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

tel-00703797, version 2 - 7 Jun 2012
3.2. Methods to solve nonlinear equations
Let (z B k )k be the sequence generated by the Broyden method. If for the starting points, there
exist δ, ɛ > 0, such that
||z B 0 − z ⋆ || ≤ δ, ||B0 − J(z ⋆ )|| ≤ ɛ,
then (zB k )k converges superlinearly to z⋆ .
Let (zLM k
)k be the sequence generated by the Levenberg-Marquardt method. If F is Lipschitz
continuously differentiable on O and ||F (x)|| provides a local error bound, then (zLM k
)k
converges superlinearly to z⋆ .
In summary, convergence theorems require (i) F is Lipschitz continuously differentiable
(LC) 1 in an open convex set around a solution z ⋆ and (ii) the Jacobian at the solution J(z ⋆ )
is nonsingular for Newton and Broyden method or the function F verifies a local error bound.
Let f be the merit function. The global convergence of line-search techniques is guaranteed
when we have the following conditions: (i) the set L = {z, f(z) ≤ f(z0)} is bounded, (ii) the
Jacobian is bounded in an open convex set around a solution z ⋆ and (iii) line-search always
satisfies the Wolfe conditions with a descent direction, see, e.g. Theorem 11.6 of Nocedal and
Wright (2006). We have seen that the backtracking algorithm satisfies the Wolfe conditions
at each step.
The convergence of trust-region strategies is proved with similar conditions and requires
also that the set L and the Jacobian are bounded. Furthermore, the approximated solution
of the quadratic local problem min mk(h) such that ||h|| < ∆k verifies two conditions: (i)
mk(0)−mk(h ⋆ ) ≥ c1||J T k Fk|| min(∆k, ||J T k Fk||/||J T k Jk||) for some c1 > 0 and (ii) ||h ⋆ || ≤ γ∆k
for some γ ≥ 1.
Overall, local methods and globalization strategies need differentiability by definition and
Lipschitz continuity and some additional conditions for convergence. In the next subsection,
we will see how these conditions can be weakened.
Toward the non-differentiable setting
Getting back to our original GNEP, we want to solve the KKT conditions (3.3) using the
complementarity reformulation of Subsection 3.1.1. Thus, the root function Φ : R n × R m ↦→
R n × R m is defined as
Φ(x, λ) =
D(x, λ)
,
φ◦(−g(x), λ)
where the first component D(x, λ) is composed of N derivatives of the Lagrangian function
Li(x, λ i ) and the second component φ◦(−g(x), λ) is the component-wise application of the
complementarity function φ on the overall constraint function g : R n ↦→ R m . Firstly, only the
bottom component has some non-differentiability problem, because most complementarity
functions φ(., .) are non-differentiable at (0, 0). In this paper, we use the minimum function
φ∧(a, b) = min(a, b) and the Fischer-Burmeister function φF B(a, b) = √ a 2 + b 2 − (a + b).
To deal with non-differentiability, Clarke (1975) introduced the generalized gradient. Few
laters, its famous book, Clarke (1990), provides a comprehensive presentation of the mathematical
fundamentals of nonsmooth analysis. We briefly present here some elements necessary
to our paper, refer to Appendix 3.5.1 for details.
Let G : R n ↦→ R m a function with component Gj. By the Rademacher theorem, a locally
Lipschitzian function is almost everywhere differentiable. We define first the limiting Jacobian,
also called B(ouligand) subdifferential, denoted by ∂BG(x).
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