The Solution Set Characterization and Error Bound for the Extended Mixed Linear Complementarity Problem

Abstract

For the extended mixed linear complementarity problem (EML CP), we first present the characterization of the solution set for the EMLCP. Based on this, its global error bound is also established under milder conditions. The results obtained in this paper can be taken as an extension for the classical linear complementarity problems.

1. Introduction

We consider that the extended mixed linear complementarity problem, abbreviated as EMLCP, is to find vector (𝑥^∗;𝑦^∗)∈𝑅^2𝑛 such that 𝐹𝑥^∗≥0,𝐺𝑥^∗,𝑦^∗≥0,𝐹𝑥^∗⊤𝐺𝑥^∗,𝑦^∗=0,𝐴𝑥^∗+𝐵𝑦^∗+𝑏≥0,𝐶𝑥^∗+𝐷𝑦^∗+𝑑=0, (1.1) where 𝐹(𝑥)=𝑀𝑥+𝑝, 𝐺(𝑥)=𝑁𝑥+𝑄𝑦+𝑞, 𝑀,𝑁,𝑄∈𝑅^𝑚×𝑛, 𝑝,𝑞∈𝑅^𝑚, 𝐴,𝐵∈𝑅^𝑠×𝑛, 𝐶, 𝐷∈𝑅^𝑡×𝑛, 𝑏∈𝑅^𝑠, 𝑑∈𝑅^𝑡. We assume that the solution set of the EMLCP is nonempty throughout this paper.

The EMLCP is a direct generalization of the classical linear complementarity problem and a special case of the generalized nonlinear complementarity problem which was discussed in the literature ([1, 2]). The extended complementarity problem plays a significant role in economics, engineering, and operation research, and so forth [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 [4].

Up to now, the issues of the solution set characterization and numerical methods for the classical linear complementarity problem or the classical nonlinear complementarity problem were fully discussed in the literature (e.g., [5–8]). On the other hand, the global error bound is also an important tool in the theoretical analysis and numerical treatment for variational inequalities, nonlinear complementarity problems, and other related optimization problems [9]. The error bound estimation for the classical linear complementarity problems (LCP) was fully analyzed (e.g., [7–12]).

Obviously, the EMLCP is an extension of the LCP, and this motivates us to extend the solution set characterization and error bound estimation results of the LCP to the EMLCP. To this end, we first detect the solution set characterization of the EMLCP under milder conditions in Section 2. Based on these, we establish the global error bound estimation for the EMLCP in Section 3. These constitute what can be taken as an extension of those for linear complementarity problems.

We end this section with 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 ‖⋅‖. 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}and(𝑥₋)_𝑖∶=max{−𝑥_𝑖,0},1≤𝑖≤𝑛, respectively. For simplicity, we use (𝑥;𝑦) for column vector (𝑥^⊤,𝑦^⊤)^⊤. We also use 𝑥≥0 to denote a nonnegative vector 𝑥∈𝑅^𝑛 if there is no confusion.

2. The Solution Set Characterization for EMLCP

In this section, we will characterize the solution set of the EMLCP. First, we can give the needed assumptions for our analysis.

Assumption 2.1. For the matrices 𝑀,𝑁,𝑄 involved in the EMLCP, we assume that the matrix 𝑀^⊤𝑁+𝑁^⊤𝑀𝑀^⊤𝑄𝑄^⊤𝑀0 is positive semidefinite.

Theorem 2.2. Suppose that Assumption 2.1 holds; the following conclusions hold.(i) If (𝑥₀;𝑦₀) is a solution of the EMLCP, then 𝑋^∗=𝑥;𝑦∈𝑋∣𝑀,0_𝑚×𝑛^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤𝑀,0_𝑚×𝑛𝑥;𝑦−𝑥₀;𝑦₀=0,𝑀,0_𝑚×𝑛^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀=0, (2.1)where 𝑋={(𝑥;𝑦)∈𝑅^2𝑛∣𝑀𝑥+𝑝≥0, 𝑁𝑥+𝑄𝑦+𝑞≥0, 𝐴𝑥+𝐵𝑦+𝑏≥0, 𝐶𝑥+𝐷𝑦+𝑑=0}, and 𝑋^∗ denotes the solution set of EMLCP.(ii) If (𝑥₁;𝑦₁) and (𝑥₂;𝑦₂) are two solutions of the EMLCP, then 𝑀𝑥₁+𝑝^⊤𝑁𝑥₂+𝑄𝑦₂+𝑞=𝑀𝑥₂+𝑝^⊤𝑁𝑥₁+𝑄𝑦₁+𝑞=0. (2.2)(iii) The solution set of EMLCP is convex.

