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.

1. Introduction

Let mappings 𝐹,𝐺∶𝑅^𝑛→𝑅^𝑚, 𝐻∶𝑅^𝑛→𝑅^𝑙, and the generalized nonlinear complementarity problem, abbreviated as GNCP, is to find vector 𝑥^∗∈𝑅^𝑛 such that 𝐹𝑥^∗∈𝒦,𝐺𝑥^∗∈𝒦⁰,𝐹𝑥^∗𝖳𝐺𝑥^∗=0,𝐻𝑥^∗=0, (1.1) where 𝒦 is a nonempty closed convex cone in 𝑅^𝑚 and 𝒦^∘ is its dual cone, that is, 𝒦^∘={𝑢∈𝑅^𝑚∣𝑢^𝖳𝑣≥0,forall𝑣∈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. 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 𝑅^𝑛 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 ⟨⋅,⋅⟩.

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^∗𝖳𝐹(𝑥)−𝐹𝑥^∗(𝑥)𝐻𝑥≥0,∀𝐹∈𝒦,_∗=0. (2.1)

Proof. Suppose that 𝑥^∗ is a solution of (2.1). Since vector 0∈𝒦, by substituting 𝐹(𝑥)=0 into (2.1), we have 𝐺(𝑥^∗)^𝖳𝐹(𝑥^∗)≤0. On the other hand, since 𝐹(𝑥^∗)∈𝒦, then 2𝐹(𝑥^∗)∈𝒦. By substituting 𝐹(𝑥)=2𝐹(𝑥^∗) into (2.1), we obtain 𝐺(𝑥^∗)^𝖳𝐹(𝑥^∗)≥0. Consequently, 𝐺(𝑥^∗)^𝖳𝐹(𝑥^∗)=0. For any 𝐹(𝑥)∈𝒦, we have 𝐺(𝑥^∗)^𝖳𝐹(𝑥)=𝐺(𝑥^∗)^𝖳[𝐹(𝑥)−𝐹(𝑥^∗)]≥0, that is, 𝐺(𝑥^∗)∈𝒦^∘. 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^∗)∈𝒦^∘, for any 𝐹(𝑥)∈𝒦, we have 𝐺(𝑥^∗)^𝖳𝐹(𝑥)≥0, and from 𝐺(𝑥^∗)^𝖳𝐹(𝑥^∗)=0, we have 𝐺(𝑥^∗)^𝖳[𝐹(𝑥)−𝐹(𝑥^∗)]≥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 𝒦⊂𝑅^𝑚 and any vector 𝑥∈𝑅^𝑚, the orthogonal projection of 𝑥 onto 𝒦, that is, argmin{‖𝑦−𝑥‖∣𝑦∈𝒦}, is denoted by 𝑃_𝒦(𝑥).

Lemma 2.2. For any 𝑢∈𝑅^𝑚,𝑣∈𝒦, then (i)⟨𝑃_𝒦(𝑢)−𝑢,𝑣−𝑃_𝒦(𝑢)⟩≥0, (ii)‖P_K(u)−P_K(v)‖≤‖u−v‖.

