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

tel-00703797, version 2 - 7 Jun 2012
3.1.1 The KKT conditions
3.1. Problem reformulations
The first reformulation uses the Karush-Kuhn-Tucker (KKT) conditions of the N optimization
subproblems. We assume that both constraints and objective functions are twice
continuously differentiable C 2 . Let x ⋆ be a solution of the GNEP. If a constraint qualification
holds for all players, then for all player i, there exists a Lagrange multiplier λ i⋆ ∈ R mi such
that
∇xi θi(x ⋆ ) +
1≤j≤mi
λ i⋆
j ∇xi gi j(x ⋆ ) = 0 (∈ R ni ).
0 ≤ λ i⋆ , −g i (x ⋆ ) ≥ 0, g i (x ⋆ ) T λ i⋆ = 0 (∈ R mi ).
Concatening the N subproblems, we get the following “extended” KKT system
D(x, λ) = 0, λ ≥ 0, g(x) ≤ 0, λ T g(x) = 0, (3.2)
where the functions D, g are defined as
⎛
∇x1
⎜
D(x, λ) = ⎝
L1(x, λ1 )
.
∇xN LN(x, λN ⎞
⎟
⎠ ∈ R
)
n ⎛
λ
⎜
, λ = ⎝
1
.
λN ⎞
⎟
⎠ ∈ R m ⎛
g
⎜
, g(x) = ⎝
1 (x)
.
gN ⎞
⎟
⎠ ∈ R
(x)
m ,
and Li is the Lagrangian function Li(x, λ i ) = θi(x)+g i (x) T λ i . The following theorem precises
the necessary and sufficient condition between the original GNEP in Equation (3.1) and the
KKT system in Equation (3.2).
Theorem. Let a GNEP with twice continuity and differentiability of objective and constraint
functions.
(i) If x ⋆ solves the GNEP at which all the player’s subproblems satisfy a constraint qualification,
then there exists λ ⋆ ∈ R m such that x ⋆ , λ ⋆ solve equation 3.2.
(ii) If x ⋆ , λ ⋆ solve equation 3.2 and that the functions θi’s are player convex and {yi, g i (yi, x−i) ≤
0} are closed convex sets, then x ⋆ solves the original GNEP.
Facchinei and Kanzow (2009) and Facchinei et al. (2009) report the previous theorem,
respectively in Theorem 4.6 and Proposition 1. Using Fritz John conditions, see, e.g., Simon
(2011) or Bazaraa et al. (2006), the player convexity of θi in item (ii) can be relaxed to player
pseudoconvexity, i.e. xi ↦→ θi(x) is pseudoconvexe.
The complementarity reformulation
A complementarity function φ : R 2 → R is a function verifying the following property
φ(a, b) = 0 ⇔ a ≥ 0, b ≥ 0, ab = 0.
Examples are φ∧(a, b) = min(a, b), φF B(a, b) = √ a 2 + b 2 − (a + b), see, e.g., Facchinei and
Pang (2003).
The complementarity reformulation of the KKT conditions is
D(x, λ)
Φ(x, λ) = 0 where Φ(x, λ) =
, (3.3)
φ◦(−g(x), λ)
where φ◦ is the component-wise version of the complementarity function φ and D defined
from the extended system. This reformulation of the GNEP is given in Facchinei et al. (2009),
Facchinei and Kanzow (2009) and Dreves et al. (2011). For a general discussion of semismooth
reformulations of optimization problems, Fukushima and Qi (1999).
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