## Some Recent Trends in Variational Inequalities and Optimization Problems with Applications

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

**Academic Editor:**Abdellah Bnouhachem

#### Abstract

We revisit the global error bound for the generalized nonlinear complementarity problem over a polyhedral cone (GNCP). By establishing a new equivalent formulation of the GNCP, we establish a sharper global error bound for the GNCP under weaker conditions, which improves the existing error bound estimation for the problem.

#### 1. Introduction

Let be a polyhedral cone in for matrices , , and let be its dual cone; that is, For continuous mappings , the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector such that 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 [2â€“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 HÃ¶lder 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 HÃ¶lder 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, 9â€“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 . We use to denote the nonnegative orthant in and use and to denote the vectors composed by elements , , , respectively. For simplicity, we use to denote vector , use to denote the identity matrix with appropriate dimension, use to denote a nonnegative vector , and use to denote the distance from point to the solution set .

#### 2. Global Error Bound for the GNCP

First, we give some concepts used in the subsequent.

*Definition 1. *The mapping is said to be(i)monotone with respect to if
(ii)-strongly -monotone with respect to if there are constants , such that

*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.

*Remark 4. *Under (A2) in the assumption, matrix has left inverse , that is, its pseudoinverse of . Certainly, the assumption on matrix is weaker than that on matrix which has full-column rank [4]. In addition, when the mappings , are both linear, then Assumption 3(A1) coincides with Assumption (A1) in [4].

In the following, we will establish a new equivalent reformulation to the GNCP. First, we give the following conclusion established in [2].

Theorem 5. *A point is a solution of the GNCP if and only if there exist , , such that
*

From Theorem 5, under Assumption 3(A2), we can transform the system into a new system in which neither nor 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.*(1)* There exist , such that .*(2)* Consider
**where .*

*Proof. *The proof follows that of Lemma 2.1 in [4], and for completeness, we include it.

Set
Now, we show that these two sets are equal.

First, for any , there exist , such that
Premultiplying (8) by gives
Combining this with (8) yields that
that is,
Recalling Lemma 6, we further have
Combining this with (9) yields that
Using (8), (12), and (13), we have
From the fact that , by (13), one has
Combining this with (14) leads to that . This shows that .

Second, for any , let
Then, , . From (14), one has
that is, . Hence, , and the desired result follows.

Combining this conclusion with Theorem 5, we can establish the following equivalent formulation of the GNCP: where

For the ease of description, we denote , . Thus, system (18) can be written as

For system (20), one has where the first equality follows from the last equality in (20), and the last equality uses the second equality in (20). Thus, system (20) can be further written as Furthermore, for any with , , it holds from (21) that

Now, consider the following optimization problem: where , . Denote the solution set of (24) by .

Lemma 8. *Under Assumption 3(A1), is a convex function.*

*Proof. *For any , , we have
where the first inequality uses Assumption 3(A1). The desired result follows.

Based on (20), combining (23) with Lemma 8, we can obtain the following conclusion.

Lemma 9. *A point is a solution of (20) 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 with , , , and , there exists a constant such that
*

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 such that
*

Before proceeding, we present the following definition introduced in [15].

*Definition 12. *The mapping is said to be strongly nonexpanding with a constant if .

By Lemma 8, is a convex function and the feasible set 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 , , respectively, and is strongly nonexpanding with constant . Then, there exists constant such that
*

*Proof. *For any , let . Then, there exists such that . A direct computation yields that
where the second inequality follows from Definition 12 with constant , the third inequality follows from Definition 1(ii) with constants , , the fourth inequality follows from the Cauchy-Schwarz inequality, the fifth inequality follows from the fact that , for all , the sixth inequality follows from Lemma 11 with constant and Lemma 9, and the seventh inequality follows from Lemma 10 with constant . By (30) and letting , then the desired result follows.

*Remark 14. * It is clear that if is -strongly -monotone and is strongly nonexpanding, then
Moreover, the conditions which both and are HÃ¶lder 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., ) 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., ) in Theorem 3.1 in [11], and compared with the condition that , is strongly monotone (i.e., ) in Theorem 3.1 in [10].

Using the following Definition 15 developed from the complementarity conditions in (22), we can further detect the error bound of the GNCP.