Proof. (i) Set 𝑊=𝑥;𝑦∈𝑋∣(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀)=0,(𝑀,0^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀=0. (2.3)
For any (𝑥;𝑦)∈𝑋^∗, since (𝑥₀;𝑦₀)∈𝑋, we have 𝑥₀;𝑦₀−𝑥;𝑦^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥;𝑦+(𝑀,0)^⊤𝑞=𝑀𝑥₀+𝑝−𝑀𝑥+𝑝^⊤𝑁=𝑥+𝑄𝑦+𝑞𝑀𝑥₀+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞−𝑀𝑥+𝑝^⊤𝑁=𝑥+𝑄𝑦+𝑞𝑀𝑥₀+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞≥0. (2.4) Since (𝑥;𝑦)∈𝑋,(𝑥₀;𝑦₀)∈𝑋^∗, using the similar arguments to that in (2.4), we have 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞≥0. (2.5) Combining (2.4) with (2.5), one has 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥;𝑦−𝑥₀;𝑦₀≤0. (2.6) By (2.6), we have 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀≤0. (2.7) By Assumption 2.1, one has 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀=𝑥;𝑦−𝑥₀;𝑦₀^⊤𝑀^⊤𝑁+𝑁^⊤𝑀𝑀^⊤𝑄𝑄^⊤𝑦−𝑥𝑀0𝑥;₀;𝑦₀≥0. (2.8) Combining (2.7) with (2.8), we have 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀=0. (2.9) That is, (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀=0. (2.10) Using (𝑥₀;𝑦₀)∈𝑋,(𝑥;𝑦)∈𝑋^∗ again, we have 𝑥₀;𝑦₀−𝑥;𝑦^⊤(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦+(𝑁,𝑄)^⊤𝑝=𝑁𝑥₀+𝑄𝑦₀+𝑞−𝑁𝑥+𝑄𝑦+𝑞^⊤𝑀=𝑥+𝑝𝑁𝑥₀+𝑄𝑦₀+𝑞^⊤𝑀𝑥+𝑝−𝑁𝑥+𝑄𝑦+𝑞^⊤𝑀=𝑥+𝑝𝑁𝑥₀+𝑄𝑦₀+𝑞^⊤𝑀𝑥+𝑝≥0. (2.11) Using (𝑥;𝑦)∈𝑋,(𝑥₀;𝑦₀)∈𝑋^∗ again, using the similar arguments to that in (2.11), we have 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑁,𝑄)^⊤𝑝≥0. (2.12) From (2.9), (2.4), and (2.11), one has 𝑥₀;𝑦₀−𝑥;𝑦^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝=𝑥₀;𝑦₀−𝑥;𝑦^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀−𝑦+𝑥𝑥;₀;𝑦₀−𝑥;𝑦^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤()𝑦+()𝑀,0𝑥;𝑀,0^⊤𝑞+(𝑁,𝑄)^⊤𝑝=𝑥₀;𝑦₀−𝑥;𝑦^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥;𝑦+(𝑀,0)^⊤𝑞+𝑥₀;𝑦₀−𝑥;𝑦^⊤(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦+(𝑁,𝑄)^⊤𝑝≥0. (2.13) Combining (2.5) with (2.12) yields 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≥0. (2.14) Combining this with (2.13) yields 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝=0. (2.15) From (2.10) and (2.15), one has (𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀=0. (2.16) By (2.10) and (2.16), we obtain that (𝑥;𝑦)∈𝑊 follows.
