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
appropriate steps hk in a “small” region around zk. Such regions are not the half-line as in
line search.
To find the root of F (x) = 0, trust-region methods minimizes a local quadratic approximation
mk of a merit function f : R n ↦→ R on a bounded subset: the trust region
{z, ||z − zk|| ≤ ∆k}. The name comes from viewing ∆k as providing a region in which we can
trust mk to adequately model f. Classic trust region methods consider the following model
function
mk(h) = f(zk) + g(xk) T h + 1
2 hT ˜ Hkh,
with g(xk) the approximate gradient of f and ˜ Hk a (ideally) positive definite matrix approximating
the Hessian of f.
To adapt the trust region radius ∆k, we define the following ratio
ρk(h) = f(xk) − f(xk + h)
mk(0) − mk(h) .
ρk(h) is the ratio of the actual reduction and the predicted reduction of the merit function for
a step h. The higher is ρk(h), the higher is the reduction of the merit function f for a given
h. Now, we can define a generic algorithm for a model function mk, see, e.g., Nocedal and
Wright (2006).
Init ∆0 = 1, η0 > 0, 0 < η1 < η2 < 1 and 0 < γ1 < 1 < γ2.
Iterate until a termination criterion is satisfied,
mk(h) (approximately),
– get hk = arg min
||h|| η2 and ||hk|| = ∆k then ∆k+1 = min(γ2∆k, ∆max) (very successful),
– else ∆k+1 = ∆k.
– next iterate
– if ρk(hk) > η0 then xk+1 = xk + hk,
– else xk+1 = xk.
end Iterate
Typical values of parameters are ∆0 = 1 or ||g0||
10 , ∆max = 10 10 for radius bounds, η0 = 10 −4 ,
η1 = 1
4 , η2 = 3
4 for ratio threshold, γ1 = 1
2 and γ2 = 2 for radius expansion coefficients.
If readers have been attentive, then they should have noticed that the algorithm cannot be
used directly. In fact, we have to determine how to compute the solution hk of the following
minimization problem
min mk(h).
||h||