A Sharper Global Error Bound for the Generalized Nonlinear Complementarity Problem over a Polyhedral Cone

and Applied Analysis 3 that is,


Introduction
Let K = {V ∈   | V ≥ 0, V = 0} be a polyhedral cone in   for matrices  ∈  × ,  ∈  × , and let K ∘ be its dual cone; that is, For continuous mappings ,  :   →   , the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector  * ∈   such that  ( * ) ∈ K, ( * ) ∈ K ∘ , ( * ) ⊤  ( * ) = 0. ( Throughout this paper, the solution set of the GNCP, denoted by  * , is assumed to be nonempty.The GNCP is a direct generalization of the classical nonlinear complementarity problem and a special case of the general variational inequalities problem [1].The GNCP was deeply discussed [2][3][4][5] after the work in [6].The GNCP plays a significant role in economics, operation research, nonlinear analysis, and so forth (see [7,8]).For example, the classical Walrasian law of competitive equilibria of exchange economies can be formulated as a generalized nonlinear complementarity problem in the price and excess demand variables (see [8]).
For the GNCP, the solution existence and the numerical solution methods for the GNCP were discussed [2,3,6].As an important tool for a mathematical problem, the global error bound estimation for GNCP with the mapping being -strongly monotone and Hölder continuous was discussed in [5], and a global error bound for the GNCP for the linear and monotonic case was established in [4].
In this paper, we will establish a global error bound for the problem (2) without the Hölder continuity of the underlying mapping.To this end, we first develop some new equivalent reformulations of the GNCP under weaker conditions and then establish a sharper global error bound for the GNCP in terms of some easier computed residual functions.The results obtained in this paper can be taken as an improvement of the existing results for GNCP and variational inequalities problem [4,5,[9][10][11].
To end this section, we give some notations used in this paper.Vectors considered in this paper are taken in the Euclidean space   equipped with the usual inner product, and the Euclidean 2-norm and 1-norm of vector in   are, respectively, denoted by ‖ ⋅ ‖ and ‖ ⋅ ‖ 1 .We use   + to denote the nonnegative orthant in   and use  + and  − to denote the vectors composed by elements ( + )  := max{  , 0}, ( − )  := max{−  , 0}, 1 ≤  ≤ , respectively.For simplicity, we use (; ) to denote vector ( ⊤ ,  ⊤ ) ⊤ , use  to denote the identity matrix with appropriate dimension, use  ≥ 0 to denote
Remark 2. Based on this definition, -strongly -monotone implies monotonicity, and if () =  + , () =  +  with ,  ∈  × , ,  ∈   , then the above Definition 1(i) is equivalent to that the matrix  ⊤  is positive semidefinite.Now, we give some assumptions for our analysis based on Definition 1. Assumption 3.For mappings ,  and matrix  involved in the GNCP, we assume that (A1) mapping  is monotone with respect to mapping ; (A2) matrix  ⊤ has full-column rank.
In the following, we will establish a new equivalent reformulation to the GNCP.First, we give the following conclusion established in [2]. ( From Theorem 5, under Assumption 3(A2), we can transform the system into a new system in which neither  1 nor  2 is involved.To this end, we need the following conclusion [12].Lemma 6.If the linear system  =  is consistent, then  =  +  is the solution with the minimum 2-norm, where  + is the pesudo-inverse of .Lemma 7. Suppose that Assumption 3(A2) holds.Then, for any  ∈   , the following statements are equivalent.
if and only if  * is a global optimal solution with the objective vanishing of (24).
In the following, we give the error bound for a polyhedral cone from [13] and error bound for a convex optimization from [14] to reach our aims.

