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

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 𝒦={𝑣∈𝑅^𝑚∣𝐴𝑣≥0,𝐵𝑣=0} be a polyhedral cone in 𝑅^𝑚 for matrices 𝐴∈𝑅^𝑠×𝑚, 𝐵∈𝑅^𝑡×𝑚, and let 𝒦^∘ be its dual cone; that is, 𝒦^∘=𝑢∈𝑅^𝑚∣𝑢=𝐴^⊤𝜆₁+𝐵^⊤𝜆₂,𝜆₁∈𝑅^𝑠₊,𝜆₂∈𝑅^𝑡. (1) For continuous mappings 𝐹,𝐺∶𝑅^𝑛→𝑅^𝑚, the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector 𝑥^∗∈𝑅^𝑛 such that 𝐹𝑥^∗∈𝒦,𝐺𝑥^∗∈𝒦^∘,𝐹𝑥^∗⊤𝐺𝑥^∗=0. (2) Throughout this paper, the solution set of the GNCP, denoted by 𝑋^∗, 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 [2–5] 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 𝛾-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, 9–11].

To end this section, we give some notations used in this paper. Vectors considered in this paper are taken in the Euclidean space 𝑅^𝑛 equipped with the usual inner product, and the Euclidean 2-norm and 1-norm of vector in 𝑅^𝑛 are, respectively, denoted by ‖⋅‖ and ‖⋅‖₁. We use 𝑅^𝑛₊ to denote the nonnegative orthant in 𝑅^𝑛 and use 𝑥₊ and 𝑥₋ to denote the vectors composed by elements (𝑥₊)_𝑖∶=max{𝑥_𝑖,0}, (𝑥₋)_𝑖∶=max{−𝑥_𝑖,0}, 1≤𝑖≤𝑛, respectively. For simplicity, we use (𝑥;𝑦) to denote vector (𝑥^⊤,𝑦^⊤)^⊤, use 𝐼 to denote the identity matrix with appropriate dimension, use 𝑥≥0 to denote a nonnegative vector 𝑥∈𝑅^𝑛, and use dist(𝑥,𝑋^∗) to denote the distance from point 𝑥 to the solution set 𝑋^∗.

2. Global Error Bound for the GNCP

First, we give some concepts used in the subsequent.

Definition 1. The mapping 𝐹∶𝑅^𝑛→𝑅^𝑚 is said to be(i)monotone with respect to 𝐺∶𝑅^𝑛→𝑅^𝑚 if 𝐹(𝑥)−𝐹𝑦,𝐺(𝑥)−𝐺𝑦≥0,∀𝑥,𝑦∈𝑅^𝑛; (3)(ii)𝛾-strongly 𝐺-monotone with respect to 𝐺∶𝑅^𝑛→𝑅^𝑚 if there are constants 𝑐₁>0, 𝛾>0 such that 𝐹(𝑥)−𝐹𝑦,𝐺(𝑥)−𝐺𝑦≥𝑐₁𝐺(𝑥)−𝐺𝑦^1+𝛾,∀𝑥,𝑦∈𝑅^𝑛. (4)

Remark 2. Based on this definition, 𝛾-strongly 𝐺-monotone implies monotonicity, and if 𝐹(𝑥)=𝑀𝑥+𝑝, 𝐺(𝑥)=𝑁𝑥+𝑞 with 𝑀,𝑁∈𝑅^𝑚×𝑛, 𝑝,𝑞∈𝑅^𝑚, then the above Definition 1(i) is equivalent to that the matrix 𝑀^⊤𝑁 is positive semidefinite.

Now, we give some assumptions for our analysis based on Definition 1.

Assumption 3. For mappings 𝐹, 𝐺 and matrix 𝐴 involved in the GNCP, we assume that(A1) mapping 𝐹 is monotone with respect to mapping 𝐺;(A2) matrix 𝐴^⊤ has full-column rank.

Remark 4. Under (A2) in the assumption, matrix 𝐴^⊤ has left inverse (𝐴𝐴^⊤)⁻¹𝐴, that is, its pseudoinverse of 𝐴^⊤. Certainly, the assumption on matrix 𝐴^⊤ is weaker than that on matrix (𝐴^⊤,𝐵^⊤) which has full-column rank [4]. In addition, when the mappings 𝐹, 𝐺 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 𝑥^∗∈𝑅^𝑛 is a solution of the GNCP if and only if there exist 𝜆^∗₁∈𝑅^𝑠, 𝜆^∗₂∈𝑅^𝑡, such that 𝐴𝐹𝑥^∗𝑥≥0,𝐵𝐹^∗𝜆=0,_∗1𝐹𝑥≥0,_∗⊤𝐺𝑥^∗𝐺𝑥=0,_∗=𝐴^⊤𝜆^∗₁+𝐵^⊤𝜆^∗₂. (5)

From Theorem 5, under Assumption 3(A2), we can transform the system into a new system in which neither 𝜆₁ nor 𝜆₂ is involved. To this end, we need the following conclusion [12].

Lemma 6. If the linear system 𝐻𝑦=𝑏 is consistent, then 𝑦=𝐻⁺𝑏 is the solution with the minimum 2-norm, where 𝐻⁺ is the pesudo-inverse of 𝐻.

Lemma 7. Suppose that Assumption 3(A2) holds. Then, for any 𝑥∈𝑅^𝑛, the following statements are equivalent.(1) There exist 𝜆₁∈𝑅^𝑠₊, 𝜆₂∈𝑅^𝑡 such that 𝐺(𝑥)=𝐴^⊤𝜆₁+𝐵^⊤𝜆₂.(2) Consider −𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹𝐺(𝑥)𝐴≥0,_⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹+𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼−𝐼𝐺(𝑥)=0, (6)where 𝐴_𝐿⁻¹=(𝐴𝐴^⊤)⁻¹𝐴.

Proof. The proof follows that of Lemma 2.1 in [4], and for completeness, we include it.
Set 𝑋₁∶=𝑥∈𝑅^𝑛∣𝐺(𝑥)=𝐴^⊤𝜆₁+𝐵^⊤𝜆₂forsome𝜆₁∈𝑅^𝑠₊,𝜆₂∈𝑅^𝑡,𝑋₂∶=𝑥∈𝑅^𝑛∣−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+×𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹𝐺(𝑥)𝐴≥0,_⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+×𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹+𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+×𝐴^⊤𝐴_𝐿⁻¹−𝐼−𝐼𝐺(𝑥)=0. (7) Now, we show that these two sets are equal.
First, for any 𝑥∈𝑋₁, there exist 𝜆₁∈𝑅^𝑠₊, 𝜆₂∈𝑅^𝑡 such that 𝐺(𝑥)=𝐴^⊤𝜆₁+𝐵^⊤𝜆₂. (8) Premultiplying (8) by 𝐴_𝐿⁻¹∶=(𝐴𝐴^⊤)⁻¹𝐴 gives 𝐴_𝐿⁻¹𝐺(𝑥)=𝜆₁+𝐴_𝐿⁻¹𝐵^⊤𝜆₂. (9) Combining this with (8) yields that 𝐺(𝑥)=𝐴^⊤𝐴_𝐿⁻¹𝐺(𝑥)−𝐴_𝐿⁻¹𝐵^⊤𝜆₂+𝐵^⊤𝜆₂=𝐴^⊤𝐴_𝐿⁻¹𝐺(𝑥)−𝐴^⊤𝐴_𝐿⁻¹𝐵^⊤−𝐵^⊤𝜆₂; (10) that is, 𝐴^⊤𝐴_𝐿⁻¹𝐵^⊤−𝐵^⊤𝜆₂=𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐺(𝑥). (11) Recalling Lemma 6, we further have 𝜆₂=𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐺(𝑥). (12) Combining this with (9) yields that 𝜆₁=−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹×𝐺(𝑥). (13) Using (8), (12), and (13), we have 𝐴^⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹+𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼−𝐼𝐺(𝑥)=0. (14) From the fact that 𝜆₁≥0, by (13), one has −𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹𝐺(𝑥)≥0. (15) Combining this with (14) leads to that 𝑥∈𝑋₂. This shows that 𝑋₁⊆𝑋₂.
Second, for any 𝑥∈𝑋₂, let 𝜆₁=−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹(𝑥),𝜆×𝐺₂=𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐺(𝑥). (16) Then, 𝜆₁∈𝑅^𝑠₊, 𝜆₂∈𝑅^𝑡. From (14), one has 𝐺(𝑥)=𝐴^⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹×𝐺(𝑥)+𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐺(𝑥)=𝐴^⊤𝜆₁+𝐵^⊤𝜆₂; (17) that is, 𝑥∈𝑋₁. Hence, 𝑋₂⊆𝑋₁, and the desired result follows.