On the other hand, for any (𝑥;𝑦)∈𝑊, then (𝑥;𝑦)∈𝑋, and (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀()=0,𝑀,0^⊤𝑞+(𝑁,𝑄)^⊤𝑝𝑥;𝑦−𝑥₀;𝑦₀=0, (2.17) and one has 0=𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝=𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑁,𝑄)^⊤𝑝=𝑀𝑥+𝑝−𝑀𝑥₀+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀+𝑁−+𝑞𝑥+𝑄𝑦+𝑞𝑁𝑥₀+𝑄𝑦₀+𝑞)^⊤𝑀𝑥₀=𝑀+𝑝𝑥+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀+𝑞+𝑁𝑥+𝑄𝑦+𝑞^⊤𝑀𝑥₀+𝑝. (2.18) Using (2.18), one has 0=𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀=2𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥;𝑦−𝑥₀;𝑦₀=2𝑀𝑥+𝑝−𝑀𝑥₀+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞−𝑁𝑥₀+𝑄𝑦₀𝑀+𝑞=2𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞−𝑀𝑥+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀−+𝑞𝑀𝑥₀+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞+𝑀𝑥₀+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀𝑀+𝑞=2𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞. (2.19) Thus, we have that (𝑥;𝑦)∈𝑋^∗.
(ii) Since (𝑥₁;𝑦₁) and (𝑥₂;𝑦₂) are two solutions of the EMLCP, by Theorem 2.2  (i), we have (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₁;𝑦₁−𝑥₂;𝑦₂=(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₁;𝑦₁−𝑥₀;𝑦₀−(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₂;𝑦₂−𝑥₀;𝑦₀=0. (2.20) Combining this with (𝑀𝑥₁+𝑝)^⊤(𝑁𝑥₁+𝑄𝑦₁+𝑞)=(𝑀𝑥₂+𝑝)^⊤(𝑁𝑥₂+𝑄𝑦₂+𝑞)=0, one has 0=𝑥₁;𝑦₁−𝑥₂;𝑦₂^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₁;𝑦₁−𝑥₂;𝑦₂=2𝑥₁;𝑦₁−𝑥₂;𝑦₂^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥₁;𝑦₁−𝑥₂;𝑦₂=2𝑀𝑥₁+𝑝−𝑀𝑥₂+𝑝^⊤𝑁𝑥₁+𝑄𝑦₁+𝑞−𝑁𝑥₂+𝑄𝑦₂+𝑞=−2𝑀𝑥₁+𝑝^⊤𝑁𝑥₂+𝑄𝑦₂+𝑞+𝑀𝑥₂+𝑝^⊤𝑁𝑥₁+𝑄𝑦₁+𝑞. (2.21) On the other hand, from 𝑀𝑥_𝑖+𝑝≥0,𝑁𝑥_𝑖+𝑄𝑦_𝑖+𝑞≥0,𝑖=1,2, we can deduce 𝑀𝑥₁+𝑝^⊤𝑁𝑥₂+𝑄𝑦₂+𝑞≥0,𝑀𝑥₂+𝑝^⊤𝑁𝑥₁+𝑄𝑦₁+𝑞≥0. (2.22) From (2.21) and (2.22), thus, we have that Theorem 2.2  (ii) holds.
(iii) If solution set of the EMLCP is single point set, then it is obviously convex. In this following, we suppose that (𝑥¹;𝑦¹) and (𝑥²;𝑦²) are two solutions of the EMLCP. By Theorem 2.2  (i), we have (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥¹;𝑦¹−𝑥₀;𝑦₀()=0,𝑀,0^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥²;𝑦²−𝑥₀;𝑦₀=0,(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥¹;𝑦¹−𝑥₀;𝑦₀()=0,𝑀,0^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥²;𝑦²−𝑥₀;𝑦₀=0. (2.23) For the vector (𝑥;𝑦)=𝜏(𝑥¹;𝑦¹)+(1−𝜏)(𝑥²;𝑦²),forall𝜏∈[0,1], by (2.23), we have (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀=(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝜏𝑥¹;𝑦¹−𝜏𝑥₀;𝑦₀+(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)(1−𝜏)𝑥²;𝑦²−(1−𝜏)𝑥₀;𝑦₀=0. (2.24) Using the similar arguments to that in (2.24), we can also obtain (𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀=0. (2.25) Combining (2.24) and (2.25) with the conclusion of Theorem 2.2 (i), we obtain the desired result.

