Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 245458, 11 pages doi:10.1155/2012/245458 Research Article Global Error Bound Estimation for the Generalized Nonlinear Complementarity Problem over a Closed Convex Cone Hongchun Sun1 and Yiju Wang2 1 2 School of Sciences, Linyi University, Shandong, 276005 Linyi, China School of Management Science, Qufu Normal University, Shandong, 276800 Rizhao, China Correspondence should be addressed to Yiju Wang, wyiju@hotmail.com Received 10 February 2012; Accepted 28 April 2012 Academic Editor: Zhenyu Huang Copyright q 2012 H. Sun and Y. Wang. 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. 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. 1. Introduction Let mappings F, G : Rn → Rm , H : Rn → Rl , and the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector x∗ ∈ Rn such that Fx∗ ∈ K, Gx∗ ∈ K0 , Fx∗ T Gx∗ 0, Hx∗ 0, 1.1 where K is a nonempty closed convex cone in Rm and K◦ is its dual cone, that is, K◦ {u ∈ Rm | uT v ≥ 0, for all v ∈ K}. 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. 2 Journal of Applied Mathematics 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 Fx x, Gx is γ-strongly monotone and H ölder 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 H ölder-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 ·, ·. 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: Gx∗ T Fx − Fx∗ ≥ 0, Hx∗ 0. ∀Fx ∈ K, 2.1 Proof. Suppose that x∗ is a solution of 2.1. Since vector 0 ∈ K, by substituting Fx 0 into 2.1, we have Gx∗ T Fx∗ ≤ 0. On the other hand, since Fx∗ ∈ K, then 2Fx∗ ∈ K. By substituting Fx 2Fx∗ into 2.1, we obtain Gx∗ T Fx∗ ≥ 0. Consequently, Gx∗ T Fx∗ 0. For any Fx ∈ K, we have Gx∗ T Fx Gx∗ T Fx − Fx∗ ≥ 0, that is, Gx∗ ∈ K◦ . Combining Hx∗ 0, thus, x∗ is a solution of 1.1. Journal of Applied Mathematics 3 On the contrary, suppose that x∗ is a solution of 1.1, since Gx∗ ∈ K◦ , for any Fx ∈ K, we have Gx∗ T Fx ≥ 0, and from Gx∗ T Fx∗ 0, we have Gx∗ T Fx − Fx∗ ≥ 0, combining Hx∗ 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 K ⊂ Rm and any vector x ∈ Rm , the orthogonal projection of x onto K, that is, argmin{y − x | y ∈ K}, is denoted by PK x. Lemma 2.2. For any u ∈ Rm , v ∈ K, then i PK u − u, v − PK u ≥ 0, ii PK u − PK v ≤ u − v. For 2.1, β > 0 is a constant, ex : Fx − PK Fx − βGx is called projection-type residual function, and let rx : ex. 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 ex 0, Hx 0. To establish the global error bound of GNCP, we also need the following definition. Definition 2.4. The mapping F : Rn → Rm is said to be 1 γ-strongly monotone with respect to G : Rn → Rm if there are constants μ > 0, γ > 1 such that 1γ Fx − F y , Gx − G y ≥ μx − y , ∀x, y ∈ Rn ; 2.2 2 H ölder-continuous if there are constants L > 0, v ∈ 0, 1 such that Fx − F y ≤ Lx − yv , ∀x, y ∈ Rn . 