Combining this conclusion with Theorem 5, we can establish the following equivalent formulation of the GNCP: (𝑥)(𝑥)(𝐹(𝑥𝐴𝐹≥0,𝐵𝐹=0,))^⊤𝐺(𝑥)(𝑥)(𝑥)=0,𝑈𝐺≥0,𝑉𝐺=0, (18) where 𝑈=−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹,𝑉=𝐴^⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+×𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹+𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹−𝐼−𝐼. (19)

For the ease of description, we denote 𝜇=𝐹(𝑥), 𝜈=𝐺(𝑥). Thus, system (18) can be written as 𝜇𝐴𝜇≥0,𝐵𝜇=0,_⊤𝜈=0,𝑈𝜈≥0,𝑉𝜈=0. (20)

For system (20), one has 𝜇^⊤𝜈=𝜇^⊤𝐴^⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+×𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹+𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹𝜈=−𝐼𝐴𝜇^⊤−𝐴_𝐿⁻¹𝐵^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+×𝐴^⊤𝐴_𝐿⁻¹−𝐼+𝐴_𝐿⁻¹𝜈+𝐵𝜇^⊤𝐴^⊤𝐴_𝐿⁻¹−𝐼𝐵^⊤+𝐴^⊤𝐴_𝐿⁻¹𝜈=−𝐼𝐴𝜇^⊤[𝑈𝜈], (21) 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 𝐴𝜇≥0,𝐵𝜇=0,𝐴𝜇^⊤(𝑈𝜈)=0,𝑈𝜈≥0,𝑉𝜈=0. (22) Furthermore, for any (𝜇,𝜈)∈𝑅^𝑚×𝑅^𝑚 with 𝐴𝜇≥0, 𝑈𝜈≥0, it holds from (21) that 𝜇^⊤𝜈≥0. (23)

Now, consider the following optimization problem: min𝑓(𝜔)=[(𝐼,0)𝜔]^⊤[(0,𝐼)𝜔]s.t.𝜔∈Ω, (24) where 𝜔=(𝜇,𝜈), Ω={𝜔∈𝑅^2𝑚∣𝐴(𝐼,0)𝜔≥0,𝐵(𝐼,0)𝜔=0,𝑈(0,𝐼)𝜔≥0,𝑉(0,𝐼)𝜔=0}. Denote the solution set of (24) by Ω^∗.

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

Proof. For any 𝜔₁,𝜔₂∈𝑅^2𝑚, 𝜏∈[0,1], we have 𝑓𝜏𝜔₁+(1−𝜏)𝜔₂−𝜏𝑓𝜔₁−(1−𝜏)𝑓𝜔₂=(𝐼,0)𝜏𝜔₁+(1−𝜏)𝜔₂^⊤×(0,𝐼)𝜏𝜔₁+(1−𝜏)𝜔₂−𝜏(𝐼,0)𝜔₁^⊤(0,𝐼)𝜔₁−(1−𝜏)(𝐼,0)𝜔₂^⊤(0,𝐼)𝜔₂=𝜏²(𝐼,0)𝜔₁^⊤(0,𝐼)𝜔₁+(1−𝜏)²(𝐼,0)𝜔₂^⊤(0,𝐼)𝜔₂+𝜏(1−𝜏)(𝐼,0)𝜔₁^⊤(0,𝐼)𝜔₂+𝜏(1−𝜏)(𝐼,0)𝜔₂^⊤(0,𝐼)𝜔₁−𝜏(𝐼,0)𝜔₁^⊤(0,𝐼)𝜔₁−(1−𝜏)(𝐼,0)𝜔₂^⊤(0,𝐼)𝜔₂=−𝜏(1−𝜏)(𝐼,0)𝜔₁^⊤(0,𝐼)𝜔₁−𝜏(1−𝜏)(𝐼,0)𝜔₂^⊤(0,𝐼)𝜔₂+𝜏(1−𝜏)(𝐼,0)𝜔₁^⊤(0,𝐼)𝜔₂+𝜏(1−𝜏)(𝐼,0)𝜔₂^⊤(0,𝐼)𝜔₁=−𝜏(1−𝜏)(𝐼,0)𝜔₁−(𝐼,0)𝜔₂^⊤×(0,𝐼)𝜔₁−(0,𝐼)𝜔₂≤0, (25) 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 𝜔^∗=(𝜇^∗,𝜈^∗)∈𝑅^2𝑚 is a solution of (20) if and only if 𝜔^∗ 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 𝑃={𝑥∈𝑅^𝑛∣𝐷₁𝑥=𝑑₁,𝐵₁𝑥≤𝑏₁} with 𝐷₁∈𝑅^𝑙×𝑛, 𝐵₁∈𝑅^𝑚×𝑛, 𝑑₁∈𝑅^𝑙, and 𝑏₁∈𝑅^𝑚, there exists a constant 𝑐₂>0 such that dist(𝑥,𝑃)≤𝑐₂𝐷₁𝑥−𝑑₁+𝐵₁𝑥−𝑏₁₊,∀𝑥∈𝑅^𝑛. (26)

Lemma 11. Let 𝑃 be a convex polyhedron in 𝑅^𝑛, and let 𝜃 be a convex quadratic function defined on 𝑅^𝑛. Let 𝑆 be the nonempty set of globally optimal solutions of the programming: min𝜃(𝑥)𝑠.𝑡.𝑥∈𝑃 (27) with 𝜃_𝑜𝑝𝑡 being the optimal value of 𝜃 on 𝑆. There exists a scalar 𝑐₃>0 such that dist(𝑥,𝑆)≤𝑐₃maxdist(𝑥,𝑃),𝜃(𝑥)−𝜃_{𝑜𝑝𝑡+},𝜃(𝑥)−𝜃_{𝑜𝑝𝑡+}^1/2,∀𝑥∈𝑅^𝑛. (28)

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

Definition 12. The mapping 𝐺∶𝑅^𝑛→𝑅^𝑚 is said to be strongly nonexpanding with a constant 𝛼>0 if ‖𝐺(𝑥)−𝐺(𝑦)‖≥𝛼‖𝑥−𝑦‖.