Lemma 10. For polyhedral cone
Lemma 11.Let  be a convex polyhedron in   , and let  be a convex quadratic function defined on   .Let  be the nonempty set of globally optimal solutions of the programming: with   being the optimal value of  on .There exists a scalar Before proceeding, we present the following definition introduced in [15].Definition 12.The mapping  :   →   is said to be strongly nonexpanding with a constant  > 0 if ‖() − ()‖ ≥ ‖ − ‖.
By Lemma 8, () is a convex function and the feasible set Ω is a polyhedral.Combining this with Lemmas 10 and 11, we immediately obtain the following conclusion.
Theorem 13.Suppose that  is -strongly -monotone with positive constants  1 , , respectively, and  is strongly nonexpanding with constant  > 0.Then, there exists constant Abstract and Applied Analysis 5 Proof.For any  ∈   , let  = (, ]) = ((), ()) ∈  2 .Then, there exists where the second inequality follows from Definition 12 with constant  > 0, the third inequality follows from Definition 1(ii) with constants  1 > 0,  > 0, the fourth inequality follows from the Cauchy-Schwarz inequality, the fifth inequality follows from the fact that (1/2)( 2 +  2 ) ≥ , for all ,  ∈ , the sixth inequality follows from Lemma 11 with constant  3 > 0 and Lemma 9, and the seventh inequality follows from Lemma 10 with constant  2 > 0. By (30) and letting (31) Moreover, the conditions which both  and  are Hölder continuous (or both  and  are Lipschitz continuous) in Theorem 13 are removed.Thus, Theorem 13 is stronger than Theorem 2.5 in [5].Furthermore, by Theorem 2.1 in [5], the GNCP can be reformulated as general variational inequalities problem, and the conditions in Theorem 13 are also weaker than those in Theorem 3.1 in [15], Theorem 3.1 in [11], Theorem 3.1 in [10], and Theorem 2 in [9], respectively.
On the other hand, the condition that  is -strongly monotone and  is strongly nonexpanding in Theorem 13 is extended compared with the condition that  is strongly monotone with respect to  (i.e.,  = 1) in Theorems 3.4 and 3.6 in [15], and it is also extended than compared with the condition  is strongly monotone with respect to  (i.e.,  = 1) in Theorem 3.1 in [11], and compared with the condition that () = , () is strongly monotone (i.e.,  = 1) in Theorem 3.1 in [10].
On the other hand, for any  ∈ , one has Since ,  0 ∈ Ω, using the similar arguments to that of (21), one has Combining this with (41) yields that From (32), we deduce that Thus, using (21), one has Hence,  ∈ Ω * .
Based on Lemma 16, we obtain the following conclusion.
In the following, we give an error bound of the Hölderian type [14].
From (50), we have Ω * = Ω ⋂  1 , where Ω is defined in (24), so for any  ∈ where the second inequality follows from (56) with constant  =  7 √ 2 + 2 + 2, the third inequality uses (59), the fifth inequality follows from (53), the sixth inequality follows from the fact that the seventh and ninth inequalities follow from the fact that and the last inequality follows by letting  8 =  max{√, √, √, 1}.
Remark 21.When  is strongly monotone with respect to , that is,  = 1, without the requirement of nondegenerate solution, the square root term in the error bound estimation is removed as stated in Theorem 20.Hence, the error estimation becomes more practical than that in Theorem 4.1 in [4].

Global Error Bound for the GLCP
In this section, we consider the linear case of the GCP such that mappings  and  are both linear; that is, () = +, () =  +  with ,  ∈  × , ,  ∈   : min  () = ( + ) ⊤ ( + ) where For problem (64), combining (18) with (23) and using a similar discussion in Lemmas 8 and 9, we also have the following conclusion.
Lemma 23.  * ∈   is a solution of the GLCP if and only if  * is global optimal solution with the objective vanishing of (64).
Based on (64), using the argument similar to that of Theorem 13, we can obtain the following conclusion.
The following result further estimates the error bound for the GLCP.
Theorem 26.Suppose that the hypotheses of Theorem 24 hold, and the GLCP has a nondegenerate solution.Then, there exists constant  5 > 0 such that Proof.From Corollary 17, we have where  0 is a nondegenerate solution of GLCP, and  is defined in (64).For any  ∈ and using the similar technique to that of ( 57 where the second inequality is by nonexpanding property of projection operator.Thus,
In fact, we consider the following four cases.
where the inequality follows from the fact that (93) In fact, we consider the following four cases.

Conclusion
In this paper, we established some global error bounds on the generalized nonlinear complementarity problems over a polyhedral cone, which improves the result obtained for variational inequalities and the GNCP [4,5,[9][10][11] by weakening the assumptions.Surely, under milder conditions, we may establish global error bounds for GNCP and use the error bounds estimation to establish quick convergence rate of the methods for the GNCP.This is a topic for future research.
[4]ark 29.In Theorem 28, without the requirement of nondegenerate solution, the square root term in the error bound estimation is removed.Hence, the error estimation becomes more practical than that in Theorem 4.1 in[4].
cannot do.