For (2.1), 𝛽>0 is a constant, 𝑒(𝑥)∶=𝐹(𝑥)−𝑃_𝒦[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∶𝑅^𝑛→𝑅^𝑚 is said to be(1)𝛾-strongly monotone with respect to 𝐺∶𝑅^𝑛→𝑅^𝑚 if there are constants 𝜇>0,𝛾>1 such that 𝐹(𝑥)−𝐹𝑦,𝐺(𝑥)−𝐺𝑦≥𝜇𝑥−𝑦^1+𝛾,∀𝑥,𝑦∈𝑅^𝑛; (2.2)(2)Hölder-continuous if there are constants 𝐿>0,𝑣∈(0,1] such that 𝐹(𝑥)−𝐹𝑦≤𝐿𝑥−𝑦^𝑣,∀𝑥,𝑦∈𝑅^𝑛. (2.3)
In this following, based on Lemmas 2.2 and 2.3, we establish error bound for GNCP in the set Ω∶={𝑥∈𝑅^𝑛∣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 𝐻𝑜𝑙̈𝑑𝑒𝑟 continuous with positive constants 𝐿₁>0,𝐿₂>0,𝑣₁,𝑣₂∈(0,1], respectively, and 𝛽𝜇≤(𝐿₁+𝐿₂𝛽)(2𝐿₁+𝐿₂𝛽) holds. Then for any 𝑥∈Ω∶={𝑥∈𝑅^𝑛∣𝐻(𝑥)=0}, there exists a solution 𝑥^∗ of (1.1) such that 𝑟(𝑥)2𝐿₁+𝐿₂𝛽^{1/min{𝑣1,𝑣2}}≤𝑥−𝑥^∗≤𝐿₁+𝐿₂𝛽𝑟(𝑥𝛽𝜇)^{1/(1+𝛾−min{𝑣1,𝑣2})},if𝑟(𝑥)≤𝛽𝜇𝐿₁+𝐿₂𝛽.𝑟(𝑥)2𝐿₁+𝐿₂𝛽^{1/max{𝑣1,𝑣2}}≤𝑥−𝑥^∗≤𝐿₁+𝐿₂𝛽𝑟(𝑥)𝛽𝜇^{1/(1+𝛾−max{𝑣1,𝑣2})},if𝑟(𝑥)≥2𝐿₁+𝐿₂𝑟(𝑥)𝛽.2𝐿₁+𝐿₂𝛽^{1/min{𝑣1,𝑣2}}≤𝑥−𝑥^∗≤𝐿₁+𝐿₂𝛽𝑟(𝑥)𝛽𝜇^{1/(1+𝛾−max{𝑣1,𝑣2})},if𝛽𝜇𝐿₁+𝐿₂𝛽<𝑟(𝑥)<2𝐿₁+𝐿₂𝛽. (2.4)(2.5)(2.6)

Proof. Since 𝐹(𝑥)−𝑒(𝑥)=𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)∈𝒦, (2.7) by the first inequality of (2.1), 𝐹(𝑥)−𝑒(𝑥)−𝐹𝑥^∗𝖳𝛽𝐺𝑥^∗≥0. (2.8) Combining 𝐹(𝑥^∗)∈𝒦 with Lemma 2.2(i), we have 𝐹𝑥^∗−𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥),𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)−𝐹(𝑥)−𝛽𝐺(𝑥)≥0. (2.9) Substituting 𝑃_𝒦[𝐹(𝑥)−𝛽𝐺(𝑥)] in (2.9) by 𝐹(𝑥)−𝑒(𝑥) leads