By Lemma 8, 𝑓(𝜔) 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 𝐹 is 𝛾-strongly 𝐺-monotone with positive constants 𝑐₁, 𝛾, respectively, and 𝐺 is strongly nonexpanding with constant 𝛼>0. Then, there exists constant 𝜌₁>0 such that dist𝑥,𝑋^∗≤𝜌₁[𝐴𝐹(𝑥)]₋+‖(𝑥+(𝑥𝐵𝐹)‖[𝑈𝐺)]₋+‖(𝑥+𝐹(𝑥)𝑉𝐺)‖_⊤𝐺(𝑥)₊+𝐹(𝑥)^⊤𝐺(𝑥)₊^{1/22/(1+𝛾)},∀𝑥∈𝑅^𝑛. (29)

Proof. For any 𝑥∈𝑅^𝑛, let 𝜔=(𝜇,𝜈)=(𝐹(𝑥),𝐺(𝑥))∈𝑅^2𝑚. Then, there exists 𝜔^∗=(𝜇^∗,𝜈^∗)=(𝐹(𝑥^∗),𝐺(𝑥^∗))∈Ω^∗ such that dist(𝜔,Ω^∗)=‖𝜔−𝜔^∗‖. A direct computation yields that dist^1+𝛾𝑥,𝑋^∗≤𝑥−𝑥^∗1+𝛾≤1𝛼^1+𝛾𝐺(𝑥)−𝐺𝑥^∗1+𝛾≤1𝑐₁𝛼^1+𝛾𝐹(𝑥)−𝐹𝑥^∗⊤𝐺(𝑥)−𝐺𝑥^∗≤1𝑐₁𝛼^1+𝛾𝐹(𝑥)−𝐹𝑥^∗𝐺(𝑥)−𝐺𝑥^∗≤12𝑐₁𝛼^1+𝛾𝐹(𝑥)−𝐹𝑥^∗2+𝐺(𝑥)−𝐺𝑥^∗2=12𝑐₁𝛼^1+𝛾𝜔−𝜔^∗2=12𝑐₁𝛼^1+𝛾dist²𝜔,Ω^∗≤12𝑐₁𝛼^1+𝛾𝑐²₃×maxdist(𝜔,Ω),𝑓(𝜔)₊,𝑓(𝜔)₊^1/22≤12𝑐₁𝛼^1+𝛾𝑐²₃×max𝑐₂[𝐴(𝐼,0)𝜔]₋+‖𝐵()𝜔‖+𝑈()𝜔]𝐼,0[0,𝐼₋+‖𝑉()𝜔,𝑓(𝜔0,𝐼‖)₊,𝑓(𝜔)₊^1/22≤12𝑐₁𝛼^1+𝛾𝑐²₃×max𝑐₂,1²𝐴𝜇₋++]𝐵𝜇[𝑈𝜈₋+‖‖+𝜇𝑉𝜈_⊤𝜈₊+𝜇^⊤𝜈₊^1/22, (30) where the second inequality follows from Definition 12 with constant 𝛼>0, the third inequality follows from Definition 1(ii) with constants 𝑐₁>0, 𝛾>0, the fourth inequality follows from the Cauchy-Schwarz inequality, the fifth inequality follows from the fact that (1/2)(𝑎²+𝑏²)≥𝑎𝑏, for all 𝑎,𝑏∈𝑅, the sixth inequality follows from Lemma 11 with constant 𝑐₃>0 and Lemma 9, and the seventh inequality follows from Lemma 10 with constant 𝑐₂>0. By (30) and letting 𝜌₁={(1/2𝑐₁𝛼^1+𝛾)𝑐²₃max{𝑐₂,1}²}^1/(1+𝛾), then the desired result follows.

Remark 14. It is clear that if 𝐹 is 𝛾-strongly 𝐺-monotone and 𝐺 is strongly nonexpanding, then 𝐹(𝑥)−𝐹𝑦,𝐺(𝑥)−𝐺𝑦≥𝑐₁𝐺(𝑥)−𝐺𝑦^1+𝛾≥𝑐₁𝛼^1+𝛾𝑥−𝑦^1+𝛾,∀𝑥,𝑦∈𝑅^𝑛. (31) Moreover, the conditions which both 𝐹 and 𝐺 are Hölder continuous (or both 𝐹 and 𝐺 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 𝐹 is 𝛾-strongly 𝐺-monotone and 𝐺 is strongly nonexpanding in Theorem 13 is extended compared with the condition that 𝐹 is strongly monotone with respect to 𝐺 (i.e., 𝛾=1) in Theorems 3.4 and 3.6 in [15], and it is also extended than compared with the condition 𝐹 is strongly monotone with respect to 𝐺 (i.e., 𝛾=1) in Theorem 3.1 in [11], and compared with the condition that 𝐹(𝑥)=𝑥, 𝐺(𝑥) is strongly monotone (i.e., 𝛾=1) 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 𝑥₀ of the GNCP is said to be nondegenerate if it satisfies 𝐴𝐹𝑥₀+𝑈𝐺𝑥₀>0. (32)

Lemma 16. Suppose that Assumptions 3(A1) and 3(A2) hold, and the GNCP has a nondegenerate solution, say 𝑥₀. Then, Ω^∗=𝜔∈Ω∣[(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀=0, (33) where 𝜔₀=(𝜇₀,𝜈₀)=(𝐹(𝑥₀),𝐺(𝑥₀)).

Proof. Since (𝐼,0)𝜔₀^⊤(0,𝐼)𝜔₀=0, (34) by Assumption 3(A1), for any 𝜔∈Ω, we have 0≤𝜇−𝜇₀^⊤𝜈−𝜈₀=(𝐼,0)𝜔−(𝐼,0)𝜔₀^⊤(0,𝐼)𝜔−(0,𝐼)𝜔₀=[(𝐼,0)𝜔]^⊤[(0,𝐼)𝜔]−[(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀−[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀; (35) that is, [(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀≤[(𝐼,0)𝜔]^⊤[(0,𝐼)𝜔]. (36) To prove the assertion, we only need to show that the solution set Ω^∗ is equal to the set 𝑊∶=𝜔∈Ω∣[(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀=0. (37)
For any 𝜔∈Ω^∗, combining Lemma 9 with (20) yields that [(𝐼,0)𝜔]^⊤[(0,𝐼)𝜔]=0. (38) Letting 𝜔=𝜔 in (36) yields that [(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀≤0. (39) Since 𝜔,𝜔₀∈Ω, using the similar technique to that of (21), we can obtain [(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀=𝜇^⊤𝜈₀+𝜈^⊤𝜇₀=𝐴𝜇^⊤𝑈𝜈₀+(𝑈𝜈)^⊤𝐴𝜇₀≥0, (40) where 𝜔=(𝜇,𝜈). Combining (39) with (40), we have Ω^∗⊆𝑊.
On the other hand, for any 𝜔∈𝑊, one has [(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀=0. (41) Since 𝜔,𝜔₀∈Ω, using the similar arguments to that of (21), one has [(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀=𝐴𝜇^⊤𝑈𝜈₀,[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀=[𝑈𝜈]^⊤𝐴𝜇₀. (42) Combining this with (41) yields that 𝐴𝜇^⊤𝑈𝜈₀+[𝑈𝜈]^⊤𝐴𝜇₀=0. (43) From (32), we deduce that 𝐴𝜇^⊤[𝑈𝜈]=0. (44) Thus, using (21), one has 𝜇^⊤𝜈=0. (45) Hence, 𝜔∈Ω^∗.

