Solving a Class of Equilibrium Problems With Equilibrium Constraints*

. An equilibrium problem with equilibrium constraints (EPEC) can be looked on as a generalized Nash equilibrium problem (GNEP) and the mathematical programs with equilibrium constraints (MPEC) whose constraints contain a parametric variational inequality or complementarity system. In this paper, we particularly consider a class of EPEC and solve its normalized stationary points where the multipliers of the leaders on the shared constraints are proportionable. We reformulate this kind of EPEC to a standard MPEC. In addition, we demonstrate the proposed approach on an EPEC model in similar products market.


Introduction
An equilibrium problem with equilibrium constraints (EPEC) is a member of a new class of mathematical programs that often arise in engineering and economics applications.
The EPEC, associated with K MPEC defined as above, is to find a Nash equilibria (x * , y * ) ∈ R n such that (x * ,ν , y * ) ∈ SOL(MPEC(x * ,−ν )), ν = 1, . . ., K. (1. 2) The EPEC has recently been studied by some researchers and used to model several problems in applications.Several EPEC models have been developed to study the strategic behavior of generating firms in deregulated electricity markets [1,3,10,18].A sequential nonlinear complementarity problem approach for solving EPEC is proposed in [21].This approach is related to the relaxation technique for solving MPECs that relaxes the complementarity conditions and drives the relaxation parameter to zero [20].Based on the strong stationarity conditions of each leader in a multi-leader-follower game, Leyffer and Munson [15] derived a family of nonlinear complementarity problem, nonlinear programming problem and MPEC formulations of the multi-leader-follower game.They also reformulated price-consistent multi-leader-follower game to a standard MPEC by imposing an additional restriction.See also various applications in economics such as Ehrenmann [2], Ehrenmann and Neuho [3], Hu [12], Murphy and Smeers [17], Su [22], Yao et al. [23]; and the algorithm investigation Su [21].Guo and Lin [7,8] reformulated various stationarities for EPECs as constrained equations and proposed a globally and superlinearly convergent algorithm to solve these constrained equations.Li [16] considered a class of EPECs which is completely separable.In [13,14], Kulkarni et al. reformulated some EPEC models as MPECs by using potential games and shared constraints.
An equilibrium problem usually has many solutions and the uniqueness of an equilibrium point is expected only under very restrictive assumptions.If there are many equilibrium points, we could try to find all equilibrium points or as many equilibrium points as possible.
An alternative approach is to single out an equilibrium point that has some special property.
The normalized equilibrium is such an equilibrium point that the Lagrange multipliers (shadow prices) associated with the shared constraints are equal among all players up to constant factors, and its uniqueness is guaranteed under appropriate conditions [19].
Note that EPEC is highly nonconvex and hence we study its stationary point.In this paper, we study normalized stationary points of a class of EPEC where the multipliers of the players on the shared constraints are proportionable.In economic terms, it means that the relative values of shadow prices associated with the common resources are identical for all players at any normalized stationary points.We reformulate this kind of EPEC to a standard MPEC by imposing some additional conditions and solve it by applying standard MPEC method, which generalizes the work of Leyffer and Munson [15] in which the multipliers are identical for all players.The EPEC model is different from the ones considered in [13,14] in which the decision variables (on the lower level) vary among different leaders decision problems, while it is assumed in our model that the leaders have a common knowledge on the solution of the lower level equilibrium problem.Moreover, from the modeling perspective, our research focuses on the leadership role which is different from the framework in [13,14].
Separability assumption of the EPEC model is also different from the ones studied in [7,8] and [16].
This paper is organized as follows.In the next section we briefly introduce several stationarity condition for EPEC and show how equilibrium points can be computed reliably for EPEC by solving nonlinear optimization problems.Section 3 present normalized stationary points of a class of equilibrium problem with equilibrium constraints and reformulate to stationary points of an associated MPEC.In the section 4, we consider the EPEC model for similar products market in the same city and demonstrate the proposed approach on the model.In addition, some concluding remarks are given in Section 5.
The following notations will be used in the paper.We denote the transposed Jacobian matrix of a differentiable function h : R s → R t at a given point x by ∇h(x) ∈ R s×t .For a real-valued function h(x, y) with the variable x ∈ R s and y ∈ R t , the partial gradients with respect to x and y are denoted by ∇ x h(x, y) ∈ R s and ∇ y h(x, y) ∈ R t , respectively.
By applying the Theorem 2 of [20] for each ν = 1, . . ., K, we can show that there exists mul- for each ν = 1, . . ., K. We can easy obtain the conclusion from the definition of C-stationary Proof.We can easy obtain the conclusion by applying the Corollary 2.1 of [24].Since its proof is similar to Proposition 2.1, we omit the proof here.
3 Normalized stationary point of a class of equilibrium problem with equilibrium constraints In this section, we will consider a special local equilibrium points of the EPEC and call it normalized stationary points of the EPEC.
Normalized equilibrium, first introduced by Rosen [19], is a special GNEP.To reduce the number of variables and constraints, which may make the problem more tractable, Leyffer and Munson [15] make a price-consistency assumption.This technique restricts the solutions considered to those for which the multipliers (prices) on the shared constraints are the same.
In economic terms, this means that the relative values of shadow prices associated with the common resources are identical for all players at any normalized equilibrium.We need the following separability assumption for objective functions.