to that 𝐹(𝑥)−𝐹𝑥^∗−𝑒(𝑥)^𝖳𝑒(𝑥)−𝛽𝐺(𝑥)≥0. (2.10) Using (2.8) and (2.10), we obtain 𝐹(𝑥)−𝐹𝑥^∗−𝑒(𝑥)^𝖳𝑒(𝑥)+𝛽𝐺𝑥^∗−𝐺(𝑥)≥0, (2.11) that is, 𝛽𝐹(𝑥)−𝐹𝑥^∗𝖳𝐺𝑥^∗−𝐺(𝑥)+𝑒(𝑥)^𝖳𝐹(𝑥)−𝐹𝑥^∗−𝛽𝐺𝑥^∗−𝐺(𝑥)−𝑒(𝑥)^𝖳𝑒(𝑥)≥0. (2.12) Base on Definition 2.4, a direct computation yields that 𝑥−𝑥^∗1+𝛾≤1𝜇𝐹(𝑥)−𝐹𝑥^∗𝖳𝐺(𝑥)−𝐺𝑥^∗≤1𝑒(𝑥)𝛽𝜇^𝖳𝐹(𝑥)−𝐹𝑥^∗+𝛽𝐺(𝑥)−𝐺𝑥^∗−𝑒(𝑥)^𝖳𝑒(𝑥≤1)𝑒(𝑥𝐹(𝑥)𝑥𝛽𝜇‖)‖−𝐹^∗+𝛽𝐺(𝑥)−𝐺𝑥^∗≤1𝑟(𝑥𝐿𝛽𝜇)₁𝑥−𝑥^∗𝑣1+𝛽𝐿₂𝑥−𝑥^∗𝑣2. (2.13) Combining this, we have ‖𝑥−𝑥^∗𝐿‖≤¹+𝐿₂𝛽𝑟(𝑥𝛽𝜇)^{1/(1+𝛾−min{𝑣1,𝑣2})},if𝑥−𝑥^∗𝐿≤1,₁+𝐿₂𝛽𝑟(𝑥)𝛽𝜇^{1/(1+𝛾−max{𝑣1,𝑣2})},if𝑥−𝑥^∗≥1. (2.14)
On the other hand, for the first inequality of (2.1), by Lemmas 2.3 and 2.2(ii), we have 𝑟(𝑥)=𝑒(𝑥)−𝑒𝑥^∗=𝐹(𝑥)−𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)−𝐹𝑥^∗+𝑃_𝒦𝐹𝑥^∗−𝛽𝐺𝑥^∗=𝐹(𝑥)−𝐹𝑥^∗+𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)−𝑃_𝒦𝐹𝑥^∗−𝛽𝐺𝑥^∗≤𝐹(𝑥)−𝐹𝑥^∗+𝐹(𝑥)−𝛽𝐺(𝑥)−𝐹𝑥^∗−𝛽𝐺𝑥^∗=2𝐹(𝑥)−𝐹𝑥^∗+𝛽𝐺(𝑥)−𝐺𝑥^∗≤2𝐿₁𝑥−𝑥^∗𝑣1+𝐿₂𝛽𝑥−𝑥^∗𝑣2. (2.15) Thus, 𝑥−𝑥^∗≥𝑟(𝑥)2𝐿₁+𝐿₂𝛽^{1/min{𝑣1,𝑣2}},if𝑥−𝑥^∗𝑟(𝑥)≤1,2𝐿₁+𝐿₂𝛽^{1/max{𝑣1,𝑣2}},if𝑥−𝑥^∗≥1. (2.16) Combining (2.14) with (2.16) for any 𝑥∈𝑅^𝑛, if ‖𝑥−𝑥^∗‖≤1, then 𝑟(𝑥)2𝐿₁+𝐿₂𝛽^{1/min{𝑣1,𝑣2}}≤𝑥−𝑥^∗𝐿≤min¹+𝐿₂𝛽𝑟(𝑥𝛽𝜇)^{1/(1+𝛾−min{𝑣1,𝑣2})},1. (2.17)
For any 𝑥∈𝑅^𝑛, if ‖𝑥−𝑥^∗‖≥1, then 𝑟(𝑥)max2𝐿₁+𝐿₂𝛽^{1/max{𝑣1,𝑣2}},1≤𝑥−𝑥^∗≤𝐿₁+𝐿₂𝛽𝑟(𝑥𝛽𝜇)^{1/(1+𝛾−max{𝑣1,𝑣2})}. (2.18)
If 𝑟(𝑥)≤𝛽𝜇/(𝐿₁+𝐿₂𝛽), then ((𝐿₁+𝐿₂𝛽)/𝛽𝜇)𝑟(𝑥)≤1, by (2.17), we have ‖𝑥−𝑥^∗‖≤1, and using (2.17) again, we obtain that (2.4) holds.