Based on Lemma 16, we obtain the following conclusion.

Corollary 17. Suppose that the hypotheses of Lemma 16 hold. Then, Ω^∗=𝜔∈Ω∣[(𝐼,0)𝜔]^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀≤0. (46)

Theorem 18. Suppose that the hypotheses of Theorem 13 hold, and the GNCP has a nondegenerate solution. Then, there exists constant 𝜌₂>0 such that dist𝑥,𝑋^∗≤𝜌₂[𝐴𝐹(𝑥)]₋+‖𝐵𝐹(𝑥)‖+[𝑈𝐺(𝑥)]₋+‖𝑉𝐺(𝑥)‖+𝐹(𝑥)^⊤𝐺(𝑥)₊^2/(1+𝛾),∀𝑥∈𝑅^𝑛. (47)

Proof. For any 𝑥∈𝑅^𝑛, let 𝜔=(𝜇,𝜈)=(𝐹(𝑥),𝐺(𝑥))∈𝑅^2𝑚. Then, there exists 𝜔^∗=(𝜇^∗,𝜈^∗)=(𝐹(𝑥^∗),𝐺(𝑥^∗))∈Ω^∗ such that dist(𝜔,Ω^∗)=‖𝜔−𝜔^∗‖. Letting 𝑥₀ be a nondegenerate solution of GNCP and letting 𝜔₀=(𝐹(𝑥₀), 𝐺(𝑥₀))∈Ω^∗, then dist^1+𝛾𝑥,𝑋^∗≤𝑥−𝑥^∗1+𝛾≤12𝑐₁𝛼^1+𝛾dist²𝜔,Ω^∗≤12𝑐₁𝛼^1+𝛾𝑐²₄×[𝐴(𝐼,0)𝜔]₋+[𝑈(0,𝐼)𝜔]₋+‖𝐵()𝜔‖+‖𝑉()𝜔‖+)𝜔]𝐼,00,𝐼[(𝐼,0^⊤(0,𝐼)𝜔₀+[(0,𝐼)𝜔]^⊤(𝐼,0)𝜔₀₊²≤12𝑐₁𝛼^1+𝛾𝑐²₄×[𝐴(𝐼,0)𝜔]₋+[𝑈(0,𝐼)𝜔]₋+‖𝐵()𝜔‖+‖𝑉()𝜔‖+{)𝜔]𝐼,00,𝐼[(𝐼,0^⊤[(0,𝐼)𝜔]₊²=12𝑐₁𝛼^1+𝛾𝑐²₄×𝐴𝜇₋+[𝑈𝜈]₋++‖‖+𝜇𝐵𝜇𝑉𝜈^⊤𝜈₊², (48) where the second equality uses the similar technique to that of (30), the third inequality follows from Corollary 17 and Lemma 10 with constant 𝑐₄>0, and the last inequality is based on (36). By (48) and letting 𝜌₂={(1/2𝑐₁𝛼^1+𝛾)𝑐²₄}^1/(1+𝛾), the desired result follows.

In the following, we give an error bound of the Hölderian type [14].

Lemma 19. For 𝑖=1,2,…,𝑚, let 𝑔_𝑖(𝑥) be a convex quadratic function. If the set 𝑆∶={𝑥∈𝑅^𝑛∣𝑔₁(𝑥)≤0,𝑔₂(𝑥)≤0,…,𝑔_𝑚(𝑥)≤0} is nonempty, then there exist a positive integer 𝑑≤𝑛+1 (called the degree of singularity of the inequality system) and a positive scalar 𝑐₅ such that dist(𝑥,𝑆)≤𝑐₅max𝑔(𝑥)₊,𝑔(𝑥)₊^1/2𝑑,∀𝑥∈𝑅^𝑛, (49) where [𝑔(𝑥)]₊=([𝑔₁(𝑥)]₊,[𝑔₂(𝑥)]₊,…,[𝑔_𝑚(𝑥)]₊). Furthermore, if 𝑆 contains an interior point, then 𝑑=0.

Based on (18) and (21), the GNCP can also be written as (𝑥)(𝑥)(𝐹(𝑥𝐴𝐹≥0,𝐵𝐹=0,))^⊤𝐺(𝑥)(𝑥)(𝑥)≤0,𝑈𝐺≥0,𝑉𝐺=0. (50)

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 𝑥∈𝑅^𝑛 such that 𝐹(𝑥)^⊤𝐺(𝑥)<0. (51) Then, there exists constant 𝜌₃>0 such that dist𝑥,𝑋^∗≤𝜌₃[𝐴𝐹(𝑥)]₋+‖(𝑥+(𝑥𝐵𝐹)‖[𝑈𝐺)]₋+‖(𝑥+𝐹(𝑥)𝑉𝐺)‖_⊤𝐺(𝑥)₊^2/(1+𝛾),∀𝑥∈𝑅^𝑛. (52)

