Generalized Nash Equilibrium Problem: existence,
uniqueness and reformulations
Didier Aussel
Univ. de Perpignan, France
CIMPA-UNESCO school, Delhi
November 25 - December 6, 2013
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Outline of the 7 lectures
Generalized Nash Equilibrium
a- Reformulations
b- Existence of equilibrium
Variational inequalities: motivations, definitions
abcd-
Motivations, definitions
Existence of solutions
Stability
Uniqueness
Quasiconvex optimization
a- Classical subdifferential approach
b- Normal approach
Whenever a set-valued map is not really set-valued...
Quasivariational inequalities
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Modelization of electricity markets:
Outline of the talk:
0- Basic data on electricity markets
I- Notations and definitions: Nash problems
II- A simple day-ahead market model
III- Variational reformulations of GNEP
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
0- The European energy markets
connecting
markets
Power | Natural Gas | Emissions | Coal
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The European energy market
This EPEX market is open to any country and concerns Austria, France,
Germany and Switzerland:
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The French part of the market
Power | Spot Market
Day-Ahead Auction | France
Day-Ahead Auction with Delivery in the French TSO Zone
Contract size
The minimum volume amounts to 1 MW for individual hours and blocks.
Prices in bid
The minimum price change is EUR 0.01 per MWh.
Underlying
Electricity traded for delivery on the following day in 24 one-hour intervals.
Place of delivery
Deliveries are made within the French transmission system managed by RTE.
Auction hours
The daily auction takes place at 12:00 am, seven days a week, year-round, including statutory holidays, and is embedded in the “CWE Price Market Coupling”,
which is itself linked to the Nordic Market through “Interim Tight Volume
Coupling”.
Time of publication
The results of the auction are made available from 12:40 am (CET).
Bid types
Individual hours
Hourly bids comprise up to 256 price/quantity combinations for every hour of the
following day. Prices must be between the technical price limits of EUR –3,000
per MWh and EUR 3,000 per MWh. The 256 prices in addition to the price ceiling
and the price floor are not necessarily the same for every hour. A volume –
whether positive, negative or nil – must be entered for the price ceiling and price
floor. A price-inelastic bid is sent by entering the same quantity for the price
ceiling and price floor.
Blocks
Block bids are used to link several hours on an all-or-none basis, which means
that either the bid is matched for all of the hours or it is rejected in its entirety.
Various standard blocks are predefined in order to facilitate entering bids. Moreover, the trading participants also have the possibility of entering user-defined blocks
combining several random hours of their choice in addition. Block bids must comprise at least two hours of the day, which do not necessarily have to be consecutive
hours.
The maximum volume for a block bid is 200 MW and a maximum of 40 block
bids per portfolio can be entered.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
I- Notations and definitions: Nash problems
A Nash equilibrium problem is a noncooperative game in which
the decision function (cost/benefit) of each player depends on the
decision of the other players.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
I- Notations and definitions: Nash problems
A Nash equilibrium problem is a noncooperative game in which
the decision function (cost/benefit) of each player depends on the
decision of the other players.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
I- Notations and definitions: Nash problems
A Nash equilibrium problem is a noncooperative game in which
the decision function (cost/benefit) of each player depends on the
decision of the other players.
Denote by N the number of players and each player i controls variables
x i ∈ Rni . The “total strategy vector” is x which will be often denoted by
x = (x i , x −i ).
where x −i is the strategy vector of the other players.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Nash Equilibrium Problem (NEP)
The strategy of player i belongs to a strategy set
x i ∈ Xi
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Nash Equilibrium Problem (NEP)
The strategy of player i belongs to a strategy set
x i ∈ Xi
Given the strategies x −i of the other players, the aim of player i is
to choose a strategy x i solving
Pi (x −i )
max
θi (x i , x −i )
s.t.
x i ∈ Xi
where θi (·, x −i ) : IRni → IR is the decision function for player i.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Nash Equilibrium Problem (NEP)
The strategy of player i belongs to a strategy set
x i ∈ Xi
Given the strategies x −i of the other players, the aim of player i is
to choose a strategy x i solving
Pi (x −i )
max
θi (x i , x −i )
s.t.
x i ∈ Xi
where θi (·, x −i ) : IRni → IR is the decision function for player i.
