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
Bonnans et al. (2006), two typical conditions are used to determine an appropriate stepsize.
Let 0 < c1 < 1/2 < c2 < 1. The Armijo condition ensures a decrease of f
φk(t) ≤ φk(0) + tc1φ ′ k (0) ⇔ f(xk + tdk) ≤ f(xk) + tc1∇f(xk) T dk.
The curvature condition ensures an increase of ∇φ, implying a decrease of f,
φ ′ k (t) ≥ c2φ ′ k (0) ⇔ ∇f(xk + tdk) T dk ≥ c2∇f(xk) T dk.
These two conditions are referred to the Wolfe conditions. In this paper, we use a backtracking
algorithm, for which the curvature condition is always satisfied. Let tk,0 = 1 be the initial
guess of the stepsize. The backtracking algorithm is defined as follows
Repeat until f(xk + tdk) ≤ f(xk) + tk,ic1∇f(xk) T dk satisfied,
– propose a new tk,i+1 using tk,i, . . . tk,0.
end Repeat
This algorithm always tests a full step with tk,0 = 1, otherwise the above algorithm tries a
new stepsize. For the backtracking line search, a classic result shows that the full step will be
eventually satisfied as zk tends to a solution. We test two stepsize proposal algorithms. The
geometric line search uses
tk,i+1 = ρ × tk,i,
with 0 < ρ < 1, whereas the quadratic line search uses a quadratic approximation of φ using
the information φk(tk,i), φk(tk,i−1), φ ′ k (tk,i−1). We get
tk,i+1 = − 1
2
φ ′ k (tk,i−1)t 2 k,i
φk(tk,i) − φk(tk,i−1) − φ ′ k (tk,i−1)tk,i
Other proposal, such as cubic approximation, are given in Chapter 3 of Bonnans et al. (2006)
or Chapter 6 of Dennis and Schnabel (1996).
Until now, we do not specify the merit function f. For nonlinear equation, we generally
choose f(z) = 1
2 ||F (z)||2 2
, sometimes referred to the residual function. This merit function has
some defficiencies, since a local minimum is not necessarily a root of the function F . We will
see later in the GNEP context, that f has still some interesting properties.
Line-search methods require to a tractable formula for the gradient ∇f(z) = JacF (z) T F (z),
when testing the Armijo condition. However, in a quasi-Newton framework, we do not necessarily
have a tractable Jacobian. One way to deal with this is to use a numerical Jacobian,
e.g., based on the forward difference. We use Dennis and Schnabel (1996)’s algorithm A5.4.1
defined by
D(F )(z) = (D1, . . . , Dn), with Dj = F (z + hjej) − F (z)
∈ R n ,
where ej is the jth unit vector and hj a small step, typically, hj = √ ɛzj where ɛ is the epsilon
machine (ɛ = 1e −16 ).
Trust-region approach
Trust-region strategies relaxe the constraint that dk is a descent direction. Line search
assumes the “best” point from zk lies on the half-line zk + R+dn. Quoting Bonnans et al.
(2006), “what is magic in this half line? answer: nothing”. Trust-region approach will look for
hj
.
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