Proof. Let 𝑆₁∶={𝜔∈𝑅^2𝑚∣𝑓(𝜔)≤0}, where 𝑓(𝜔)=[(𝐼,0)𝜔]^⊤[(0,𝐼)𝜔]. By Lemma 8, we have 𝑓(𝜔) is a convex quadratic function. Combining this with (51), using Lemma 19 with 𝑑=0, this yields the following result dist𝜔,𝑆₁≤𝑐₆𝑓(𝜔)₊,∀𝜔∈𝑅^2𝑚, (53) where 𝑐₆ is a positive constant.
Obviously, 𝑆₁ is a closed convex set. Thus, for any 𝜔∈𝑅^2𝑚, there exists a vector 𝜔∈𝑆₁ such that ‖𝜔−𝜔‖=dist𝜔,𝑆₁. (54)
For convenience, we also let Ψ(𝜔)=(()()()()()()𝜔)−𝐴𝐼,0𝜔,−𝐵𝐼,0𝜔,−𝑈0,𝐼𝜔,−𝑉0,𝐼𝜔,𝐵𝐼,0𝜔,𝑉0,𝐼₊. (55) From (50), we have Ω^∗=Ω⋂𝑆₁, where Ω is defined in (24), so for any 𝜔∈𝑆₁, combining Lemma 10, one has dist𝜔,Ω^∗≤𝑐₇(−𝐴(𝐼,0)𝜔)₊+(−𝑈(0,𝐼)𝜔)₊+‖𝐵(𝐼,0)𝜔‖+‖𝑉(0,𝐼)𝜔‖=𝑐₇(−𝐴(𝐼,0)𝜔)₊+(−𝑈(0,𝐼)𝜔)₊+(𝐵(𝐼,0)𝜔)₊+(−𝐵(𝐼,0)𝜔)₊+(𝑉(0,𝐼)𝜔)₊+(−𝑉(0,𝐼)𝜔)₊≤𝑐₇(−𝐴(𝐼,0)𝜔)₊₁+(−𝑈(0,𝐼)𝜔)₊₁+(𝐵(𝐼,0)𝜔)₊₁+(−𝐵(𝐼,0)𝜔)₊₁+(𝑉(0,𝐼)𝜔)₊₁+−(𝑉(0,𝐼)𝜔)₊₁=𝑐₇‖Ψ(𝜔)‖₁≤𝑐₇√2𝑠+2𝑡+2𝑚‖Ψ(𝜔)‖, (56) where 𝑐₇ is a positive constant, and the second and third inequalities follow from the fact that ‖𝑥‖≤‖𝑥‖₁≤√𝑛‖𝑥‖, for all 𝑥∈𝑅^𝑛.
Furthermore, ‖Ψ(𝜔)−Ψ(𝜔=()()()()()()𝜔))‖‖(−𝐴𝐼,0𝜔,−𝐵𝐼,0𝜔,−𝑈0,𝐼𝜔,−𝑉0,𝐼𝜔,𝐵𝐼,0𝜔,𝑉0,𝐼₊−(−𝐴(𝐼,0)𝜔,−𝐵(𝐼,0)𝜔,−𝑈(0,𝐼)()𝜔,−𝑉0,𝐼𝜔,𝐵(𝐼,0)𝜔,𝑉(0,𝐼)𝜔)₊=𝑃_𝑅+^{2𝑠+2𝑡+2𝑚}()()()()𝐵()()𝜔{(−𝐴𝐼,0𝜔,−𝐵𝐼,0𝜔,−𝑈0,𝐼𝜔,−𝑉0,𝐼𝜔,𝐼,0𝜔,𝑉0,𝐼)}−𝑃_𝑅+^{2𝑠+2𝑡+2𝑚}{(−𝐴(𝐼,0)𝜔,−𝐵(𝐼,0)()𝜔,−𝑈0,𝐼𝜔,−𝑉(0,𝐼)𝐵()𝜔,𝐼,0𝜔,𝑉(0,𝐼)𝜔≤()()()()()()𝜔−())}‖{(−𝐴𝐼,0𝜔,−𝐵𝐼,0𝜔,−𝑈0,𝐼𝜔,−𝑉0,𝐼𝜔,𝐵𝐼,0𝜔,𝑉0,𝐼)}{(−𝐴𝐼,0𝜔,−𝐵(𝐼,0)𝜔,−𝑈(0,𝐼)()𝜔,−𝑉0,𝐼𝜔,𝐵(𝐼,0)𝜔,𝑉(0,𝐼)𝜔≤‖𝐴()()}‖𝐼,0𝜔−𝐴𝐼,0𝜔‖+2‖𝐵(𝐼,0)𝜔−𝐵(𝐼,0)𝜔‖+‖𝑈(0,𝐼)𝜔−𝑈(0,𝐼)𝜔‖+2‖𝑉(0,𝐼)𝜔−𝑉(0,𝐼)𝜔‖≤𝐴(‖𝐵(+‖𝑈(‖𝑉(×‖(‖𝐼,0)‖+2𝐼,0)‖0,𝐼)‖+20,𝐼)‖)𝜔−𝜔‖=(‖𝐴(𝐼,0)‖+2‖𝐵(𝐼,0)‖+‖𝑈(0,𝐼)‖+2‖𝑉(0,𝐼)‖)×dist𝜔,𝑆₁, (57) where the second equality follows from the fact that min{𝑎,𝑏}=𝑎−𝑃_𝑅+(𝑎−𝑏),∀𝑎,𝑏∈𝑅, (58) and the first inequality is by nonexpanding property of projection operator. Thus, ‖Ψ(𝜔≤‖Ψ(𝜔+𝐴(‖𝐵(+‖𝑈(‖𝑉()‖)‖(‖𝐼,0)‖+2𝐼,0)‖0,𝐼)‖+20,𝐼)‖)dist𝜔,𝑆₁. (59) Combining (56) with (59), for any 𝜔∈𝑅^2𝑚, we have dist𝜔,Ω^∗≤dist𝜔,𝑆₁+dist𝜔,Ω^∗≤dist𝜔,𝑆₁+𝜎‖Ψ(𝜔)‖≤dist𝜔,𝑆₁‖Ψ(𝜔+𝐴(‖𝐵(+‖𝑈(‖𝑉(+𝜎)‖(‖𝐼,0)‖+2𝐼,0)‖0,𝐼)‖+20,𝐼)‖)dist𝜔,𝑆₁‖Ψ(𝜔+[𝜎𝐴(‖𝐵(+‖𝑈(‖𝑉(]≤𝜎)‖(‖𝐼,0)‖+2𝐼,0)‖0,𝐼)‖+20,𝐼)‖)+1×dist𝜔,𝑆₁‖Ψ(𝜔+[𝜎𝐴(‖𝐵(+‖𝑈(‖𝑉(]𝑐≤𝜎)‖(‖𝐼,0)‖+2𝐼,0)‖0,𝐼)‖+20,𝐼)‖)+1₆×𝑓(𝜔)₊≤𝜂‖Ψ(𝜔)‖+𝑓(𝜔)₊≤𝜂‖Ψ(𝜔)‖₁+𝑓(𝜔)₊≤𝜂(−𝐴(𝐼,0)𝜔)₊₁+(−𝑈(0,𝐼)𝜔)₊₁+‖𝐵(𝐼,0)𝜔‖₁+‖𝑉(0,𝐼)𝜔‖₁+𝑓(𝜔)₊≤𝜂√𝑠(−𝐴(𝐼,0)𝜔)₊+√𝑠(−𝑈(0,𝐼)𝜔)₊+√𝑡‖𝐵(𝐼,0)𝜔‖+√𝑚‖𝑉()𝜔‖+𝑓(𝜔0,𝐼)₊≤𝑐₈(−𝐴(𝐼,0)𝜔)₊+(−𝑈(0,𝐼)𝜔)₊+‖𝐵()𝜔‖+‖𝑉()𝜔‖+𝑓(𝜔𝐼,00,𝐼)₊, (60) where the second inequality follows from (56) with constant 𝜎=𝑐₇√2𝑠+2𝑡+2𝑚, the third inequality uses (59), the fifth inequality follows from (53), the sixth inequality follows from the fact that {[𝜎𝐴(‖𝐵(+‖𝑈(‖𝑉(]𝑐𝜂=max𝜎,(‖𝐼,0)‖+2𝐼,0)‖0,𝐼)‖+20,𝐼)‖)+1₆, (61) the seventh and ninth inequalities follow from the fact that ‖𝑥‖≤‖𝑥‖₁≤√𝑛‖𝑥‖,∀𝑥∈𝑅^𝑛, (62) and the last inequality follows by letting 𝑐₈=𝜂max{√𝑠,√𝑡,√𝑚,1}.
For any 𝑥∈𝑅^𝑛, letting 𝜔∶=(𝜇,𝜈)=(𝐹(𝑥),𝐺(𝑥))∈𝑅^2𝑚, then there exists 𝜔^∗=(𝜇^∗,𝜈^∗)=(𝐹(𝑥^∗),𝐺(𝑥^∗))∈Ω^∗ such that dist(𝜔,Ω^∗)=‖𝜔−𝜔^∗‖, and a direct computation yields that dist^1+𝛾𝑥,𝑋^∗≤𝑥−𝑥^∗1+𝛾≤12𝑐₁𝛼^1+𝛾dist²𝜔,Ω^∗≤12𝑐₁𝛼^1+𝛾𝑐²₈×[𝐴(𝐼,0)𝜔]₋+[𝑈(0,𝐼)𝜔]₋+‖𝐵()𝜔‖+‖𝑉()𝜔‖+)𝜔]𝐼,00,𝐼[(𝐼,0^⊤[(0,𝐼)𝜔]₊²=12𝑐₁𝛼^1+𝛾𝑐²₈𝐴𝜇₋+[𝑈𝜈]₋++‖‖+𝜇𝐵𝜇𝑉𝜈^⊤𝜈₊², (63) 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 𝜌₃={(1/2𝑐₁𝛼^1+𝛾)𝑐²₈}^1/(1+𝛾), then the desired result follows.