A vector x̄ is a Nash Equilibrium if
for any i,
x̄ i
Didier Aussel
solves
Pi (x̄ −i ).
Generalized Nash Equilibrium Problem: existence, uniqueness and
Generalized Nash Equilibrium Problem
A generalized Nash equilibrium problem (GNEP) is a
noncooperative game in which the decision function and
strategy set of each player depend on the decision of the other
players.
The strategy of player i belongs to a strategy set
i
x ∈ Xi (x
−i
)
which depends on the decision variables of the other players.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Generalized Nash Equilibrium Problem
A generalized Nash equilibrium problem (GNEP) is a
noncooperative game in which the decision function and
strategy set of each player depend on the decision of the other
players.
The strategy of player i belongs to a strategy set
i
x ∈ Xi (x
−i
)
which depends on the decision variables of the other players.
Given the strategies x −i of the other players, the aim of player i is to choose a strategy x i solving
Pi (x −i )
max
θi (x i , x −i )
s.t.
x i ∈ Xi (x −i )
where θi (·, x −i ) : IRni → IR is the decision function for player i.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Generalized Nash Equilibrium Problem
A generalized Nash equilibrium problem (GNEP) is a
noncooperative game in which the decision function and
strategy set of each player depend on the decision of the other
players.
The strategy of player i belongs to a strategy set
i
x ∈ Xi (x
−i
)
which depends on the decision variables of the other players.
Given the strategies x −i of the other players, the aim of player i is to choose a strategy x i solving
Pi (x −i )
max
θi (x i , x −i )
s.t.
x i ∈ Xi (x −i )
where θi (·, x −i ) : IRni → IR is the decision function for player i.
A vector x̄ is a Generalized Nash Equilibrium if
for any i,
x̄
Didier Aussel
i
solves
Pi (x̄
−i
).
Generalized Nash Equilibrium Problem: existence, uniqueness and
II
A simple day-ahead market model
X. Hu & D. Ralph, Operations Research 55 (2007), 809-827.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
In this electricity market model, there are three main actors: the
productors, the consumers and the ISO. The market is represented by a
set of nodes, denoted by N = {1, · · · , N}, and a set of lines, denoted by
L. Each node can be producer and/or consumer. The market is
centralized by an Independant System Operator (ISO) which allocates
the production of each producer in order to minimize the global cost of
production.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
In this electricity market model, there are three main actors: the
productors, the consumers and the ISO. The market is represented by a
set of nodes, denoted by N = {1, · · · , N}, and a set of lines, denoted by
L. Each node can be producer and/or consumer. The market is
centralized by an Independant System Operator (ISO) which allocates
the production of each producer in order to minimize the global cost of
production.
Each producer bids at the ISO his cost of production.
Each consumer bids at the ISO his utility function.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
In this electricity market model, there are three main actors: the
productors, the consumers and the ISO. The market is represented by a
set of nodes, denoted by N = {1, · · · , N}, and a set of lines, denoted by
L. Each node can be producer and/or consumer. The market is
centralized by an Independant System Operator (ISO) which allocates
the production of each producer in order to minimize the global cost of
production.
Each producer bids at the ISO his cost of production.
Each consumer bids at the ISO his utility function.
The ISO evaluates an equilibrium.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Notations
In this electricity market model, there are three main actors: the
productors, the consumers and the ISO. The market is represented by a
set of nodes, denoted by N = {1, · · · , N}, and a set of lines, denoted by
L. Each node can be producer and/or consumer. The market is
centralized by an Independant System Operator (ISO) which allocates
the production of each producer in order to minimize the global cost of
production.
Each producer bids at the ISO his cost of production.
Each consumer bids at the ISO his utility function.