If 𝑟(𝑥)≥𝐿₂𝛽+2𝐿₁, then 𝑟(𝑥)/(2𝐿₁+𝐿₂𝛽)≥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+𝐿₂𝛽)<𝑟(𝑥)<2𝐿₁+𝐿₂𝛽, then 𝐿₁+𝐿₂𝛽𝑟(𝑥)𝑟(𝑥)𝛽𝜇>1,𝐿₂𝛽+2𝐿₁<1. (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 𝑅^𝑛 if there are positive constants 𝜎>0,𝛼>1 such that 𝑥−𝑦,𝐻(𝑥)−𝐻𝑦≥𝜎𝑥−𝑦^1+𝛼,∀𝑥,𝑦∈𝑅^𝑛. (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∈𝑅^𝑛∣𝐻(𝑥)=0} is convex. Then, there exists a constant 𝜌>0, such that, for any 𝑥∈𝑅^𝑛, there exists 𝑥^∗∈𝑋^∗ such that 𝑥−𝑥^∗≤𝜌‖𝐻(𝑥)‖^1/𝛼+𝑅(𝑥),∀𝑥∈𝑅^𝑛, (2.21) where 1𝜌=max𝜎,𝐿max¹+𝐿₂𝛽,𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽𝜎𝛽𝜇^{−min{𝑣1,𝑣2}1/(1+𝛾−min{𝑣1,𝑣2})},𝐿max¹+𝐿₂𝛽,𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽𝜎𝛽𝜇^{−max{𝑣1,𝑣2}1/(1+𝛾−min{𝑣1,𝑣2})},𝐿max¹+𝐿₂𝛽,𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽𝜎𝛽𝜇^{−min{𝑣1,𝑣2}1/(1+𝛾−max{𝑣1,𝑣2})},𝐿max¹+𝐿₂𝛽,𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽𝜎𝛽𝜇^{−max{𝑣1,𝑣2}1/(1+𝛾−max{𝑣1,𝑣2})},𝑅(𝑥)‖‖,=max𝑟(𝑥)+𝐻(𝑥)^{min{𝑣1,𝑣2}/𝛼1/(1+𝛾−min{𝑣1,𝑣2})}𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}/𝛼1/(1+𝛾−min{𝑣1,𝑣2})}𝑟(𝑥)+‖𝐻(𝑥)‖^{min{𝑣1,𝑣2}/𝛼1/(1+𝛾−max{𝑣1,𝑣2})}𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}/𝛼1/(1+𝛾−max{𝑣1,𝑣2})}. (2.22)(2.23)

Proof. For given 𝑥∈𝑅^𝑛, we only need to first project 𝑥 to Ω∶={𝑥∈𝑅^𝑛∣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 ‖𝑥−𝑥‖^1+𝛼≤1𝜎⟨𝑥−𝑥,𝐻(𝑥)−𝐻(𝑥≤1)⟩𝜎‖𝑥−𝑥‖‖𝐻(𝑥)−𝐻(𝑥=1)‖𝜎‖𝑥−𝑥‖‖𝐻(𝑥)‖, (2.24) that is, dist(𝑥,Ω)=‖𝑥−𝑥‖≤(1/𝜎)‖𝐻(𝑥)‖^1/𝛼.
Since 𝑟(𝑥)−𝑟(𝑥)≤‖𝑒(𝑥)−𝑒(𝑥=𝐹(𝑥))‖−𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)−𝐹(𝑥)−𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥≤‖𝐹(𝑥)()−𝐹𝑥)‖+𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)−𝑃_𝒦𝐹(𝑥)−𝛽𝐺(𝑥)≤‖𝐹(𝑥)−𝐹(𝑥)‖+𝐹(𝑥)−𝛽𝐺(𝑥)−𝐹(𝑥)−𝛽𝐺(𝑥‖𝐹(𝑥)()≤2−𝐹𝑥)‖+𝛽‖𝐺(𝑥)−𝐺(𝑥)‖≤2𝐿₁‖𝑥−𝑥‖^𝜈1+𝛽𝐿₂‖𝑥−𝑥‖^𝜈2≤2𝐿₁+𝛽𝐿₂‖𝑥−𝑥‖^{min{𝑣1,𝑣2}},if‖𝑥−𝑥‖<12𝐿₁+𝛽𝐿₂‖𝑥−𝑥‖^{max{𝑣1,𝑣2}},if‖𝑥−𝑥‖=≥12𝐿₁+𝛽𝐿₂dist(𝑥,Ω)^{min{𝑣1,𝑣2}},if‖𝑥−𝑥‖≤12𝐿₁+𝛽𝐿₂dist(𝑥,Ω)^{max{𝑣1,𝑣2}},if‖𝑥−𝑥‖≥1, (2.25) Combining this, we have 𝑟(𝑥)≤𝑟(𝑥)+2𝐿₁+𝛽𝐿₂dist(𝑥,Ω)^{min{𝑣1,𝑣2}},if‖𝑥−𝑥‖<12𝐿₁+𝛽𝐿₂dist(𝑥,Ω)^{max{𝑣1,𝑣2}},if‖𝑥−𝑥‖≥1. (2.26) Combining (2.26) with Theorem 2.5, we have the following results.