Definition 3.1 We say that the EPEC (1.2) is relatively separable if the general constraints consist of a set of constraints independent of other decision variables and a set of constraints
common across all players, that is, and the objective function consists of a separable term and a term relatively common across all players, that is, where β ν > 0.
The following theorem relates normalized stationary point of relatively separable EPEC to a standard MPEC.
Note that the definition of C-(M-,S-) stationary point of EPEC, we can easy get the conclusion.
Obviously, solving normalized stationary points of relatively separable EPEC is a generalization of the price-consistent multi-leader-follower games in [15].By introducing the relatively separable assumption and finding normalized stationary points, we produce a model that may be easier to solve than the original standard EPEC.The following examples of Nash game shows the results for solving the normalized stationary points: Example 1 (Leyffer's example) This problem is taken from [15].
where a, b, and c are parameters.The NCP formulation of solving normalized stationary points for this problem is the system of equations where (aλ 0 , bλ 0 ) is Lagrange multiplier vectors for shared constraint.This problem have It is equivalent to computing a first-order critical point for the single optimization problem: min Example 2 (Harker's example) This problem is taken from [9].There are two players and they solve the following problems: This is a GNEP with one shared constraint and the solution set is given by If ( 8 3 λ 0 , 5 4 λ 0 ), λ 0 ≥ 0 is Lagrange multiplier vectors for shared constraint.This problem has normalized equilibrium (5,9) and ( 477 , 58 7 ), which λ 0 = 0 and λ 0 = 4 7 .It is equivalent to computing a first-order critical point for the single optimization problem: min

Applications
In this section, we first discuss normalized stationary points of relatively separable EPEC arising from competition of manufacturer for similar products in the same city.

Model in similar products market
Since similar products have some same function but not exactly the same, similar products have different prices.We consider an oligopoly consisting of K + F manufacturers that produce similar products noncooperatively before the market demand is realized.The first K manufacturers (herein leaders) have no capacity installed and thus have to decide now what their future output will be before the demand function is realized.The remaining F manufacturers (followers) have sufficient capacity installed and thus do not have to make a decision today, but instead they can wait to observe the quantities supplied by the K leaders as well as the realized demand function before making a decision on their supply quantities.
The market demand are characterized by inverse demand functions p ν (x, y), ν = 1, . . ., K+ F , where p ν (x, y) is the market price of the product made by the manufacturer ν, x = (x i ) K i=1 , x i is the supply quantity of the leader i, and y = (y j ) F j=1 , y j is the supply quantity of the follower j.
Before market demand is realized, leader i chooses his quantity x i .The leader's profit can be formulated as where X −i denotes the total bids by the other leaders, Y * = (y * j ) K j=1 , where y * j is the strategies of the j follower.x i p i (x i , X −i , Y * ) means the total revenue for leader i, and C i (x i ) denotes the cost function of leader i.The jth leader's decision problem is to choose the supply quantity x i that maximizes its profit; that is, max where X i := {x i ∈ [0, +∞) | g i (x i ) ≤ 0, g(x, y) ≤ 0} is nonempty and bounded convex set, for each i = 1, . . ., K.
The jth follower chooses its supply quantity after observing the aggregate leaders supply X.Thus, the total revenue of the jth follower is y j p j (X * , y j , Y −j ), and its total cost is C j (y j ).
Consequently, the jth follower profit is where Y −j denotes the total bids by the other followers, X * = (x * i ) K i=1 , where x * i is the strategies of the leader i.The jth follower decision problem is max y j ≥0 R j (X * , y j , Y −j ) = y j p j (X * , y j , Y −j ) − C j (y j ). (4.4) Note that the existence of the multi-leader-follower games can be obtained under the following assumptions: ∀ν = 1, 2, . . ., K + F , (A1) p ν (•) is twice continuously differentiable and decreasing, (A2) There holds p ν ′ q (q) + qp ν ′′ qq (q) ≤ 0 for any q ≥ 0, (A3) The cost functions C ν (q ν ), are twice continuously differentiable and their first and second derivatives are nonnegative for all q ν ≥ 0.
Under Assumptions (A1)-(A3), one can easily show that R ν (x, y) is concave, which guarantees the existence of multi-leader-follower games of the model.Additionally, we suppose that the multipliers of the leaders on the shared constraints are proportionable; that is, • The constraint functions of the leaders are given by where Then the problem can be written as follows: x The results reveal that the proposed methods were able to solve normalized stationary points of relatively separable EPEC successfully.
Although the example is basic, it is thought to be competent to show the difficulties in this field of research and efficiency of our method.

Conclusions
We introduce relatively separable EPECs and solve its normalized stationary points, that result in a standard MPEC.In this approach, the special equilibrium problem with equilibrium constraints is solved by a single optimization problem, unlike traditional approach that solve a sequence of related optimization problems.Additionally, we provide numerical results and application demonstrating that our new approach.

Declarations
We would like to submit the enclosed manuscript entitled "Solving a class of equilibrium problems with equilibrium constraints", which we wish to be considered for publication in "Soft Computing".No conflict of interest exits in the submission of this manuscript, and manuscript is approved by author for publication.We would like to declare that the work described was original research that has not been published previously, and not under consideration for publication elsewhere, in whole or in part.The conception and design of the study owe to Pei-yu li, while the main contributions of Ke-wei Ding are analysis and interpretation of data.

. 6 )
By Theorem 3.1, the normalized C-(M-,S-) stationary point of the EPEC (4.6) is the standard C-(M-,S-) stationary point of the following MPEC: min x,y