The ISO evaluates an equilibrium.
The Parameters
* Ai qi + Bi qi2 is the real cost of generation of the node i,
i = 1, n.
* Ai (−qi ) − Bi (−qi2 ) is the utility for consumption of the node i,
i = n + 1, 2n.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Why quadratic cost?
In fact, in many markets, the bids are done by “boxes functions” or
“step functions”
But the bid offer has an increasing shape!
=⇒ a quadratic approximation makes sense.
Prix
Production
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The producer’s problem
Knowing a−i = (aj )j6=i , the node i computes ai , (qj )j∈N in order to solve the following optimization problem:
(Pi )
max
ai ,bi ,q
such that
(ai + 2Bi qi )qi − (Ai qi + Bi qi2 )
(
Ai ≤ ai ≤ Āi
(qj )j∈N ∈ Q(a)
with 0 ≤ Ai ≤ Āi .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The producer’s problem
Knowing a−i = (aj )j6=i , the node i computes ai , (qj )j∈N in order to solve the following optimization problem:
(Pi )
max
ai ,bi ,q
such that
(ai + 2Bi qi )qi − (Ai qi + Bi qi2 )
(
Ai ≤ ai ≤ Āi
(qj )j∈N ∈ Q(a)
with 0 ≤ Ai ≤ Āi .
The set Q(a) denotes the set of the solutions of the ISO’s problem which is:
min
P
q
(
such that
i∈N (ai qi
+ Bi qi2 )
qi ≥ 0 , ∀i ∈ N
q1 + · · · + q2n = 0
Didier Aussel
(λ)
Generalized Nash Equilibrium Problem: existence, uniqueness and
The producer’s problem
Knowing a−i = (aj )j6=i , the node i computes ai , (qj )j∈N in order to solve the following optimization problem:
Optimistic way
(Pi )
max max
ai ,bi
q
such that
(ai + 2bi qi )qi − (Ai qi + Bi qi2 )
(
Ai ≤ ai ≤ Āi
(qj )j∈N ∈ Q(a)
with 0 ≤ Ai ≤ Āi .
The set Q(a) denotes the set of the solutions of the ISO’s problem which is:
min
P
q
(
such that
i∈N (ai qi
+ Bi qi2 )
qi ≥ 0 , ∀i ∈ N
q1 + · · · + q2n = 0
Didier Aussel
(λ)
Generalized Nash Equilibrium Problem: existence, uniqueness and
The producer’s problem
Knowing a−i = (aj )j6=i , the node i computes ai , (qj )j∈N in order to solve the following optimization problem:
Pessimistic way
(Pi )
max min
ai ,bi
q
such that
(ai + 2bi qi )qi − (Ai qi + Bi qi2 )
(
Ai ≤ ai ≤ Āi
(qj )j∈N ∈ Q(a)
with 0 ≤ Ai ≤ Āi .
The set Q(a) denotes the set of the solutions of the ISO’s problem which is:
min
P
q
(
such that
i∈N (ai qi
+ Bi qi2 )
qi ≥ 0 , ∀i ∈ N
q1 + · · · + q2n = 0
Didier Aussel
(λ)
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence
Theorem
If Bi > 0, for any i, then the ISO’s problem (Q(a)) admits a unique
solution.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence
Theorem
If Bi > 0, for any i, then the ISO’s problem (Q(a)) admits a unique
solution.
Theorem
If, for any possible bid a, the ISO’s response q ∗ (a) is such that, qi∗ > 0,
for any i, then the Generalized Nash game admits at least a solution.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence
Theorem
If Bi > 0, for any i, then the ISO’s problem (Q(a)) admits a unique
solution.
Theorem
If, for any possible bid a, the ISO’s response q ∗ (a) is such that, qi∗ > 0,
for any i, then the Generalized Nash game admits at least a solution.