Corollary 2.3. Suppose that Assumption 2.1 holds. Then, the solution set for EMLCP has the following characterization: 𝑋^∗=𝑥;𝑦∈𝑋∣(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀−𝑥=0,𝑥;𝑦₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≤0.. (2.26)

Proof. Set 𝑊=𝑥;𝑦∈𝑋∣(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀−𝑥=0,𝑥;𝑦₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≤0. (2.27) For any (𝑥;𝑦)∈𝑊, then (𝑥;𝑦)∈𝑋, combining this with (𝑥₀;𝑦₀)∈𝑋^∗. Using the similar arguments to that in (2.5) and (2.12), we have 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≥0. (2.28) Combining this with (𝑥;𝑦)∈𝑊, one has 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝=0. (2.29) From ((𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0))((𝑥;𝑦)−(𝑥₀;𝑦₀))=0, we have (𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀=0. (2.30) Thus, by Theorem 2.2(i), one has (𝑥;𝑦)∈𝑋^∗.
On the other hand, for any (𝑥;𝑦)∈𝑋^∗, by Theorem 2.2  (i), we have (𝑥;𝑦)∈𝑋, ((𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0))((𝑥;𝑦)−(𝑥₀;𝑦₀))=0, and ((𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝)^⊤((𝑥;𝑦)−(𝑥₀;𝑦₀))=0, that is, 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝=0. (2.31) Thus, (𝑥;𝑦)∈𝑊.

Using the following definition developed from EMLCP, we can further detect the solution structure of the EMLCP.

Definition 2.4. A solution (𝑥;𝑦) of the EMLCP is said to be nondegenerate if it satisfies 𝑀𝑥+𝑝+𝑁𝑥+𝑄𝑦+𝑞>0. (2.32)

Theorem 2.5. Suppose that Assumption 2.1 holds, and the EMLCP has a nondegenerate solution, say (𝑥₀;𝑦₀). Then, the following conclusions hold.(i) The solution set of EMLCP 𝑋^∗=𝑥;𝑦∈𝑋∣𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≤0. (2.33)(ii) If the matrices 𝑀_𝛼 and 𝑄_𝛼 are the full-column rank, where 𝛼={𝑖∣(𝑀𝑥₀+𝑝)_𝑖>0,𝑖=1,2,…,𝑚}, 𝛼={𝑖∣𝑖=1,2,…,𝑚,𝑖∉𝛼}, then (𝑥₀;𝑦₀) is the unique nondegenerate solution of EMLCP.

Proof. (i) Set 𝑊=𝑥;𝑦∈𝑋∣𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≤0. (2.34) From Corollary 2.3, one has 𝑋^∗⊆𝑊. In this following, we will show that 𝑊⊆𝑋^∗. For any (𝑥;𝑦)∈𝑊, then (𝑥;𝑦)∈𝑋, combining this with (𝑥₀;𝑦₀)∈𝑋^∗. Using the similar arguments to that in (2.14), we have 𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝≥0. (2.35) Combining this with (𝑥;𝑦)∈𝑊, one has 0=𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝=𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑁,𝑄)^⊤𝑝=𝑀𝑥+𝑝−𝑀𝑥₀+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀+−+𝑞𝑁𝑥+𝑄𝑦+𝑞𝑁𝑥₀+𝑄𝑦₀+𝑞^⊤𝑀𝑥₀=+𝑝𝑀𝑥+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀+𝑞+𝑁𝑥+𝑄𝑦+𝑞^⊤𝑀𝑥₀+𝑝. (2.36) Combining 𝑀𝑥+𝑝≥0,𝑁𝑥+𝑄𝑦+𝑞≥0 with (2.36), one has 𝑀𝑥+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀+𝑞=𝑀𝑥₀+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞=0. (2.37) Since (𝑥₀;𝑦₀) is a nondegenerate solution, combining this with (2.37), we have (𝑀𝑥+𝑝)^⊤(𝑁𝑥+𝑄𝑦+𝑞)=0. That is, (𝑥;𝑦)∈𝑋^∗.
(ii) Let (𝑥;𝑦) be any nondegenerate solution. Since (𝑥₀;𝑦₀) is a nondegenerate solution, then we have 𝑀𝑥₀+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀+𝑞=0,𝑀𝑥₀+𝑝+𝑁𝑥₀+𝑄𝑦₀+𝑞>0. (2.38)(2.39) Combining (2.38) with (2.39), we have 𝑁𝑥₀+𝑄𝑦₀+𝑞_𝑖=0,∀𝑖∈𝛼. (2.40) If 𝑖∉𝛼, then (𝑁𝑥₀+𝑄𝑦₀+𝑞)_𝑖>0 by (2.39). By (2.38) again, we can deduce that 𝑀𝑥₀+𝑝_𝑖=0,∀𝑖∉𝛼. (2.41) On the other hand, for the (𝑥₀;𝑦₀) and (𝑥;𝑦) which are solutions of EMLCP, and combining Theorem 2.2  (ii), we have (𝑀𝑥+𝑝)^⊤(𝑁𝑥₀+𝑄𝑦₀+𝑞)=0. Using (𝑁𝑥₀+𝑄𝑦₀+𝑞)_𝑖>0,forall𝑖∉𝛼, we can deduce that 𝑀𝑥+𝑝_𝑖=0,∀𝑖∉𝛼. (2.42) Combining Theorem 2.2  (ii) again, we also have 𝑀𝑥₀+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞=0. (2.43) For any 𝑖∈𝛼, that is, (𝑀𝑥₀+𝑝)_𝑖>0, and combining (2.43), we obtain 𝑁𝑥+𝑄𝑦+𝑞_𝑖=0,∀𝑖∈𝛼. (2.44) Combining this with the fact that (𝑀𝑥+𝑝)+(𝑁𝑥+𝑄𝑦+𝑞)>0, we can deduce that 𝑀𝑥+𝑝_𝑖>0,∀𝑖∈𝛼. (2.45) From (2.41) and (2.42), we obtain 𝑀_𝛼𝑥−𝑥₀=0. (2.46) Thus, 𝑥=𝑥₀ by the full-column rank assumption on 𝑀_𝛼. Using 𝑥=𝑥₀, combining (2.40) with (2.44), we can deduce that 𝑄_𝛼𝑦=−𝑁_𝛼𝑥−𝑞=−𝑁_𝛼𝑥₀−𝑞=𝑄_𝛼𝑦₀. (2.47) That is, 𝑦=𝑦₀ by the full-column rank assumption on 𝑄_𝛼. Thus, the desired result follows.