Remark 21. When 𝐹 is strongly monotone with respect to 𝐺, that is, 𝛾=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].

3. Global Error Bound for the GLCP

In this section, we consider the linear case of the GCP such that mappings 𝐹 and 𝐺 are both linear; that is, 𝐹(𝑥)=𝑀𝑥+𝑝, 𝐺(𝑥)=𝑁𝑥+𝑞 with 𝑀,𝑁∈𝑅^𝑚×𝑛, 𝑝,𝑞∈𝑅^𝑚: min𝐻(𝑥)=𝑀𝑥+𝑝^⊤𝑁𝑥+𝑞s.t.𝑥∈𝑋, (64) where 𝑋=𝑥∈𝑅^𝑛∣𝐴𝐵𝑀𝑥+𝑝≥0,𝑈𝑁𝑥+𝑞≥0,𝑀𝑥+𝑝=0,𝑉𝑁𝑥+𝑞=0. (65)

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), 𝐻(𝑥) is a convex function.

Lemma 23. 𝑥^∗∈𝑅^𝑛 is a solution of the GLCP if and only if 𝑥^∗ 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 𝐹 and 𝐺 are both linear, there exists constant 𝜌₄>0 such that dist𝑥,𝑋^∗≤𝜌₄[𝐴𝐹(𝑥)]₋+‖𝐵𝐹(𝑥)‖+[𝑈𝐺(𝑥)]₋+‖𝑉𝐺(𝑥)‖+𝐹(𝑥)^⊤𝐺(𝑥)₊+𝐹(𝑥)^⊤𝐺(𝑥)₊^1/2,∀𝑥∈𝑅^𝑛. (66)

Proof. For any 𝑥∈𝑅^𝑛, a direct computation yields that dist𝑥,𝑋^∗≤𝑐₉maxdist(𝑥,𝑋),[𝐻(𝑥)]₊,[𝐻(𝑥)]₊^1/2≤𝑐₉max𝑐₁₀[𝐴𝐹(𝑥)]₋+‖(𝑥+(𝑥𝐵𝐹)‖[𝑈𝐺)]₋+‖(𝑥,𝐻(𝑥𝑉𝐺)‖[)]₊,[𝐻(𝑥)]₊^1/2≤𝑐₉max𝑐₁₀,1[𝐴𝐹(𝑥)]₋+‖(𝑥+(𝑥𝐵𝐹)‖[𝑈𝐺)]₋+‖𝑉𝐺(𝑥)‖+(𝐹(𝑥))^⊤𝐺(𝑥)₊+(𝐹(𝑥))^⊤𝐺(𝑥)₊^1/2, (67) where the first inequality follows from Lemma 11 with constant 𝑐₉>0 and Lemma 23, and the second inequality uses Lemma 10 with constant 𝑐₁₀>0. By (67) and letting 𝜌₄=𝑐₉max{𝑐₁₀,1}, 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 𝜌₅>0 such that dist𝑥,𝑋^∗≤𝜌₅[𝐴𝐹(𝑥)]₋+‖𝐵𝐹(𝑥)‖+[𝑈𝐺(𝑥)]₋+‖𝑉𝐺(𝑥)‖+𝐹(𝑥)^⊤𝐺(𝑥)₊,∀𝑥∈𝑅^𝑛. (68)

Proof. From Corollary 17, we have 𝑋^∗=𝑥∈𝑋∣𝑀𝑥+𝑝^⊤𝑁𝑥₀++𝑞𝑁𝑥+𝑞^⊤𝑀𝑥₀+𝑝≤0, (69) where 𝑥₀ is a nondegenerate solution of GLCP, and 𝑋 is defined in (64). For any 𝑥∈𝑅^𝑛, a direct computation yields that dist𝑥,𝑋^∗≤𝑐₁₁𝐴𝑀𝑥+𝑝₋+𝑈𝑁𝑥+𝑞₋+𝐵+𝑉+𝑀𝑥+𝑝𝑁𝑥+𝑞𝑀𝑥+𝑝^⊤𝑁𝑥₀++𝑞𝑁𝑥+𝑞^⊤𝑀𝑥₀+𝑝₊≤𝑐₁₁𝐴𝑀𝑥+𝑝₋+𝑈𝑁𝑥+𝑞₋+𝐵+𝑉+𝑀𝑥+𝑝𝑁𝑥+𝑞𝑀𝑥+𝑝^⊤𝑁𝑥+𝑞₊, (70) where the first inequality follows from Lemma 10 with constant 𝑐₁₁>0, and the second inequality uses (36). Letting 𝜌₅=𝑐₁₁, 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 𝑥∈𝑅^𝑛 such that (51) holds. Then there exists constant 𝜌₆>0 such that dist𝑥,𝑋^∗≤𝜌₆[𝐴𝐹(𝑥)]₋+‖𝐵𝐹(𝑥)‖+[𝑈𝐺(𝑥)]₋+‖𝑉𝐺(𝑥)‖+𝐹(𝑥)^⊤𝐺(𝑥)₊,∀𝑥∈𝑅^𝑛. (71)