Case 1 (if 𝑟(𝑥)≤𝛽𝜇/(𝐿₁+𝐿₂𝛽) and ‖𝑥−𝑥‖≤1). Combining (2.4) with the first inequality in (2.26), we can obtain that 𝑥−𝑥^∗≤dist(𝑥,Ω)+𝑥−𝑥^∗()+𝐿≤dist𝑥,Ω¹+𝐿₂𝛽𝛽𝜇𝑟(𝑥)^{1/(1+𝛾−min{𝑣1,𝑣2})}()+𝐿≤dist𝑥,Ω₁+𝐿₂𝛽𝑟(𝑥)+𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽()𝛽𝜇dist𝑥,Ω^{min{𝑣1,𝑣2}1/(1+𝛾−min{𝑣1,𝑣2})}≤1𝜎‖𝐻(𝑥)‖^1/𝛼+𝜂₁𝑟(𝑥)+‖𝐻(𝑥)‖^{min{𝑣1,𝑣2}/𝛼1/(1+𝛾−min{𝑣1,𝑣2})}≤𝜌₁‖𝐻(𝑥)‖^1/𝛼+𝑟(𝑥)+‖𝐻(𝑥)‖^{min{𝑣1,𝑣2}/𝛼1/(1+𝛾−min{𝑣1,𝑣2})}, (2.27) where 𝜂₁=(max{(𝐿₁+𝐿₂𝛽)/𝛽𝜇,((2𝐿₁+𝛽𝐿₂)(𝐿₁+𝐿₂𝛽)/𝛽𝜇)𝜎^{−min{𝑣1,𝑣2}}})^{1/(1+𝛾−min{𝑣1,𝑣2})},𝜌₁=max{1/𝜎,𝜂₁}.
Case 2 (If 𝑟(𝑥)≤𝛽𝜇/(𝐿₁+𝐿₂𝛽) and ‖𝑥−𝑥‖≥1). Combining (2.4) with the second inequality in (2.26), we can also obtain that 𝑥−𝑥^∗≤dist(𝑥,Ω)+𝑥−𝑥^∗()+𝐿≤dist𝑥,Ω¹+𝐿₂𝛽𝛽𝜇𝑟(𝑥)^{1/(1+𝛾−min{𝑣1,𝑣2})}()+𝐿≤dist𝑥,Ω₁+𝐿₂𝛽𝑟(𝑥)+𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽()𝛽𝜇dist𝑥,Ω^{max{𝑣1,𝑣2}1/(1+𝛾−min{𝑣1,𝑣2})}≤1𝜎‖𝐻(𝑥)‖^1/𝛼+𝜂₂𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}/𝛼1/(1+𝛾−min{𝑣1,𝑣2})}≤𝜌₂‖𝐻(𝑥)‖^1/𝛼+𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}/𝛼1/(1+𝛾−min{𝑣1,𝑣2})}, (2.28) where 𝜂₂=(max{(𝐿₁+𝐿₂𝛽)/𝛽𝜇,((2𝐿₁+𝛽𝐿₂)(𝐿₁+𝐿₂𝛽)/𝛽𝜇)𝜎^{−max{𝑣1,𝑣2}}})^{1/(1+𝛾−min{𝑣1,𝑣2})},𝜌₂=max{1/𝜎,𝜂₂}.
