Global Error Bound Estimation for the Generalized Nonlinear Complementarity Problem over a Closed Convex Cone

Abstract

The global error bound estimation for the generalized nonlinear complementarity problem over a closed convex cone (GNCP) is considered. To obtain a global error bound for the GNCP, we first develop an equivalent reformulation of the problem. Based on this, a global error bound for the GNCP is established. The results obtained in this paper can be taken as an extension of previously known results.

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Hongchun Sun 1 and Yiju Wang 2

Recommended by Zhenyu Huang

1, School of Sciences, Linyi University, Shandong, 276005 Linyi, China
2, School of Management Science, Qufu Normal University, Shandong, 276800 Rizhao, China

Received 10 February 2012; Accepted 28 April 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let mappings F,G:^Rn [arrow right]^Rm , H:^Rn [arrow right]^Rl , and the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector ^x* ∈^Rn such that [figure omitted; refer to PDF] where ...A6; is a nonempty closed convex cone in ^Rm and ^{...A6;[composite function]} is its dual cone, that is, ^{...A6;[composite function]} ={u∈^Rm |"^uT v...5;0,for all v∈...A6;} . We denote the solution set of the GNCP by ^X* , and assume that it is nonempty throughout this paper.

The GNCP is a direct generalization of the classical nonlinear complementarity problem which finds applications in engineering, economics, finance, and robust optimization operations research [1-3]. For example, the balance of supply and demand is central to all economic systems; mathematically, this fundamental equation in economics is often described by a complementarity relation between two sets of decision variables. Furthermore, 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 [2]. Up to now, the issues of numerical methods and existence of the solution for the problem were discussed in the literature [4].

Among all the useful tools for theoretical and numerical treatment to variational inequalities, nonlinear complementarity problems, and other related optimization problems, the global error bound, that is, an upper bound estimation of the distance from a given point in ^Rn to the solution set of the problem in terms of some residual functions, is an important one [5, 6]. The error bound estimation for the generalized linear complementarity problems over a polyhedral cone was analyzed by Sun et al. [7]. Using the natural residual function, Pang [8] obtained a global error bound for the strongly monotone and Lipschitz continuous classical nonlinear complementarity problem with a linear constraint set. Xiu and Zhang [9] also presented a global error bound for general variational inequalities with the mapping being strongly monotone and Lipschitz continuous in terms of the natural residual function. If F(x)=x , G(x) is γ -strongly monotone and Hölder continuous, the local error bound for classical variational inequality problems was given by Solodov [6].

To our knowledge, the global error bound for the problem (1.1) with the mapping being γ -strongly monotone and Hölder -continuous hasn't been investigated. Motivated by this fact, The main contribution of this paper is to establish a global error bound for the GNCP via the natural residual function under milder conditions than those needed in [6, 8, 9]. The results obtained in this paper can be taken as an extension of the previously known results in [6, 8, 9].

We give some notations used in this paper. Vectors considered in this paper are all taken in Euclidean space equipped with the standard inner product. The Euclidean norm of vector in the space is denoted by ||·|| . The inner product of vector in the space is denoted by Y9;·,·YA; .

2. The Global Error Bound for GNCP

In this section, we would give error bound for GNCP, which can be viewed as extensions of previously known results. To this end, we will in the following establish an equivalent reformulation of the GNCP and state some well-known properties of the projection operator which is crucial to our results.

In the following, we first give the equivalent reformulation of the GNCP.

Theorem 2.1.

A point ^x* is a solution of (1.1) if and only if ^x* is a solution of the following problem: [figure omitted; refer to PDF]

Proof.

Suppose that ^x* is a solution of (2.1). Since vector 0∈...A6; , by substituting F(x)=0 into (2.1), we have G(^x*)T F(^x* )...4;0 . On the other hand, since F(^x* )∈...A6; , then 2F(^x* )∈...A6; . By substituting F(x)=2F(^x* ) into (2.1), we obtain G(^x*)T F(^x* )...5;0 . Consequently, G(^x*)T F(^x* )=0 . For any F(x)∈...A6; , we have G(^x*)T F(x)=G(^x*)T [F(x)-F(^x* )]...5;0 , that is, G(^x* )∈^{...A6;[composite function]} . Combining H(^x* )=0 , thus, ^x* is a solution of (1.1).

