ) are nonsingular, then (zk)k converges
superlinearly. If in addition F is strongly semismooth at z ⋆ , then it converges quadratically.
The attentive reader may have noticed that in global convergence not all stationary points
z ⋆ of the merit function f are not necessarily a solution of the equation F (z) = 0. This
is already the case in the smooth setting, e.g. F (x) = sin(5x) − x has only three roots (0
and ±0.519148..) but (sin(5x) − x) 2 /2 has many local minima, see Figures 11.1 and 11.2 of
Nocedal and Wright (2006).
Now, we focus on the second globalization strategy: the trust-region approach. We present
first a convergence result of Jiang and Ralph (1998), based on a result of Jiang et al. (1998)
in the context of complementarity problems.
Theorem. Consider F (z) = 0 with F semismooth and a C 1 merit function f? Let (zk)
be generated by the trust-region algorithm with the submodel given mk(h) = J T k F (xk)h +
1
2 hT J T k Jkh. If z ⋆ is an accumulation point of the sequence, then z ⋆ is a stationary point of
f. If for all element of ∂BF (z ⋆ ) are nonsingular, then the entire sequence converges to x ⋆
superlinearly. If in addition, F is strongly semismooth at z ⋆ , then the convergence rate is
quadratic.
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