Case 3 (if 𝑟(𝑥)>𝛽𝜇/(𝐿₁+𝐿₂𝛽)and ‖𝑥−𝑥‖≤1). Combining (2.5)-(2.6) with the first inequality in (2.26), we can obtain that 𝑥−𝑥^∗≤dist(𝑥,Ω)+𝑥−𝑥^∗()+𝐿≤dist𝑥,Ω¹+𝐿₂𝛽𝛽𝜇𝑟(𝑥)^{1/(1+𝛾−max{𝑣1,𝑣2})}()+𝐿≤dist𝑥,Ω₁+𝐿₂𝛽𝑟(𝑥)+𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽()𝛽𝜇dist𝑥,Ω^{min{𝑣1,𝑣2}1/(1+𝛾−max{𝑣1,𝑣2})}≤1𝜎‖𝐻(𝑥)‖^1/𝛼+𝜂₃𝑟(𝑥)+‖𝐻(𝑥)‖^{min{𝑣1,𝑣2}/𝛼1/(1+𝛾−max{𝑣1,𝑣2})}≤𝜌₁‖𝐻(𝑥)‖^1/𝛼+𝑟(𝑥)+‖𝐻(𝑥)‖^{min{𝑣1,𝑣2}/𝛼1/(1+𝛾−max{𝑣1,𝑣2})}, (2.29) where 𝜂₃=(max{(𝐿₁+𝐿₂𝛽)/𝛽𝜇,((2𝐿₁+𝛽𝐿₂)(𝐿₁+𝐿₂𝛽)/𝛽𝜇)𝜎^{−min{𝑣1,𝑣2}}})^{1/(1+𝛾−max{𝑣1,𝑣2})},𝜌₃=max{1/𝜎,𝜂₃}.
Case 4 (If 𝑟(𝑥)>𝛽𝜇/(𝐿₁+𝐿₂𝛽)and ‖𝑥−𝑥‖≥1). Combining (2.5)-(2.6) with the second inequality in (2.26), we can also obtain that 𝑥−𝑥^∗≤dist(𝑥,Ω)+𝑥−𝑥^∗()+𝐿≤dist𝑥,Ω¹+𝐿₂𝛽𝛽𝜇𝑟(𝑥)^{1/(1+𝛾−max{𝑣1,𝑣2})}()+𝐿≤dist𝑥,Ω₁+𝐿₂𝛽𝑟(𝑥)+𝛽𝜇2𝐿₁+𝛽𝐿₂𝐿₁+𝐿₂𝛽()𝛽𝜇dist𝑥,Ω^{max{𝑣1,𝑣2}1/(1+𝛾−max{𝑣1,𝑣2})}≤1𝜎‖𝐻(𝑥)‖^1/𝛼+𝜂₄𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}/𝛼1/(1+𝛾−max{𝑣1,𝑣2})}≤𝜌₄‖𝐻(𝑥)‖^1/𝛼+𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}/𝛼1/(1+𝛾−max{𝑣1,𝑣2})}, (2.30) where 𝜂₄=(max{(𝐿₁+𝐿₂𝛽)/𝛽𝜇,((2𝐿₁+𝛽𝐿₂)(𝐿₁+𝐿₂𝛽)/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∈𝑅^𝑛∣𝐷₁𝑥=𝑑₁,𝐷₂𝑥≤𝑑₂} with 𝐷₁∈𝑅^𝑙×𝑛,𝐷₂∈𝑅^𝑚×𝑛,𝑑₁∈𝑅^𝑙,and𝑑₂∈𝑅^𝑚, there exists a constant 𝑐₁>0 such that dist(𝑥,𝑃)≤𝑐₁𝐷₁𝑥−𝑑₁+𝐷₂𝑥−𝑑₂₊∀𝑥∈𝑅^𝑛. (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∈𝑅^𝑛, there exists 𝑥^∗∈𝑋^∗ such that 𝑥−𝑥^∗≤𝜇{‖𝐻(𝑥)‖+𝑅(𝑥)},∀𝑥∈𝑅^𝑛, (2.32) where 𝑅(𝑥)𝑟(𝑥)+‖𝐻(𝑥=max)‖^{min{𝑣1,𝑣2}1/(1+𝛾−min{𝑣1,𝑣2})}𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}1/(1+𝛾−min{𝑣1,𝑣2})}𝑟(𝑥)+‖𝐻(𝑥)‖^{min{𝑣1,𝑣2}1/(1+𝛾−max{𝑣1,𝑣2})}𝑟(𝑥)+‖𝐻(𝑥)‖^{max{𝑣1,𝑣2}1/(1+𝛾−max{𝑣1,𝑣2})}. (2.33)

Proof. For given 𝑥∈𝑅^𝑛, 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‖≤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∈𝑅^𝑛∣‖𝑥−𝑥^∗‖≤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).