*Definition 15. *A solution of the GNCP is said to be nondegenerate if it satisfies

Lemma 16. *Suppose that Assumptions 3(A1) and 3(A2) hold, and the GNCP has a nondegenerate solution, say . Then,
**
where .*

*Proof. *Since
by Assumption 3(A1), for any , we have
that is,
To prove the assertion, we only need to show that the solution set is equal to the set

For any , combining Lemma 9 with (20) yields that
Letting in (36) yields that
Since , using the similar technique to that of (21), we can obtain
where . Combining (39) with (40), we have .

On the other hand, for any , one has
Since , 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, .

Based on Lemma 16, we obtain the following conclusion.

Corollary 17. *Suppose that the hypotheses of Lemma 16 hold. Then,
*

Theorem 18. *Suppose that the hypotheses of Theorem 13 hold, and the GNCP has a nondegenerate solution. Then, there exists constant such that
*

*Proof. *For any , let . Then, there exists such that . Letting be a nondegenerate solution of GNCP and letting , , then
where the second equality uses the similar technique to that of (30), the third inequality follows from Corollary 17 and Lemma 10 with constant , and the last inequality is based on (36). By (48) and letting , the desired result follows.

In the following, we give an error bound of the HÃ¶lderian type [14].

Lemma 19. *For , let be a convex quadratic function. If the set is nonempty, then there exist a positive integer (called the degree of singularity of the inequality system) and a positive scalar such that
**
where . Furthermore, if contains an interior point, then . *

Based on (18) and (21), the GNCP can also be written as

From Lemma 19, we can establish the following global error bound for GNCP.

Theorem 20. *Suppose that the hypotheses of Theorem 13 hold, and there exists point such that
**
Then, there exists constant such that
*

*Proof. *Let , where . By Lemma 8, we have is a convex quadratic function. Combining this with (51), using Lemma 19 with , this yields the following result
where is a positive constant.

Obviously, is a closed convex set. Thus, for any , there exists a vector such that

For convenience, we also let
From (50), we have , where is defined in (24), so for any , combining Lemma 10, one has
where is a positive constant, and the second and third inequalities follow from the fact that , for all .

Furthermore,
where the second equality follows from the fact that
and the first inequality is by nonexpanding property of projection operator. Thus,
Combining (56) with (59), for any , we have
where the second inequality follows from (56) with constant , 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 .

For any , letting , then there exists such that , and a direct computation yields that
where the deduction of the second equality uses the similar technique to that of (30), and the third inequality is by (60). By (63) and letting , then the desired result follows.

*Remark 21. * When is strongly monotone with respect to , that is, , 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].

#### 3. 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 , : 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 22. *Under Assumption 3(A1), is a convex function.*

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.

Theorem 24. * Under Assumptions 3(A1) and 3(A2), and that mappings and are both linear, there exists constant such that
*

*Proof. *For any , a direct computation yields that
where the first inequality follows from Lemma 11 with constant and Lemma 23, and the second inequality uses Lemma 10 with constant . By (67) and letting , the desired result follows.

*Remark 25. * Obviously, Assumption 3(A2) in Theorem 24 is weaker than Assumption (A2) in Theorem 4.1 in [4], Assumption 3(A1) coincides with Assumption (A1) in [4]. In addition, Theorem 24 is sharper than Theorem 4.1 in [4].

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 such that
*

*Proof. * From Corollary 17, we have
where is a nondegenerate solution of GLCP, and is defined in (64). For any , a direct computation yields that
where the first inequality follows from Lemma 10 with constant , and the second inequality uses (36). Letting , the desired result follows.

*Remark 27. *The condition in Theorem 26 is weaker than that in Theorem 4.2 in [4].

Theorem 28. *Suppose that the hypotheses of Theorem 24 hold, and there exists point such that (51) holds. Then there exists constant such that
*

*Proof. *Let , where . By Lemma 22, is a convex quadratic function, and is a closed convex set. For any , there exists a vector such that
Combining (51) and applying Lemma 19 yield the following result:
where is a positive constant. For convenience, we let
From (50), we have , where is defined in (64). So for any , combining Lemma 10 and using the similar technique to that of (56), one has
where is a positive constant.

Using the fact that
and using the similar technique to that of (57), one has
where the second inequality is by nonexpanding property of projection operator. Thus,
Combining (75) with (78), we know that for any , it holds that