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
In our GNEP context, the root function Φ is given in Equation (3.3) and the merit function
is defined as
f(z) = 1
D(x, λ) 2
2
.
φ◦(−g(x), λ)
The gradient of f can be expressed as
Jac
∇f(z) =
xD(x, λ) diag ∇xigi (x) T
D(x, λ)
i
.
−Da(z)Jac xg(x) Db(z) φ◦(−g(x), λ)
using Equation (3.20). As mentioned in Dreves et al. (2011), the gradient ∇f is single-valued
since only the bottom part of the generalized Jacobian ∂Φ(z) contains multi-valued expressions
(Da and Db when gi j (x) = λij = 0) and the bottom part of Φ(x, λ) has zero entries in that
case. Hence, f is C1 as long as the objective and constraint functions are C2 . Therefore, the
line-search is well defined for our GNEP reformulation.
The nonsingularity assumption of elements in the limiting Jacobian ∂BΦ was already
studied in the previous subsection for local convergence. Furthermore, Dreves et al. (2011)’s
theorem 3 analyzes the additional conditions for a stationary point of f to be a solution of
the nonsmooth equation, hence of the original GNEP. The condition is that Jac xD(x, λ) is
nonsingular and the matrix
M = Jac xg(x)Jac xD(x, λ) −1 diag ∇xigi (x)
(3.28)
is a P0-matrix ∗ . The nonsingularity condition (3.28) for local convergence requires that
Db(z) + MDa(z) is nonsingular. If diagonal matrices Da and Db have non zero terms, then M
is P0-matrix implies that Db(z) + MDa(z) is nonsingular. Thus, Conditions (3.28) and (3.26)
are closely related.
3.2.4 Specific methods for constrained equations
This subsection aims to present methods specific to solve constrained (nonlinear) equations,
first proposed by Dreves et al. (2011) in the GNEP context. The KKT system can be
reformulated as a constrained equation, see Equation (3.4) of Subsection 3.1.1. Techniques to
solve such equation may provide good alternatives to standard optimization procedures. In a
VI problem context, Facchinei and Pang (2003) devotes a chapter to interior-point methods
for solving such constrained equations (CE). Here, we focus on the method of Monteiro and
Pang (1999) providing a general framework for CE problems.
A constrained equation is defined as
2
H(z) = 0, z ∈ Ω, (3.29)
where Ω is a closed subset of Rn . Generally, the constraint set Ω has a simple structure, e.g.
the nonnegative orthant Ω = Rn + or a hypercube Ω = [l, u]. As mentioned in Wang et al.
(1996), in practice, the root function H has also a structured form
F (z)
H(z) = .
G(z)
∗. A m × m square matrix M is a P0-matrix if for all subscript set α ⊂ {1, . . . , m} the determinant
det(Mαα) ≥ 0.
i
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