Observe that the marginal price λ = ai + 2Bi qi∗ is the same for any
producer.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
III
Variational reformulation
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Classical reformulations
−ν
Suppose that for any ν and any x −ν ∈ IRn , function θν (·, x −ν ) is
continuously differentiable and convex and Xν (x −ν ) is convex.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Classical reformulations
−ν
Suppose that for any ν and any x −ν ∈ IRn , function θν (·, x −ν ) is
continuously differentiable and convex and Xν (x −ν ) is convex.
Denoting by
and
X (x) =
Y
Xν (x −ν ),
∀ x ∈ IRn
F (x) = (∇x 1 θ1ν(x), . . . , ∇x N θN (x))
∈ IRn
we have the reformulation
x̄ gene. Nash equil. ⇔
x̄ ∈ X (x̄) and
hF (x̄), y − x̄i ≥ 0,
Didier Aussel
∀ y ∈ X (x̄)
Generalized Nash Equilibrium Problem: existence, uniqueness and
Classical reformulations
−ν
Suppose that for any ν and any x −ν ∈ IRn , function θν (·, x −ν ) is
continuously differentiable and convex and Xν (x −ν ) is convex.
Denoting by
and
X (x) =
Y
Xν (x −ν ),
∀ x ∈ IRn
F (x) = (∇x 1 θ1ν(x), . . . , ∇x N θN (x))
∈ IRn
we have the reformulation
x̄ gene. Nash equil. ⇔
x̄ ∈ X (x̄) and
hF (x̄), y − x̄i ≥ 0,
∀ y ∈ X (x̄)
that is a quasi-variational inequality.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The convex nondifferentiable case
We suppose that the functions θi (·, x −i ) are convex. We denote
S(x) =
n
Y
Xi (x −i )
i=1
and
F (x) =
n
Y
∂x i θi (x i , x −i )
i=1
Then
x̄ is a GNEP ⇐⇒ if x̄ solves QVI (F , S).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
The convex nondifferentiable case
We suppose that the functions θi (·, x −i ) are convex. We denote
S(x) =
n
Y
Xi (x −i )
i=1
and
F (x) =
n
Y
∂x i θi (x i , x −i )
i=1
Then
x̄ is a GNEP ⇐⇒ if x̄ solves QVI (F , S).
What about nonconvex loss functions?
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Pseudoconvex reformulations
F. Facchinei, A. Fischer and V. Piccialli, On generalized Nash games and variational inequalities, Operations
Research Letters, Vol 35, 2007, pp 159-164.
Proposition
−ν
Suppose that for any ν and any x −ν ∈ IRn , function θν (·, x −ν ) is
continuously differentiable and pseudoconvex.
Assume the Rosen’s law : there exists a nonempty convex set X of IRn
such that,
∀ ν,
Xν (x −ν ) = {x ν ∈ IRn
ν
: (x ν , x −ν ) ∈ X }.
Then any solution of S(F , X ) is a solution of the GNEP.
where
F (x) = (∇x 1 θ1 (x), . . . , ∇x N θN (x))
Didier Aussel
∈ IRn
Generalized Nash Equilibrium Problem: existence, uniqueness and
Reverse implication?
The reverse is not true in general:
Let us consider, in R2 , the convex subset
X = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0
and 2x1 + x2 ≤ 1}.
and the two players generalized Nash problem defined by X and
the loss θ1 (x1 , x2 ) = (x1 − 2)2 and θ2 (x1 , x2 ) = (x2 − 2)2 .
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Reverse implication?
The reverse is not true in general:
Let us consider, in R2 , the convex subset
X = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0
and 2x1 + x2 ≤ 1}.
and the two players generalized Nash problem defined by X and
the loss θ1 (x1 , x2 ) = (x1 − 2)2 and θ2 (x1 , x2 ) = (x2 − 2)2 .
The solution set of the GNEP is
GNEP = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0
Didier Aussel
and
2x1 + x2 = 1}
Generalized Nash Equilibrium Problem: existence, uniqueness and
Reverse implication?