Proof. Let 𝑆₂∶={𝑥∈𝑅^𝑛∣𝐻(𝑥)≤0}, where 𝐻(𝑥)=(𝑀𝑥+𝑝)^⊤(𝑁𝑥+𝑞). By Lemma 22, 𝐻(𝑥) is a convex quadratic function, and 𝑆₂ is a closed convex set. For any 𝑥∈𝑅^𝑛, there exists a vector 𝑥∈𝑆₂ such that ‖𝑥−𝑥‖=dist𝑥,𝑆₂. (72) Combining (51) and applying Lemma 19 yield the following result: dist𝑥,𝑆₂≤𝑐₁₂[𝐻(𝑥)]₊,∀𝑥∈𝑅^𝑛, (73) where 𝑐₁₂ is a positive constant. For convenience, we let 𝜑(𝑥)=((𝑥)(𝑥)(𝑥)(𝑥)(𝑥)(𝑥−𝐴𝐹,−𝐵𝐹,−𝑈𝐺−𝑉𝐺,𝐵𝐹,𝑉𝐺))₊. (74) From (50), we have 𝑋^∗=𝑋⋂𝑆₂, where 𝑋 is defined in (64). So for any 𝑥∈𝑆₂, combining Lemma 10 and using the similar technique to that of (56), one has dist𝑥,𝑋^∗≤𝑐₁₃(−𝐴𝐹(𝑥))₊+(−𝑈𝐺(𝑥))₊+‖𝐵𝐹(𝑥)‖+‖𝑉𝐺(𝑥)‖≤𝑐₁₃√2𝑠+2𝑡+2𝑚𝜑(𝑥), (75) where 𝑐₁₃ is a positive constant.
Using the fact that min{𝑎,𝑏}=𝑎−𝑃_𝑅+(𝑎−𝑏),∀𝑎,𝑏∈𝑅, (76) and using the similar technique to that of (57), one has 𝜑(𝑥)−𝜑(𝑥=(𝑥)(𝑥)(𝑥)(𝑥),(𝑥)(𝑥)(−𝐴𝐹,−𝐵𝐹,−𝑈𝐺,−𝑉𝐺𝐵𝐹,𝑉𝐺))₊−(−𝐴𝐹(𝑥),−𝐵𝐹(𝑥),−𝑈𝐺(𝑥),−𝑉𝐺(𝑥),𝐵𝐹(𝑥),𝑉𝐺(𝑥))₊≤‖𝐴𝐹(𝑥)−𝐴𝐹(𝑥)‖+2‖𝐵𝐹(𝑥)−𝐵𝐹(𝑥+‖(𝑥)()‖𝑈𝐺−𝑈𝐺𝑥)‖+2‖𝑉𝐺(𝑥)−𝑉𝐺(𝑥≤‖‖‖+‖‖‖)‖(‖𝐴𝑀+2𝐵𝑀𝑈𝑁+2𝑉𝑁‖)‖𝑥−𝑥‖=‖‖‖+‖‖‖(‖𝐴𝑀+2𝐵𝑀𝑈𝑁+2𝑉𝑁‖)dist𝑥,𝑆₂, (77) where the second inequality is by nonexpanding property of projection operator. Thus, 𝜑(𝑥≤𝜑(𝑥+‖‖‖+‖‖‖))(‖𝐴𝑀+2𝐵𝑀𝑈𝑁+2𝑉𝑁‖)dist𝑥,𝑆₂. (78) Combining (75) with (78), we know that for any 𝑥∈𝑅^𝑛, it holds that dist𝑥,𝑋^∗≤dist𝑥,𝑆₂+dist𝑥,𝑋^∗≤dist𝑥,𝑆₂+𝜎₁𝜑(𝑥)≤dist𝑥,𝑆₂+𝜎₁𝜑(𝑥+‖‖‖+‖‖‖)(‖𝐴𝑀+2𝐵𝑀𝑈𝑁+2𝑉𝑁‖)dist𝑥,𝑆₂≤𝜎₁𝜑(𝑥+𝜎₁)‖‖‖+‖‖‖](‖𝐴𝑀+2𝐵𝑀𝑈𝑁+2𝑉𝑁‖)+1dist𝑥,𝑆₂≤𝜎₁𝜑(𝑥+𝜎₁)‖‖‖+‖‖‖]𝑐(‖𝐴𝑀+2𝐵𝑀𝑈𝑁+2𝑉𝑁‖)+1₁₂[𝐻(𝑥)]₊≤𝜂₁𝜑(𝑥)+[𝐻(𝑥)]₊≤𝜂₁𝜑(𝑥)₁+[𝐻(𝑥)]₊≤𝜂₁(−𝐴𝐹(𝑥))₊₁+(−𝑈𝐺(𝑥))₊₁+‖𝐵𝐹(𝑥)‖₁+‖𝑉𝐺(𝑥)‖₁+[𝐻(𝑥)]₊≤𝜂₁√𝑠(−𝐴𝐹(𝑥))₊+√𝑠(−𝑈𝐺(𝑥))₊+√𝑡‖𝐵𝐹(𝑥)‖+√𝑚‖𝑉𝐺(𝑥)‖+[𝐻(𝑥)]₊≤𝜌₆(−𝐴𝐹(𝑥))₊+(−𝑈𝐺(𝑥))₊+‖(𝑥+‖(𝑥+𝐻(𝑥𝐵𝐹)‖𝑉𝐺)‖[)]₊, (79) where the second inequalities follows from (75) with constant 𝜎₁=𝑐₁₃√2𝑠+2𝑡+2𝑚, the third inequality follows from (78), the fifth inequality follows from (73), the sixth inequality follows by letting 𝜂₁=max{𝜎₁,[𝜎₁(‖𝐴𝑀‖+2‖𝐵𝑀‖+‖𝑈𝑁‖+2‖𝑉𝑁‖)+1]𝑐₁₂}, and the seventh and ninth inequality follow from the fact that ‖𝑥‖≤‖𝑥‖₁≤√𝑛‖𝑥‖,∀𝑥∈𝑅^𝑛. (80) By (79) and letting 𝜌₆=𝜂₁max{√𝑠,√𝑡,√𝑚,1}, the desired result follows.

Remark 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].

4. Comparison with Existing Error Bound

In the end of this paper, we will present an example to compare Theorem 13 and Theorem 2.5 in [5]. Furthermore, we will present two examples to show the conclusion in Theorem 13 can provide a global error bound for the GNCP, while the conclusion in Theorem 2.5 in [5] cannot do.

Example 30. When 𝒦=𝑅^𝑚₊, (2) reduces to the generalized nonlinear complementarity problem of finding vector 𝑥^∗∈𝑅^𝑛 such that 𝐹𝑥^∗≥0,𝐺𝑥^∗≥0,𝐹𝑥^∗⊤𝐺𝑥^∗=0. (81)

For (81), using Theorem 13 with 𝛾=1, we have dist𝑥,𝑋^∗≤𝜌𝜑(𝑥),∀𝑥∈𝑅^𝑛, (82) where 𝜑(𝑥)=∶‖[𝐹(𝑥)]₋‖+‖[𝐺(𝑥)]₋‖+|[𝐹(𝑥)^⊤𝐺(𝑥)]₊|+|[𝐹(𝑥)^⊤𝐺(𝑥)]₊|^1/2.

Using Theorem 2.5 in [5] with 𝛾=1, 𝑣₁=𝑣₂=1, and 𝛽=1, we have that there exists constant 𝜌>0 such that dist𝑥,𝑋^∗≤𝜌𝑟(𝑥), (83) where 𝑟(𝑥)=∶‖min{𝐹(𝑥),𝐺(𝑥)}‖. In addition, 𝜑(𝑥)≤2√𝑚+‖max{𝐹(𝑥),𝐺(𝑥)}‖𝑟(𝑥)+𝑟(𝑥)^1/2. (84) In particular, when ‖𝑥‖≤𝑐₁₄ with constant 𝑐₁₄>0, then there exists positive constant 𝑐₁₅ such that 𝜑(𝑥)≤𝑐₁₅𝑟(𝑥)+𝑟(𝑥)^1/2. (85)