The solution set characterization obtained in Theorem 2.2  (i) coincides with that of Lemma  2.1 in [7], and the solution set characterization obtained in Theorem 2.5  (i) coincides with that of Lemma  2.2 in [8] for the linear complementarity problem.

3. Global Error Bound for the EMLCP

In this following, we will present a global error bound for the EMLCP based on the results obtained in Corollary 2.3 and Theorem 2.5  (i). Firstly, we can give the needed error bound for a polyhedral cone from [13] and following technical lemmas to reach our claims.

Lemma 3.1. 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 𝑐₁>0 such that dist(𝑥,𝑃)≤𝑐₁𝐷₁𝑥−𝑑₁+𝐵₁𝑥−𝑏₁₊∀𝑥∈𝑅^𝑛; (3.1)

Lemma 3.2. Suppose that (𝑥₀;𝑦₀) is a solution of EMLCP, and let 𝜔=(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝, (3.2) then, there exists a constant 𝜏>0, such that for any (𝑥;𝑦)∈𝑅^2𝑛, one has 𝜔^⊤𝑥;𝑦−𝑥₀;𝑦₀₋≤𝜏𝑀𝑥+𝑝₋+𝑁𝑥+𝑄𝑦+𝑞₋+𝐴𝑥+𝐵𝑦+𝑏₋+𝐶𝑥+𝐷𝑦+𝑑. (3.3)