The reverse is not true in general:
Let us consider, in R2 , the convex subset
X = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0
and 2x1 + x2 ≤ 1}.
and the two players generalized Nash problem defined by X and
the loss θ1 (x1 , x2 ) = (x1 − 2)2 and θ2 (x1 , x2 ) = (x2 − 2)2 .
The solution set of the GNEP is
GNEP = {(x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0
and
2x1 + x2 = 1}
whereas the variational inequality S(F , X ) has the point (x̄1 , x̄2 ) = (0, 1)
as unique solution.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
b- An extended reformulation
To simplify the notations, we will denote, for any ν and any x ∈ Rn , by Sν (x)
and Aν (x −ν ) the subsets of Rnν
Sν (x) = Sθaν (·,x −ν ) (x ν )
and
Aν (x −ν ) = arg min
θν (·, x −ν ).
n
R ν
In order to construct the variational inequality problem we define the following
n
set-valued map Nθa : Rn → 2R which is described,
for any x = (x 1 , . . . , x p ) ∈ Rn1 × . . . × Rnp , by
Nθa (x) = F1 (x) × . . . × Fp (x),
(
B ν (0, 1)
if x ν ∈ Aν (x −ν )
where Fν (x) =
co(Nθaν (x ν ) ∩ Sν (0, 1)) otherwise
The set-valued map Nθa has nonempty convex compact values.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Sufficient condition
In the following we assume that X is a given nonempty subset X of Rn , such that for any i, the set Xi (x −i ) is
given as
−i
i
n
i
−i
Xi (x ) = {x ∈ R i : (x , x ) ∈ X }.
Theorem
Let us assume that, for any i, the function θi is continuous and
quasiconvex with respect to the i-th variable. Then every solution of
S(Nθa , X ) is a solution of the GNEP.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Sufficient condition
In the following we assume that X is a given nonempty subset X of Rn , such that for any i, the set Xi (x −i ) is
given as
−i
i
n
i
−i
Xi (x ) = {x ∈ R i : (x , x ) ∈ X }.
Theorem
Let us assume that, for any i, the function θi is continuous and
quasiconvex with respect to the i-th variable. Then every solution of
S(Nθa , X ) is a solution of the GNEP.
Note that the link between GNEP and variational inequality is valid even
if the constraint set X is neither convex nor compact.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Lemma
Let ν ∈ {1, . . . , p}. If the function θν is continuous quasiconvex with
respect to the ν-th variable, then,
0 ∈ Fν (x̄) ⇐⇒ x̄ ν ∈ Aν (x̄ −ν ).
Proof. It is sufficient to consider the case of a point x̄ such that x̄ ν 6∈ Aν (x̄ −ν ). Since θν (·, x̄ −ν ) is continuous
at x̄ ν , the interior of Sν (x̄) is nonempty. Let us denote by Kν the convex cone
a
ν ◦
ν
Kν = Nθ (x̄ ) = (Sν (x̄) − x̄ ) .
ν
By quasiconvexity of θν , Kν is not reduced to {0}. Let us first observe that, since Sν (x̄) has a nonempty interior,
Kν is a pointed cone, that is Kν ∩ (−Kν ) = {0}.
Now let us suppose that 0 ∈ Fν (x̄). By Caratheodory theorem, there exist vectors vi ∈ [Kν ∩ Sν (0, 1)],
i = 1, . . . , n + 1 and scalars λi ≥ 0, i = 1, . . . , n + 1 with
n+1
X
λi = 1 and 0 =
i=1
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n+1
X
λi vi .
i=1
Generalized Nash Equilibrium Problem: existence, uniqueness and
Since there exists at least one r ∈ {1, . . . , n + 1} such that λr > 0 we have
vr = −
n+1
X
λi
λ
i=1,i6=r r
vi
which clearly shows that vr is an element of the convex cone −Kν . But vr ∈ Sν (0, 1) and thus vr 6= 0. This
contradicts the fact that Kν is pointed and the proof is complete.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proof of necessary condition
Proof. Let us consider x̄ to be a solution of S(Nθa , X ). There exists v ∈ Nθa (x̄) such that
hv , y − x̄i ≥ 0,
∀ y ∈ X.