2.3 In this following, based on Lemmas 2.2 and 2.3, we establish error bound for GNCP in the set Ω : {x ∈ Rn | Hx 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 H ölder continuous with positive constants L1 > 0, L2 > 0, v1 , v2 ∈ 0, 1, 4 Journal of Applied Mathematics respectively, and βμ ≤ L1 L2 β2L1 L2 β holds. Then for any x ∈ Ω : {x ∈ Rn | Hx 0}, there exists a solution x∗ of 1.1 such that rx 2L1 L2 β 1/ min{v1 ,v2 } ≤ x − x∗ ≤ L1 L2 β rx βμ 1/1γ−min{v1 ,v2 } , βμ . if rx ≤ L1 L2 β rx 2L1 L2 β 1/ max{v1 ,v2 } ≤ x − x∗ ≤ L1 L2 β rx βμ 2.4 1/1γ−max{v1 ,v2 } , 2.5 if rx ≥ 2L1 L2 β. rx 2L1 L2 β 1/ min{v1 ,v2 } ≤ x − x∗ ≤ L1 L2 β rx βμ 1/1γ−max{v1 ,v2 } , βμ < rx < 2L1 L2 β. if L1 L2 β 2.6 Proof. Since Fx − ex PK Fx − βGx ∈ K, 2.7 by the first inequality of 2.1, Fx − ex − Fx∗ T βGx∗ ≥ 0. 2.8 Combining Fx∗ ∈ K with Lemma 2.2i, we have Fx∗ − PK Fx − βGx , PK Fx − βGx − Fx − βGx ≥ 0. 2.9 Substituting PK Fx − βGx in 2.9 by Fx − ex leads to that Fx − Fx∗ − exT ex − βGx ≥ 0. 2.10 Using 2.8 and 2.10, we obtain Fx − Fx∗ − exT ex βGx∗ − Gx ≥ 0, 2.11 Journal of Applied Mathematics 5 that is, βFx − Fx∗ T Gx∗ − Gx exT Fx − Fx∗ − βGx∗ − Gx − exT ex ≥ 0. 2.12 Base on Definition 2.4, a direct computation yields that x − x∗ 1γ ≤ ≤ 1 Fx − Fx∗ T Gx − Gx∗ μ 1 exT Fx − Fx∗ βGx − Gx∗ − exT ex βμ 1 ≤ ex Fx − Fx∗ βGx − Gx∗ βμ ≤ 2.13 1 rx L1 x − x∗ v1 βL2 x − x∗ v2 . βμ Combining this, we have ⎧ 1/1γ−min{v1 ,v2 } ⎪ L1 L2 β ⎪ ⎪ rx , ⎪ ⎨ βμ x − x∗ ≤ 1/1γ−max{v1 ,v2 } ⎪ ⎪ L1 L 2 β ⎪ ⎪ rx , ⎩ βμ if x − x∗ ≤ 1, 2.14 if x − x∗ ≥ 1. On the other hand, for the first inequality of 2.1, by Lemmas 2.3 and 2.2ii, we have rx ex − ex∗ Fx − PK Fx − βGx − Fx∗ PK Fx∗ − βGx∗ Fx − Fx∗ PK Fx − βGx − PK Fx∗ − βGx∗ ≤ Fx − Fx∗ Fx − βGx − Fx∗ − βGx∗ 2.15 2Fx − Fx∗ βGx − Gx∗ ≤ 2L1 x − x∗ v1 L2 βx − x∗ v2 . Thus, ⎧ 1/ min{v1 ,v2 } ⎪ rx ⎪ ⎪ , ⎨ 2L L2 β x − x∗ ≥ 1 1/ max{v1 ,v2 } ⎪ rx ⎪ ⎪ , ⎩ 2L1 L2 β if x − x∗ ≤ 1, ∗ if x − x ≥ 1. 2.16 6 Journal of Applied Mathematics Combining 2.14 with 2.16 for any x ∈ Rn , if x − x∗ ≤ 1, then rx 2L1 L2 β 1/1γ−min{v1 ,v2 } L1 L 2 β rx ,1 . βμ 2.17 1/1γ−max{v1 ,v2 } 1/ max{v1 ,v2 } L1 L2 β ∗ rx , 1 ≤ x − x ≤ . βμ 2.18 1/ min{v1 ,v2 } ∗ ≤ x − x ≤ min For any x ∈ Rn , if x − x∗ ≥ 1, then max rx 2L1 L2 β If rx ≤ βμ/L1 L2 β, then L1 L2 β/βμrx ≤ 1, by 2.17, we have x − x∗ ≤ 1, and using 2.17 again, we obtain that 2.4 holds. If rx ≥ L2 β 2L1 , then rx/2L1 L2 β ≥ 1, combining this with 2.18, we have x − x∗ ≥ 1, and using 2.18 again, we conclude that 2.5 holds. If βμ/L1 L2 β < rx < 2L1 L2 β, then L1 L 2 β rx > 1, βμ rx < 1. L2 β 2L1 2.19 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 1α x − y, Hx − H y ≥ σ x − y , ∀x, y ∈ Rn . 