Proof. Similar to the proof of (2.14), we can obtain 𝜔^⊤𝑥;𝑦−𝑥₀;𝑦₀≥0,∀𝑥;𝑦∈𝑋. (3.4) We consider the following linear programming problems min𝜔^⊤𝑥;𝑦s.t.𝑀𝑥+𝑝≥0,𝑁𝑥+𝑄𝑦+𝑞≥0,𝐴𝑥+𝐵𝑦+𝑏≥0,𝐶𝑥+𝐷𝑦+𝑑=0. (3.5) From the assumption, we know that (𝑥₀,𝑦₀) is an optimal point of the linear programming problem. Thus, there exist optimal Lagrange multipliers 𝜆₁,𝜆₂∈𝑅^𝑚₊, 𝜆₃∈𝑅^𝑠₊, and 𝜆₄∈𝑅^𝑡 such that 𝜔=(𝑀,0)^⊤𝜆₁+(𝑁,𝑄)^⊤𝜆₂+(𝐴,𝐵)^⊤𝜆₃+(𝐶,𝐷)^⊤𝜆₄,𝑀𝑥₀+𝑝≥0,𝑁𝑥₀+𝑄𝑦₀+𝑞≥0,𝐴𝑥₀+𝐵𝑦₀+𝑏≥0,𝐶𝑥₀+𝐷𝑦₀)𝑥+𝑑=0,(𝑀,0₀;𝑦₀+𝑝^⊤𝜆₁=0,𝑁𝑥₀+𝑄𝑦₀+𝑞^⊤𝜆₂=0,𝐴𝑥₀+𝐵𝑦₀+𝑏^⊤𝜆₃=0. (3.6) From (3.6), we can easily deduce that 𝜔^⊤𝑥₀;𝑦₀=(𝑀,0)^⊤𝜆₁+(𝑁,𝑄)^⊤𝜆₂+(𝐴,𝐵)^⊤𝜆₃+(𝐶,𝐷)^⊤𝜆₄^⊤𝑥₀;𝑦₀=𝜆^⊤₁(𝑀,0)𝑥₀;𝑦₀+𝜆^⊤₂(𝑁,𝑄)𝑥₀;𝑦₀+𝜆^⊤₃(𝐴,𝐵)𝑥₀;𝑦₀+𝜆^⊤₄(𝐶,𝐷)𝑥₀;𝑦₀=−𝜆^⊤₁𝑝−𝜆^⊤₂𝑞−𝜆^⊤₃𝑏−𝜆^⊤₄𝑑. (3.7) Thus, for any (𝑥;𝑦)∈𝑅^2𝑛, from the first equation in (3.6), we have 𝜔^⊤𝑥;𝑦−𝑥₀;𝑦₀₋=𝜆^⊤₁(𝑀,0)𝑥;𝑦+𝑝+𝜆^⊤₂(𝑁,𝑄)𝑥;𝑦+𝑞+𝜆^⊤₃(𝐴,𝐵)𝑥;𝑦+𝑏+𝜆^⊤₄(𝐶,𝐷)𝑥;𝑦+𝑑₋≤𝜆^⊤₁(𝑀,0)𝑥;𝑦+𝑝₋+𝜆^⊤₂(𝑁,𝑄)𝑥;𝑦+𝑞₋+𝜆^⊤₃(𝐴,𝐵)𝑥;𝑦+𝑏₋+𝜆^⊤₄(𝐶,𝐷)𝑥;𝑦+𝑑₋≤𝜆^⊤₁(𝑀,0)𝑥;𝑦+𝑝₋+𝜆^⊤₂(𝑁,𝑄)𝑥;𝑦+𝑞₋+𝜆^⊤₃(𝐴,𝐵)𝑥;𝑦+𝑏₋+𝜆₄^⊤₋(𝐶,𝐷)𝑥;𝑦+𝑑₊+𝜆₄^⊤₊(𝐶,𝐷)𝑥;𝑦+𝑑₋≤𝜆₁(𝑀,0)𝑥;𝑦+𝑝₋+𝜆₂(𝑁,𝑄)𝑥;𝑦+𝑞₋+𝜆₃(𝐴,𝐵)𝑥;𝑦+𝑏₋+𝜈(𝐶,𝐷)𝑥;𝑦+𝑑, (3.8) Where 𝜈≥0 is a constant. Let 𝜏=max{‖𝜆₁‖,‖𝜆₂‖,‖𝜆₃‖,𝜈}, then the desired result follows.

Now, we are at the position to state our results.

Theorem 3.3. Suppose that Assumption 2.1 holds. Then, there exists a constant 𝜂>0 such that for any (𝑥;𝑦)∈𝑅^2𝑛, there exists (𝑥^∗;𝑦^∗)∈𝑋^∗ such that 𝑥;𝑦−𝑥^∗;𝑦^∗≤𝜂𝑠𝑥,𝑦+𝑠𝑥,𝑦^1/2, (3.9) where 𝑠𝑥,𝑦=𝑀𝑥+𝑝₋+𝑁𝑥+𝑄𝑦+𝑞₋+𝐴𝑥+𝐵𝑦+𝑏₋+𝐶𝑥+𝐷𝑦+𝑑+𝑀𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞₊. (3.10)