(∗)
Let ν ∈ {1, . . . , p}.
If x̄ ν ∈ Aν (x̄ −ν ) then obviously x̄ ν ∈ Solν (x̄ −ν ).
Otherwise v ν ∈ Fν (x̄) = co(Nθa (x̄ ν ) ∩ Sν (0, 1)). Thus, according to Lemma 2, there exist λ > 0 and
ν
u ν ∈ Nθa (x̄ ν ) \ {0} satisfying v ν = λu ν .
ν
ν
−ν
Now for any x ∈ Xν (x̄
), consider y = x̄ 1 , . . . , x̄ ν−1 , x ν , x̄ ν+1 , . . . , x̄ p .
From (∗) one immediately obtains that hu ν , x ν − x̄ ν i ≥ 0. Since x ν is an arbitrary element of Xν (x̄ −ν ), we
have that x̄ ν is a solution of S(Nθa \ {0}, Xν (x̄ −ν )) and therefore, according to Prop. 4,
ν
x̄
ν
∈ Solν (x̄
−ν
)
Since ν was arbitrarily chosen we conclude that x̄ solves the GNEP.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Necessary and sufficient condition
Theorem
Let us suppose that, for any i, the loss function θi is continuous and
semistrictly quasiconvex with respect to the i-th variable. Further assume
that the set X is a nonempty convex subset of RN . Then
any solution of the variational inequality S(Nθa , X ) is a solution
of the GNEP
and
any solution of the GNEP is a solution of the quasi-variational
inequality QVI (Nθa , X )
where X stands for the set-valued map defined on R2 by
X (x) =
p
Y
Xi (x −i )
i=1
.
D.A. & J. Dutta, Oper. Res. Letters, 2008.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Existence of GNEP
Based on the previous developments it is possible to obtain an existence
result for general GNEP in an elegant and direct way.
Theorem
Assume that for every player ν, the loss function θν is continuous on Rn
and semistrictly quasiconvex with respect to the ν-th variable. If the set
X involved in the Rosen law is nonempty, convex and compact, then the
generalized Nash equilibrium problem admits at least a solution.
Proof: According to sufficient optimality condition, it is sufficient to show that the variational inequality VI (Nθa , X )
admits a solutions. But X is nonempty convex compact and the operator Nθa has nonempty convex compact values
included in the compact set B 1 (0, 1) × · · · × B p (0, 1). Therefore, according to classical existence result, the
variational inequality VI (Nθa , X ) admits at least one solution, provided that the operator Nθa is closed.
Looking to the definition of Nθa , we thus need to show that, for any ν, the set-valued map Fν defined in (??) has a
closed graph. So let ν ∈ {1, . . . , p} and consider a sequence (xk )k of X converging to x̄ and, for any k ∈ N,
vkν ∈ Fν (xk ) be such that the sequence (vkν )k converge (to v̄ ν ).
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Proof (continued)
If x̄ ν is an element of Aν (x̄ −ν ) then obviously v̄ ν , as a limit of elements of the closed set B ν (0, 1), is an
element of Fν (x̄) = B ν (0, 1).
If x̄ ν 6∈ Aν (x̄ −ν ) then, due to the continuity of the loss function θν on Rn , one have, for k large enough,
xkν 6∈ Aν (xk−ν ). This implies that vkν ∈ conv(Nθa (xkν ) ∩ Sν (0, 1)) and therefore, by Caratheodory theorem,
ν
there exist λik ∈ [0, 1], i = 1, . . . , nν + 1 and uki ∈ Nθa (xkν ) ∩ Sν (0, 1), i = 1, . . . , nν + 1 such that
ν
ν
vk =
nX
ν +1
i
i
λk uk
nX
ν +1
and
i=1
i
λk = 1.