On the contrary, suppose that ^x* is a solution of (1.1), since G(^x* )∈^{...A6;[composite function]} , for any F(x)∈...A6; , we have G(^x*)T F(x)...5;0 , and from G(^x*)T F(^x* )=0 , we have G(^x*)T [F(x)-F(^x* )]...5;0 , combining H(^x* )=0 . Therefor, ^x* is a solution of (2.1).

Now, we give the definition of projection operator and some related properties [10]. For nonempty closed convex set ...A6;⊂^Rm and any vector x∈^Rm , the orthogonal projection of x onto ...A6; , that is, argmin{||y-x|||"y∈...A6;} , is denoted by _P...A6; (x) .

Lemma 2.2.

For any u∈^Rm , v∈...A6; , then

(i) Y9;_P...A6; (u)-u,v-_P...A6; (u)YA;...5;0 ,

(ii) ||_PK (u)-_PK (v)||...4;||u-v|| .

For (2.1), β>0 is a constant, e(x):=F(x)-_P...A6; [F(x)-βG(x)] is called projection-type residual function, and let r(x):=||e(x)|| . The following conclusion provides the relationship between the solution set of (2.1) and that of projection-type residual function [11], which is due to Noor [11].

Lemma 2.3.

x is a solution of (2.1) if and only if e(x)=0, H(x)=0 .

To establish the global error bound of GNCP, we also need the following definition.

Definition 2.4.

The mapping F:^Rn [arrow right]^Rm is said to be

(1) γ -strongly monotone with respect to G:^Rn [arrow right]^Rm if there are constants μ>0, γ>1 such that [figure omitted; refer to PDF]

(2) Hölder -continuous if there are constants L>0, v∈(0,1] such that [figure omitted; refer to PDF]

In this following, based on Lemmas 2.2 and 2.3, we establish error bound for GNCP in the set Ω:={x∈^Rn |"H(x)=0} .

Theorem 2.5.

Suppose that F is γ -strongly monotone with respect to G and with positive constants μ,γ , both F and G are Ho¨lder continuous with positive constants _L1 >0, _L2 >0, _v1 , _v2 ∈(0,1] , respectively, and βμ...4;(_L1 +_L2 β)(2_L1 +_L2 β) holds. Then for any x∈Ω:={x∈^Rn |"H(x)=0} , there exists a solution ^x* of (1.1) such that [figure omitted; refer to PDF]

Proof.

Since [figure omitted; refer to PDF] by the first inequality of (2.1), [figure omitted; refer to PDF] Combining F(^x* )∈...A6; with Lemma 2.2(i), we have [figure omitted; refer to PDF] Substituting _P...A6; [F(x)-βG(x)] in (2.9) by F(x)-e(x) leads to that [figure omitted; refer to PDF] Using (2.8) and (2.10), we obtain [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Base on Definition 2.4, a direct computation yields that [figure omitted; refer to PDF] Combining this, we have [figure omitted; refer to PDF]

On the other hand, for the first inequality of (2.1), by Lemmas 2.3 and 2.2(ii), we have [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Combining (2.14) with (2.16) for any x∈^Rn , if ||x-^x* ||...4;1 , then [figure omitted; refer to PDF]

For any x∈^Rn , if ||x-^x* ||...5;1 , then [figure omitted; refer to PDF]

If r(x)...4;βμ/(_L1 +_L2 β) , then ((_L1 +_L2 β)/βμ)r(x)...4;1 , by (2.17), we have ||x-^x* ||...4;1 , and using (2.17) again, we obtain that (2.4) holds.

If r(x)...5;_L2 β+2_L1 , then r(x)/(2_L1 +_L2 β)...5;1 , combining this with (2.18), we have ||x-^x* ||...5;1 , and using (2.18) again, we conclude that (2.5) holds.