2.20 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 | Hx 0} is convex. Then, there exists a constant ρ > 0, such that, for any x ∈ Rn , there exists x∗ ∈ X ∗ such that x − x∗ ≤ ρ Hx1/α Rx , ∀x ∈ Rn , 2.21 Journal of Applied Mathematics 7 where ⎧ 1/1γ−min{v1 ,v2 } ⎫ ⎪ ⎪ L1 L2 β 2L1 βL2 L1 L2 β − min{v1 ,v2 } 1 ⎪ ⎪ ⎪ ⎪ ⎪ , max , σ ,⎪ ⎪ ⎪ ⎪ ⎪ σ βμ βμ ⎪ ⎪ ⎪ ⎪ ⎪ 1/1γ−min{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ L1 L2 β 2L1 βL2 L1 L2 β − max{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ max , σ , ⎪ ⎪ ⎨ ⎬ βμ βμ ρ max , 1/1γ−max{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ L1 L2 β 2L1 βL2 L1 L2 β − min{v1 ,v2 } ⎪ ⎪ ⎪ max , σ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ βμ βμ ⎪ ⎪ ⎪ ⎪ ⎪ 1/1γ−max{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ ⎪ βL L β 2L L ⎪ ⎪ L1 L 2 β 1 2 1 2 ⎪ ⎪ − max{v ,v } 1 2 ⎪ ⎪ max , σ , ⎩ ⎭ βμ βμ 2.22 ⎫ ⎧ 1/1γ−min{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ rx Hxmin{v1 ,v2 }/α ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/1γ−min{v ,v } ⎪ ⎪ 1 2 ⎪ ⎪ ⎬ ⎨ rx Hxmax{v1 ,v2 }/α 2.23 Rx max 1/1γ−max{v1 ,v2 } . ⎪ ⎪ min{v1 ,v2 }/α ⎪ ⎪ rx Hx ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ rx Hxmax{v1 ,v2 }/α 1/1γ−max{v1 ,v2 } ⎪ Proof. For given x ∈ Rn , we only need to first project x to Ω : {x ∈ Rn | Hx 0}, that is, there exists a vector x ∈ Ω such that x − x distx, Ω. By Definition 2.6, there exist constants σ > 0, α > 0 such that x − x1α ≤ 1 x − x, Hx − Hx σ ≤ 1 x − xHx − Hx σ 1 x − xHx, σ that is, distx, Ω x − x ≤ 1/σHx1/α . Since rx − rx ≤ ex − ex Fx − PK Fx − βGx − Fx − PK Fx − βGx ≤ Fx − Fx PK Fx − βGx − PK Fx − βGx ≤ Fx − Fx Fx − βGx − Fx − βGx ≤ 2Fx − Fx βGx − Gx 2.24 8 Journal of Applied Mathematics ≤ 2L1 x − xν1 βL2 x − xν2 2L1 βL2 x − xmin{v1 ,v2 } , if ≤ 2L1 βL2 x − xmax{v1 ,v2 } , if 2L1 βL2 distx, Ωmin{v1 ,v2 } , 2L1 βL2 distx, Ωmax{v1 ,v2 } , x − x < 1 x − x ≥ 1 if x − x ≤ 1 if x − x ≥ 1, 2.25 Combining this, we have 2L1 βL2 distx, Ωmin{v1 ,v2 } , if x − x < 1 rx ≤ rx 2L1 βL2 distx, Ωmax{v1 ,v2 } , if x − x ≥ 1. 2.26 Combining 2.26 with Theorem 2.5, we have the following results. Case 1 if rx ≤ βμ/L1 L2 β and x − x ≤ 1. Combining 2.4 with the first inequality in 2.26, we can obtain that x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β rx βμ 1/1γ−min{v1 ,v2 } ≤ distx, Ω 1/1γ−min{v1 ,v2 } 2L1 βL2 L1 L2 β L 1 L2 β min{v1 ,v2 } rx distx, Ω βμ βμ 1/1γ−min{v1 ,v2 } 1 Hx1/α η1 rx Hxmin{v1 ,v2 }/α σ 1/1γ−min{v1 ,v2 } 1/α min{v1 ,v2 }/α ≤ ρ1 Hx rx Hx , ≤ 2.27 where η1 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − min{v1 ,v2 } } ρ1 max{1/σ, η1 }. 