i=1
Considering possibly subsequences, for any i = 1, . . . , nν + 1, the sequences (λik )k and (uki )k converge,
respectively, to λ̄i ∈ [0, 1] and to ū i ∈ Sν (0, 1). Moreover
v̄
ν
=
nX
ν +1
i i
λ̄ ū
nX
ν +1
and
i=1
i
λ̄k = 1.
i=1
In order to prove that v̄ ν is an element of Fν (x̄) = conv(Nθa (x̄ ν ) ∩ Sν (0, 1)) it is sufficient to show that, for
ν
any i, ū i ∈ Nθa (x̄ ν ). But this is an immediate consequence of the continuity of θν . Indeed, for any y ν ∈ Rnν
ν
such that θν (y ν , x̄ −ν ) < θν (x̄ ν , x̄ −ν ), one have θν (y ν , xk−ν ) < θν (xkν , xk−ν ) for k large enough.
Therefore, by definition of uki and (??), huki , y ν − xkν i ≤ 0. Taking the limit as k → ∞, hū i , y ν − x̄ ν i ≤ 0
which shows that ū i ∈ Nθa (x̄ ν ) proving the claim and completing the proof.
ν
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
IV - Reformalution of GNEP
using gap functions
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Classical reformulation
−i
Suppose that for any i and any x −i ∈ IRn , function θi (·, x −i ) is
continuously differentiable.
Let us consider a special form of the sets Xi (x −i ). This form was
originally used by Rosen in his fundamental paper (1965):
Given a nonempty convex subset X of Rn , for any ν, the set Xi (x −i ) is
given as
Xi (x −i ) = {x i ∈ Rni : (x i , x −i ) ∈ X }.
Define the Nikaido-Isoda function Ψ : IRn × IRn → IR by
Ψ(x, y ) =
N
X
θi (x i , x −i ) − θi (y i , x −i )
i=1
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Theorem
We suppose that the loss functions are continous onRn and that θi
are convex with respect to the i-th variable. Let α > 0 and
n h
i
X
α
Ψα (x, y ) =
θi (x i , x −i ) − θi (y i , x −i ) + kx i − y i k2
2
i=1
i) For all x ∈ Rn , the application Ψα (x, ·) admits an unique
minimizer over S(x), denoted by y (x).
ii) x̄ is a solution of GNEP ⇔ y (x̄) = x̄.
iii) Let Vα (x) = − inf Ψα (x, y ). Then Vα is a gap function for
y ∈S(x)
GNEP.
This function Ψα is called regularized Nikaido-Isoda-function.
Fukushima, M.: A class of gap functions for quasi-variational inequality problems, J. Ind. Manag. Optim. 3,
165–171 (2007)
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Case of quasiconvex loss functions
If the functions θi are not convex then the above function Vα is no
longer a function gap for GNEP.
Let N = 2, n1 = n2 = 1, θ1 (x1 , x2 ) = x1 |x1 |, θ2 = 0 and
X1 (x2 ) = [−1; 1], X2 (x1 ) = R. Then Vα (0, 0) = 0 but
θ1 (−1, 0) = −1 < θ1 (0, 0), which prove that (0, 0) isn’t a solution
of GNEP.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and
Theorem
We suppose that the loss functions θi are continous on RN and
semistrictly quasiconvex with respect to the i-th variable. Let
α > 0 and
Ψα (x, y ) =
sup hx ∗ , y − xi +
x ∗ ∈Nθa (x)
α
kx − y k2
2
i) For all x ∈ Rn , the application Ψα (x, ·) admits an unique
minimizer over S(x), denoted by y (x).
ii) x is a solution of GNEP iif y (x) = x.
iii) Let Vα (x) = − inf Ψα (x, y ). For all x ∈ X , Vα (x) ≥ 0, and
y ∈S(x)
x̄ is a solution of GNEP ⇔ Vα (x) = 0.
D. Aussel, R. Correa & M. Marechal Gap functions for quasivariational inequalities and generalized Nash
equilibrium problems, JOTA 151, No. 3 (2011), 474488.
Didier Aussel
Generalized Nash Equilibrium Problem: existence, uniqueness and