A quadratic convergence method for the management equilibrium model

in this paper, we study a class of methods for solving the management equilibrium model. We first give an estimate of the error bound for the model, and then, based on the estimate of the error bound, propose a method for solving the model. We prove that our algorithm is quadratically convergent without the requirement of existence of a non-degenerate solution. Keywords—Management equilibrium model; estimation of

Where ,, mn M N R   Applications of complementarity problems from the field of economics include general Walrasian equilibrium, spatial price equilibria, invariant capital stock, market equilibrium, optimal stopping, and game-theoretic models, In engineering, the complementarity problems also plays a significant role in contact mechanics problems, structural mechanics problems, obstacle problems mathematical physics, Elastohydrodynamic lubrication problems, traffic equilibrium problems(such as a pathbased formulation problem, a multicommodity formulation problem, network design problems),etc. [1,2]For example, the equilibrium of supply and demand in an economic system is often depicted as a complementary model between two decision variables.As another example, the typical Walras' Law of competition equilibrium in economic transactions can also be converted to complementary model between price and excess demand [3] .
Recently, many effective methods have been proposed to solve (1)   [ 4 6]    .The basic idea of these methods is to convert (1) into an unconstrained or a simply constrained optimization problem.As we known, if the Jacobian matrix at a solution to (1) is non-singular, then it is guaranteed that the Levenberg-Marquardt (L-M) algorithm is quadratically convergent [5,6] .Lately, Yamashita and Fukushima have proved that the condition for the local error bound to hold is weaker than the non-singularity of the Jacobian matrix [7] .This motivates the establishment of an error bound for (1).The establishment of LCP error bound has been extensively studied (see literature review [8] ).For example, Mangasarian and Ren have given an error bound under the 0 R -matrix condition [9] .Clearly, (1) is a generalization of LCP, which prompts whether or not the LCP error bound can be generalized to (1).For this reason, we focus on the establishment of an error bound for (1), design a smooth algorithm for solving (1) using the error bound, and analyze the convergence of the algorithm as well as the rate of convergence.
In section 2, we give primarily an equivalent conversion of (1).In section 3, using a new residual function, we establish an error bound for (1) under more general conditions.In section 4, based on the established error bound, we propose a smooth algorithm for solving (1), and prove that the given algorithm is quadratically convergent without the requirement of existence of a non-degenerate solution.Compared with the convergence of algorithms in [5,6], the condition is weaker.Now we give some notations.The inner product of vectors , n x y R  is written as T xy .Let || ||  be the Euclidean norm.For ease of presentation, we write ( , , ) x y z for column vector T T T T ( , , ) x y z , and use * ( , ) dist  for the shortest distance from vector  to a closed convex set *  .

II. EQUIVALENT CONVERSION OF THE MANAGEMENT EQUILIBRIUM MODEL
We give in this section an equivalent conversion of (1).For convenience, let ( , , ) . Then, (1) can be converted equivalently to the following problem ： Where ( ,0,0), (0, ,0) Proof.Assume that the theorem does not hold.Then there exists a sequence {} k  ，such that for any positive integer k ( Where,  is a positive constant. On the other hand，from (3)we have This contradicts with (4)，hence the theorem is proved.
We give in the following the error bound established by Hoffman [10] .
Lemma 2.1 For a polyhedral cone Now, we also give the definition of projection operator and its related properties [11] .
Where the second inequality is based on Lemma 2.2.Combined with the above formula, we have From ( 5) and Theorem 2.1, we have www.ijacsa.thesai.org In the following we using Fischer function ( [12]) to establish another error bound.Define In addition， Tseng [13] gives the following conclusion.
Clearly，using Lemma 2.3 and Theorem 2.2，it is easy to have the following result.Theorem 2.3 For any given positive constant  ，there exists a constant 3 0 Where 0 t  is a smooth parameter.For ease of presentation, let 11 ( , , ) ( ( , ),..., ( , )) T (0, ) (0, ) Therefore we construct a smooth method to solve ( , ) 0 Ft   , and assume that the set of solutions to ( , ) 0 First we give the following properties of ( , , ) p a b t [14,15] .
Ft  is continuously differentiable, locally Lipschitz continuous, and strongly semi-smooth, that is, there exist constants 1 0, , Proof.The result of (i) follows from Lemma 2.1 directly.

IV. ALGORITHM AND CONVERGENCE
In this section ， we give a smooth and convergent algorithm for solving (1), and using the error bound established in section 2, prove the quadratic convergence of the given smooth algorithm without the condition of existence of a non-degenerate solution.
In the following convergence analysis, assume that Algorithm 3.1 generates an infinite sequence.We have the following result.
In the following we prove the theorem in three steps.
First we prove the following result.

Let the closest point in
Let As k d is the globally optimal solution to(9)，we have 8) and( 12)，we have Using ( 12) -( 14) and ( 7)，together with the definition of . Then (10) holds.
From the definition of () In addition， from (13)， (7)and the definition of From (6)，we also have


where, from (10), the last inequality holds.In addition, since  NOTE: Theorem 3.1shows that the given smooth algorithm has the property of quadratic convergence without the condition of existence of a non-degenerate solution.This is a new result.

V. CONCLUSIONS
In this paper, we propose an algorithm for solving the management equilibrium model.Under without the requirement of nondegenerate solution, we also show that the algorithm is quadratic convergence based on error bound estimation instead of the nonsingular assumption just as was done in [5,6].This conclusion can be viewed as extension of previously known result in [5, 6].How to use the algorithm to solve the practical management based on the computer, this is a topic for future research.
 .The model originated from equilibrium problems in economic management, engineering, etc.

Theorem 3 . 1 .{
Assume that Algorithm 3.1 generates a sequence{( , )} kk t If the initial value is close sufficiently to * From the above formula, and the way  is chosen,