In fact, we have 𝜑(𝑥)≤(𝐹(𝑥))₋₁+(𝐺(𝑥))₋₁+(𝐹(𝑥))^⊤𝐺(𝑥)₊+(𝐹(𝑥))^⊤𝐺(𝑥)₊^1/2≤2√𝑚‖{𝐹(𝑥)(𝑥+𝐹(𝑥min,𝐺)}‖())^⊤𝐺(𝑥)₊+(𝐹(𝑥))^⊤𝐺(𝑥)₊^1/2≤2√𝑚‖{𝐹(𝑥)(𝑥+min,𝐺)}‖_𝑚𝑖=1(𝐹(𝑥))_𝑖(𝐺(𝑥))_𝑖++^𝑚_𝑖=1(𝐹(𝑥))_𝑖(𝐺(𝑥))_𝑖+^1/2≤2√𝑚‖{𝐹(𝑥)(𝑥+min,𝐺)}‖_𝑚𝑖=1min{𝐹(𝑥),𝐺(𝑥)}_𝑖⋅max{𝐹(𝑥),𝐺(𝑥)}_𝑖+^𝑚_𝑖=1min{𝐹(𝑥),𝐺(𝑥)}_𝑖⋅max{𝐹(𝑥),𝐺(𝑥)}_𝑖^1/2≤2√𝑚‖{𝐹(𝑥)(𝑥+‖{𝐹(𝑥)(𝑥⋅‖{𝐹(𝑥)(𝑥+{𝐹(𝑥)(𝑥⋅‖{𝐹(𝑥)(𝑥min,𝐺)}‖min,𝐺)}‖max,𝐺)}‖{‖min,𝐺)}‖max,𝐺)}‖}^1/2=2√‖{𝐹(𝑥)(𝑥𝑟(𝑥)+‖{𝐹(𝑥)(𝑥𝑚+max,𝐺)}‖max,𝐺)}‖^1/2𝑟(𝑥)^1/2≤2√𝑚+‖max{𝐹(𝑥),𝐺(𝑥)}‖𝑟(𝑥)+𝑟(𝑥)^1/2, (86) where the first inequality follows from the fact that ‖𝑥‖≤‖𝑥‖₁, for all 𝑥∈𝑅^𝑚, the second inequality follows from the fact that 𝑎₋≤|min{𝑎,𝑏}|, for all 𝑎,𝑏∈𝑅, the third inequality follows from the fact that (𝑎+𝑏)₊≤𝑎₊+𝑏₊, for all 𝑎,𝑏∈𝑅, the fourth inequality follows from the fact that (𝑎𝑏)₊≤|min{𝑎,𝑏}|⋅|max{𝑎,𝑏}|, for all 𝑎,𝑏∈𝑅, and the fifth inequality follows from the Cauchy-Schwarz inequality.

Example 31. For mappings 𝐹,𝐺∶𝑅₊→𝑅 involved in problem (81), we set 𝐹(𝑥)=(𝑥+1)²,𝐺(𝑥)=√𝑥. (87)

It is easy to see that the solution set 𝑋^∗={0}, and one has 𝐹(𝑥)−𝐹𝑦,𝐺(𝑥)−𝐺𝑦=𝑥+𝑦+2√𝑥+√𝑦𝑥−𝑦²≥𝑥−𝑦², (88) where the first inequality follows from the fact that 𝑥+𝑦+2≥√𝑥+√𝑦.

In fact, we consider the following four cases.

Case 1 (𝑥≥1 and 𝑦≥1). Then, 𝑥≥√𝑥 and 𝑦≥√𝑦, and the desired result follows.

Case 2 (0≤𝑥≤1 and 0≤𝑦≤1). Then, √𝑥≤1 and √𝑦≤1, and the desired result follows.

Case 3 (0≤𝑥≤1 and 𝑦≥1). Then, √𝑥≤1 and √𝑦≤𝑦, and the desired result follows.

Case 4 (𝑥≥1 and 0≤𝑦≤1). Then, √𝑥≤𝑥 and √𝑦≤1, and the desired result follows.

For any 𝑥(𝜖)∶=𝜖,𝜖≥0. By Theorem 13 with 𝛾=1, we can obtain ‖𝑥(𝜖)−0‖𝜑(𝑥(𝜖=𝜖))(𝜖+1)²√𝜖+(𝜖+1)²√𝜖⟶0 (89) as 𝜖→+∞. Thus, Theorem 13 provides a global error bound for the GNCP. Using Theorem 2.5 in [5], for 𝑥(𝜖), we have ‖𝑥(𝜖)−0‖𝑟(𝑥(𝜖=𝜖))min(𝜖+1)²,√𝜖=𝜖√𝜖=√𝜖⟶+∞ (90) as 𝜖→+∞. Thus, Theorem 2.5 in [5] fails in providing an error bound for this GNCP.

Example 32. For mappings 𝐹,𝐺∶𝑅→𝑅 involved in problem (81), we set 𝐹(𝑥)=13𝑥³+𝑥,𝐺(𝑥)=𝑥. (91)

It is easy to see that the solution set 𝑋^∗={0}. Without loss of generality, we let 𝑥>𝑦, and one has 𝐹(𝑥)−𝐹𝑦≥𝑥−𝑦², (92) where the inequality follows from the fact that 13𝑥³−1+𝑥3𝑦³+𝑦−𝑥−𝑦²=1𝑥−𝑦3𝑥²+13𝑦²+13𝑥𝑦+1+𝑥−𝑦≥0. (93) In fact, we consider the following four cases.

Case 1 (𝑥>𝑦≥0). Then, (1/3)𝑥²+(1/3)𝑦²+(1/3)𝑥𝑦+1+(𝑥−𝑦)≥0, and the desired result follows.

Case 2 (0≥𝑥>𝑦). Then, (1/3)𝑥²+(1/3)𝑦²+(1/3)𝑥𝑦+1+(𝑥−𝑦)≥0, and the desired result follows.

Case 3 (𝑥≥0,𝑦<0 and 𝑥+𝑦≥0). Then, 13𝑥²+13𝑦²+13=1𝑥𝑦+1+𝑥−𝑦3𝑥+1𝑥+𝑦3𝑦²+1+𝑥−𝑦≥0, (94) and the desired result follows.

Case 4 (𝑥≥0,𝑦<0, and 𝑥+𝑦≤0). Then, 13𝑥²+13𝑦²+13=1𝑥𝑦+1+𝑥−𝑦3𝑥²+13𝑦𝑥+𝑦+1+𝑥−𝑦≥0, (95) and the desired result follows.

Thus, we obtain 𝐹(𝑥)−𝐹𝑦,𝐺(𝑥)−𝐺𝑦≥𝑥−𝑦³. (96)

For any 𝑥(𝜖)∶=𝜖, 𝜖≥0. By Theorem 13 with 𝛾=2, we can obtain ‖𝑥(𝜖)−0‖𝜑(𝑥(𝜖=𝜖))(1/3)𝜖³+𝜖𝜖+(1/3)𝜖³+𝜖𝜖^2/3⟶0 (97) as 𝜖→+∞. Thus, Theorem 13 provides a global error bound for the GNCP.

On the other hand, using Theorem 2.5 in [5], for 𝑥(𝜖), we have ‖𝑥(𝜖)−0‖𝑟(𝑥(𝜖))^𝛿=𝜖min(1/3)𝜖³+𝜖,𝜖^𝛿=𝜖𝜖^𝛿⟶+∞ (98) as 𝜖→+∞, where 𝛿 is a constant with 1/3<𝛿≤1/2. Thus, Theorem 2.5 in [5] fails in providing an error bound for this GNCP.

5. 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, 9–11] 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.

Acknowledgments

The authors wish to give their sincere thanks to the associated editor and two anonymous referees for their valuable suggestions and helpful comments which improve the presentation of the paper. This work was supported by the Natural Science Foundation of China (nos. 11171180 and 11101303), the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002), the Shandong Provincial Natural Science Foundation (ZR2010AL005), the Shandong Province Science and Technology Development Projects (2013GGA13034), and the Applied Mathematics Enhancement Program of Linyi University.