If βμ/(_L1 +_L2 β)<r(x)<2_L1 +_L2 β , then [figure omitted; refer to PDF] Combining (2.17) with (2.18), we conclude that (2.6) holds.

Definition 2.6.

The mapping H involved in the GNCP is said to be α -strongly monotone in ^Rn if there are positive constants σ>0, α>1 such that [figure omitted; refer to PDF]

Base on Theorem 2.5, we are at the position to state our main results in the following.

Theorem 2.7.

Suppose that the hypotheses of Theorem 2.5 hold, H is α -strongly monotone, and the set Ω:={x∈^Rn |"H(x)=0} is convex. Then, there exists a constant ρ>0 , such that, for any x∈^Rn , there exists ^x* ∈^X* such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

For given x∈^Rn , we only need to first project x to Ω:={x∈^Rn |"H(x)=0} , that is, there exists a vector x¯ ∈Ω such that ||x-x¯||=dist (x,Ω) . By Definition 2.6, there exist constants σ>0, α>0 such that [figure omitted; refer to PDF] that is, dist (x,Ω)=||x-x-||...4;(1/σ)^||H(x)||1/α .

Since [figure omitted; refer to PDF] Combining this, we have [figure omitted; refer to PDF] Combining (2.26) with Theorem 2.5, we have the following results.

Case 1 (if r(x¯)...4;βμ/(_L1 +_L2 β) and ||x-x¯||...4;1 ). Combining (2.4) with the first inequality in (2.26), we can obtain that [figure omitted; refer to PDF] where _η1 =^{(max {(_L1 +_L2 β)/βμ,((2_L1 +β_L2 )(_L1 +_L2 β)/βμ)σ-min {_v1 ,_v2 } })1/(1+γ-min {_v1 ,_v2 })} , _ρ1 =max {1/σ,_η1 } .

Case 2 (If r(x¯)...4;βμ/(_L1 +_L2 β) and ||x-x¯||...5;1 ). Combining (2.4) with the second inequality in (2.26), we can also obtain that [figure omitted; refer to PDF] where _η2 =^{(max {(_L1 +_L2 β)/βμ,((2_L1 +β_L2 )(_L1 +_L2 β)/βμ)σ-max {_v1 ,_v2 } })1/(1+γ-min {_v1 ,_v2 })} , _ρ2 =max {1/σ,_η2 } .

Case 3 (if r(x¯)>βμ/(_L1 +_L2 β) and ||x-x¯||...4;1 ). Combining (2.5)-(2.6) with the first inequality in (2.26), we can obtain that [figure omitted; refer to PDF] where _η3 =^{(max {(_L1 +_L2 β)/βμ,((2_L1 +β_L2 )(_L1 +_L2 β)/βμ)σ-min {_v1 ,_v2 } })1/(1+γ-max {_v1 ,_v2 })} , _ρ3 =max {1/σ,_η3 } .

Case 4 (If r(x¯)>βμ/(_L1 +_L2 β) and ||x-x¯||...5;1 ). Combining (2.5)-(2.6) with the second inequality in (2.26), we can also obtain that [figure omitted; refer to PDF] where _η4 =^{(max {(_L1 +_L2 β)/βμ,((2_L1 +β_L2 )(_L1 +_L2 β)/βμ)σ-max {_v1 ,_v2 } })1/(1+γ-max {_v1 ,_v2 })} , _ρ4 =max {1/σ,_η4 } .

By (2.27)-(2.30), we can deduce that (2.21) holds.

Based on Theorem 2.7, we can further establish a global error bound for the GNCP. First, we give that the needed result from [12] mainly discusses the error bound for a polyhedral cone to reach our claims.

Lemma 2.8.

For polyhedral cone P={x∈^Rn |"_D1 x=_d1 ,_D2 x...4;_d2 } with _D1 ∈^Rl×n , _D2 ∈^Rm×n , _d1 ∈^Rl , and _d2 ∈^Rm , there exists a constant _c1 >0 such that [figure omitted; refer to PDF]

Theorem 2.9.