Proof. Using Corollary 2.3 and Lemma 3.1, there exists a constant 𝜇₁>0, for any (𝑥;𝑦)∈𝑅^2𝑛, and there exists (𝑥^∗;𝑦^∗)∈𝑋^∗ such that 𝑥;𝑦−𝑥^∗;𝑦^∗≤𝜇₁𝑀𝑥+𝑝₋+𝑁𝑥+𝑄𝑦+𝑞₋+𝐴𝑥+𝐵𝑦+𝑏₋++)𝐶𝑥+𝐷𝑦+𝑑(𝑀,0^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀₊+(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀, (3.11) Where (𝑥₀;𝑦₀) is a solution of EMLCP. Now, we consider the right-hand-side of expression (3.11).
Firstly, by Assumption 2.1, we obtain that 𝐻𝑥,𝑦=𝑀𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞 (3.12) is a convex function. For any (𝑥;𝑦)∈𝑅^2𝑛, we have 𝐻𝑥,𝑦−𝐻𝑥₀;𝑦₀≥(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀. (3.13) Combining this with 𝐻(𝑥₀;𝑦₀)=0, we can deduce that (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀₊≤𝑀𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞₊. (3.14)
Secondly, we consider the last item in (3.11). By Assumption 2.1, there exists a constant 𝜇₂>0 such that for any (𝑥;𝑦)∈𝑅^2𝑛, (𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀²≤𝜇₂𝑥;𝑦−𝑥₀;𝑦₀^⊤(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥;𝑦−𝑥₀;𝑦₀=2𝜇₂𝑀𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞−𝑀𝑥₀+𝑝^⊤𝑁𝑥₀+𝑄𝑦₀−()+𝑞𝑀,0^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤×𝑥;𝑦−𝑥₀;𝑦₀≤𝜇₂𝑀𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞₊+2𝜇₂(𝑀,0)^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀₋≤2𝜇₂𝑀𝑥+𝑝^⊤𝑁𝑥+𝑄𝑦+𝑞₊+2𝜇₂𝜏𝑀𝑥+𝑝₋+𝑁𝑥+𝑄𝑦+𝑞₋+𝐴𝑥+𝐵𝑦+𝑏₋+𝐶𝑥+𝐷𝑦+𝑑, (3.15) where the first equality is based on the Taylor expansion of function 𝐻(𝑥,𝑦) on (𝑥₀;𝑦₀) point, the second inequality follows from the fact that (𝑥₀;𝑦₀) is a solution of EMLCP and the fact that 𝑎+𝑏≤𝑎₊+𝑏₊ for any 𝑎,𝑏∈𝑅, and the last inequality is based on Lemma 3.2. By (3.11)–(3.15), we have that (3.9) holds.

The error bound obtained in Theorem 3.3 coincides with that of Theorem  2.4 in [11] for the linear complementarity problem, and it is also an extension of Theorem  2.7 in [7] and Corollary  2 in [14].

Theorem 3.4. Suppose that the assumption of Theorem 2.5 holds. Then, there exists a constant 𝜂₁>0, such that for any (𝑥;𝑦)∈𝑅^2𝑛, there exists a solution (𝑥^∗;𝑦^∗)∈𝑋^∗ such that 𝑥;𝑦−𝑥^∗;𝑦^∗≤𝜂₁𝑠𝑥,𝑦, (3.16) where 𝑠(𝑥,𝑦) is defined in Theorem 3.3.

Proof. From Theorem 2.5, using the proof technique is similar to that of Theorem 3.3. For any (𝑥;𝑦)∈𝑅^2𝑛, there exist (𝑥^∗;𝑦^∗)∈𝑋^∗ and a constant 𝜇₄>0 such that 𝑥;𝑦−𝑥^∗;𝑦^∗≤𝜇₄𝑀𝑥+𝑝₋+𝑁𝑥+𝑄𝑦+𝑞₋+𝐴𝑥+𝐵𝑦+𝑏₋++()𝐶𝑥+𝐷𝑦+𝑑𝑀,0^⊤(𝑁,𝑄)+(𝑁,𝑄)^⊤(𝑀,0)𝑥₀;𝑦₀+(𝑀,0)^⊤𝑞+(𝑁,𝑄)^⊤𝑝^⊤𝑥;𝑦−𝑥₀;𝑦₀₊. (3.17) Combining this with (3.14), we can deduce that (3.16) holds.

4. Conclusion

In this paper, we presented the solution Characterization, and also established global error bounds on the extended mixed linear complementarity problems which are the extensions of those for the classical linear complementarity problems. Surely, we may use the error bound estimation to establish quick convergence rate of the noninterior path following method for solving the EMLCP just as was done in [14], and this is a topic for future research.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant no. 11171180,11101303), Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002), and Shandong Provincial Natural Science Foundation (ZR2010AL005, ZR2011FL017).