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
The KKT reformulation of the GNEP (3.2) falls within this framework
Let ˚ Ω denote the interior of Ω. In the contrained equation literature, we assume that (i) Ω
is a closed subset with nonempty interior ˚ Ω, (ii) there exists a closed convex set S ⊂ R n , such
that 0 ∈ S, H −1 ( ˚ S) ∩ ˚ Ω is nonempty but H −1 ( ˚ S) ∩ bd Ω is empty and (iii) H is C 1 function.
The set S contains zero, so that the set H −1 ( ˚ S) contains a solution to Equation (3.29) and
local points around. The second assumption requires that such points should not be on the
boundary of Ω. The third assumption is just differentiability. In the following, these three
assumptions will be referenced as the constrained equation blanket assumptions.
Potential reduction algorithms with line-search
Monteiro and Pang (1999) build a framework for potential functions, where these functions
play a major role. A potential function ˚ S ↦→ R satisfies the following properties:
1. For all sequences (uk)k in ˚ S such that either ||uk|| tends to infinity or (uk) tends to a
point on the boundary bd S \ {0}, then p(uk) tends to infinity.
2. p is C 1 function on its domain and the curvature condition u T ∇p(u) > 0 for all nonzero
vectors.
3. There exists a pair (a, ¯σ) in R n ×]0, 1], such that for all u ∈ ˚ S, we have ||a|| 2 u T ∇p(u) ≥
¯σ(a T u)(a T ∇p(u)).
The potential function has the dual objective to keep the sequences (H(xk))k away from the
set bd S \ {0} and to help the convergence to the zero vector. The parameter a, known as the
central vector, will play a crucial role to generate iterates in the constrained set Ω.
For example, if the subset S is the nonnegative orthant Rn +, then a typical potential
function is
p(u) = ζ log ||u|| 2 n
2 − log ui for u > 0.
i=1
Monteiro and Pang (1999) prove that this function verifies the above conditions when ζ > n/2
and with the pair (a, ¯σ) = ( 1n, 1), 1n being the n-dimensional one vector.
In the GNEP context, the subset S is Rn. × R 2m.
+ where n. =
i ni is the total number
of player variables and m. =
i mi is the total number of constraints, i.e. n = n. + m.. The
function H has components F and G given by
g(x) + w
F (z) = D(x, λ) and G(z) =
,
λ ◦ w
where z = (x, λ, w), see, e.g., Wang et al. (1996). Dreves et al. (2011) propose the following
potential function
p (u) = ζ log ||u1|| 2 2 + ||u2|| 2 2m.
2 − log(u2i),
where u = (u1, u2) ∈ Rn. × R 2m.
+ and ζ > m. in order to enter the potential framework. The
pair of constants is (a, ¯σ) = ((0n., 1m.), 1).
The difficulty, compared to a classical nonlinear equation, is to ensure that all the iterates
remains in the constrained set Ω. In order to solve Equation (3.29), Monteiro and Pang
(1999) use a modified Newton globalized with a backtracking line-search. We report below
their potential reduction Newton algorithm. The algorithm is divided into two parts: (i)
160
i=1