1/1γ−min{v1 ,v2 } , Journal of Applied Mathematics 9 Case 2 If rx ≤ βμ/L1 L2 β and x − x ≥ 1. Combining 2.4 with the second inequality in 2.26, we can also obtain that x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β rx βμ 1/1γ−min{v1 ,v2 } ≤ distx, Ω 1/1γ−min{v1 ,v2 } 2L1 βL2 L1 L2 β L 1 L2 β max{v1 ,v2 } rx distx, Ω βμ βμ 1/1γ−min{v1 ,v2 } 1 Hx1/α η2 rx Hxmax{v1 ,v2 }/α σ 1/1γ−min{v1 ,v2 } ≤ ρ2 Hx1/α rx Hxmax{v1 ,v2 }/α , ≤ 2.28 1/1γ−min{v1 ,v2 } where η2 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − max{v1 ,v2 } } , ρ2 max{1/σ, η2 }. Case 3 if rx > βμ/L1 L2 β and x −x ≤ 1. Combining 2.5-2.6 with the first inequality in 2.26, we can obtain that x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β rx βμ 1/1γ−max{v1 ,v2 } ≤ distx, Ω 1/1γ−max{v1 ,v2 } 2L1 βL2 L1 L2 β L 1 L2 β min{v1 ,v2 } rx distx, Ω βμ βμ 1/1γ−max{v1 ,v2 } 1 Hx1/α η3 rx Hxmin{v1 ,v2 }/α σ 1/1γ−max{v1 ,v2 } ≤ ρ1 Hx1/α rx Hxmin{v1 ,v2 }/α , ≤ 2.29 where η3 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − min{v1 ,v2 } } ρ3 max{1/σ, η3 }. 1/1γ−max{v1 ,v2 } , 10 Journal of Applied Mathematics Case 4 If rx > βμ/L1 L2 β and x − x ≥ 1. Combining 2.5-2.6 with the second inequality in 2.26, we can also obtain that x − x∗ ≤ distx, Ω x − x∗ ≤ distx, Ω L1 L2 β rx βμ 1/1γ−max{v1 ,v2 } ≤ distx, Ω 1/1γ−max{v1 ,v2 } 2L1 βL2 L1 L2 β L 1 L2 β max{v1 ,v2 } rx distx, Ω βμ βμ 1/1γ−max{v1 ,v2 } 1 Hx1/α η4 rx Hxmax{v1 ,v2 }/α σ 1/1γ−max{v1 ,v2 } 1/α max{v1 ,v2 }/α ≤ ρ4 Hx rx Hx , ≤ 2.30 where η4 max{L1 L2 β/βμ, 2L1 βL2 L1 L2 β/βμσ − max{v1 ,v2 } } ρ4 max{1/σ, η4 }. By 2.27–2.30, we can deduce that 2.21 holds. 1/1γ−max{v1 ,v2 } , 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 ≤ d2 } with D1 ∈ Rl×n , D2 ∈ Rm×n , d1 ∈ Rl , and d2 ∈ Rm , there exists a constant c1 > 0 such that distx, P ≤ c1 D1 x − d1 D2 x − d2 ∀x ∈ Rn . 2.31 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 x − x∗ ≤ μ{Hx Rx}, ∀x ∈ Rn , 2.32 where ⎧ 1/1γ−min{v1 ,v2 } ⎫ ⎪ ⎪ ⎪ ⎪ rx Hxmin{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ ⎪ 1/1γ−min{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ rx Hxmax{v1 ,v2 } ⎬ Rx max 1/1γ−max{v1 ,v2 } . ⎪ ⎪ ⎪ ⎪ rx Hxmin{v1 ,v2 } ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1/1γ−max{v ,v } ⎪ 1 2 ⎪ ⎩ rx Hxmax{v1 ,v2 } ⎭ 2.33 Journal of Applied Mathematics 11 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 distx, Ω. By Lemma 2.8, there exists a constant τ > 0 such that distx, Ω x − x ≤ τHx. 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 Hx 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 Hx 0 is removed in 1.1 and Fx x, Gx 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 Hx 0 is removed in 1.1 and Fx x, Gx is γstrongly monotone in set {x ∈ Rn | x − x∗ ≤ 1} and H ölder 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 and 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 and J. S. 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