Suppose that the hypotheses of Theorem 2.5 hold, and H is linear mapping. Then, there exists a constant μ>0 , such that, for any x∈^Rn , there exists ^x* ∈^X* such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Proof.

For given x∈^Rn , we only need to first project x to Ω , that is, there exists a vector x¯ ∈Ω such that ||x-x¯||=dist (x,Ω) . By Lemma 2.8, there exists a constant τ>0 such that dist (x,Ω)=||x-x¯||...4;τ||H(x)|| . In the following, the proof is similar to that of Theorem 2.7, and we can deduce that (2.32) holds.

Remark 2.10.

If the constraint condition H(x)=0 is removed in (1.1), F is strongly monotone with respect to G (i.e., γ=1 ), and both F and G are Lipschitz continuous (i.e., _v1 =_v2 =1 ), the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 3.1 in [9].

If the constraint condition H(x)=0 is removed in (1.1) and F(x)=x , G(x) is strongly monotone (i.e., γ=1 ) and Lipschitz continuous (i.e., _v1 =_v2 =1 ), the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 3.1 in [8].

If the constraint condition H(x)=0 is removed in (1.1) and F(x)=x , G(x) is γ -strongly monotone in set {x∈^Rn |"||x-^x* ||...4;1} and Hölder continuous, the error bound in Theorems 2.5, 2.7, and 2.9 reduces to result of Theorem 2 in [6].

Acknowledgments

The authors wish to give their sincere thanks to the anonymous referees for their valuable suggestions and helpful comments which improved the presentation of the paper. This work was supported by the Natural Science Foundation of China (Grant nos. 11171180, 11101303), Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002), and Shandong Provincial Natural Science Foundation (ZR2010AL005, ZR2011FL017).

References

[1] M. C. Ferris, J. S. Pang, "Engineering and economic applications of complementarity problems," Society for Industrial and Applied Mathematics , vol. 39, no. 4, pp. 669-713, 1997.

[2] L. Walras Elements of Pure Economics , Allen and Unwin, London, UK, 1954.

[3] L. P. Zhang, "A nonlinear complementarity model for supply chain network equilibrium," Journal of Industrial and Management Optimization , vol. 3, no. 4, pp. 727-737, 2007.

[4] F. Facchinei, J. S. Pang Finite-Dimensional Variational Inequality and Complementarity Problems , Springer, New York, NY, USA, 2003.

[5] J. S. Pang, "Error bounds in mathematical programming," Mathematical Programming , vol. 79, no. 1-3, pp. 299-332, 1997.

[6] M. V. Solodov, "Convergence rate analysis of iteractive algorithms for solving variational inquality problems," Mathematical Programming , vol. 96, no. 3, pp. 513-528, 2003.

[7] H. C. Sun, Y. J. Wang, L. Q. Qi, "Global error bound for the generalized linear complementarity problem over a polyhedral cone," Journal of Optimization Theory and Applications , vol. 142, no. 2, pp. 417-429, 2009.

[8] J. S. Pang, "A posteriori error bounds for the linearly-constrained variational inequality problem," Mathematics of Operations Research , vol. 12, no. 3, pp. 474-484, 1987.

[9] N. H. Xiu, J. Z. Zhang, "Global projection-type error bounds for general variational inequalities," Journal of Optimization Theory and Applications , vol. 112, no. 1, pp. 213-228, 2002.

[10] E. H. Zarantonello Projections on Convex Sets in Hilbert Space and Spectral Theory, Contributions to Nonlinear Functional Analysis , Academic Press, New York, NY, USA, 1971.

[11] M. A. Noor, "General variational inequalities," Applied Mathematics Letters , vol. 1, no. 2, pp. 119-121, 1988.

[12] A. J. Hoffman, "On approximate solutions of systems of linear inequalities," Journal of Research of the National Bureau of Standards , vol. 49, pp. 263-265, 1952.