A Superlinearly Convergent Method for the Generalized Complementarity Problem over a Polyhedral Cone

Abstract

Making use of a smoothing NCP-function, we formulate the generalized complementarity problem (GCP) over a polyhedral cone as an equivalent system of equations. Then we present a Newton-type method for the equivalent system to obtain a solution of the GCP. Our method solves only one linear system of equations and performs only one line search at each iteration. Under mild assumptions, we show that our method is both globally and superlinearly convergent. Compared to the previous literatures, our method has stronger convergence results under weaker conditions.

1. Introduction

The generalized complementarity problem, denoted by GCP, is to find a vector 𝑥^∗∈𝑅^𝑛 such that 𝐹𝑥^∗∈𝒦,𝐺𝑥^∗∈𝒦^∘,𝐹𝑥^∗⊤𝐺𝑥^∗=0, (1) where 𝐹 and 𝐺 are continuous functions from 𝑅^𝑛 to 𝑅^𝑚, 𝒦 is a nonempty closed convex cone in 𝑅^𝑚, and 𝒦^∘ denotes the polar cone of 𝒦.

This problem has many interesting applications and its solution using special techniques has been considered extensively in the literature. See [1–3] and references therein. In particular, if 𝒦=𝑅^𝑛₊ and 𝐺(𝑥)=𝑥, then the GCP reduces to the classical nonlinear complementarity problem [4]. Furthermore, the GCP is closely related to the variational inequality problem in the sense that 𝑥^∗ is a solution of GCP if and only if 𝐹(𝑥^∗) is a solution of 𝑉𝐼(𝐺∘𝐹⁻¹,𝒦) if 𝐹 is invertible (see Lemma  6 in [5]).

To solve the GCP, one usually reformulates it as a minimization problem over a simple set or an unconstrained optimization problem; see [3] for the case that 𝒦 is a general cone and see [1, 2] for the case that 𝒦=𝑅^𝑛₊. The conditions under which a stationary point of the reformulated optimization is a solution of GCP were provided in this literature.

In this paper, we consider the case that 𝑚=𝑛, 𝐹 and 𝐺 are both continuously differentiable on 𝑅^𝑛, and 𝒦 is a polyhedral cone in 𝑅^𝑛; that is, there exist 𝐴∈𝑅^𝑠×𝑛,𝐵∈𝑅^𝑡×𝑛 such that 𝒦=𝑢∈𝑅^𝑛∣𝐴𝑢≥0,𝐵𝑢=0, (2) where 𝑠 and 𝑡 are both positive integers. It is easy to verify that its polar cone 𝒦^∘ has the following representation: 𝒦^∘=𝑣∈𝑅^𝑛∣𝑣=𝐴^𝑇𝜆₁+𝐵^𝑇𝜆₂,𝜆₁≥0,𝜆₁∈𝑅^𝑠,𝜆₂∈𝑅^𝑡. (3)

Obviously, if 𝐴 is an identity matrix and 𝐵=0, then this version of the GCP reduces to the case considered in [2].

From now on, the GCP is specialized over a polyhedral cone.

For the GCP, Wang et al. and Zhang et al. [6, 7], respectively, established some constrained or unconstrained optimization reformulation for the case 𝐹 and 𝐺 are both linear functions and gave some Newton-type method for the problem. Later, Ma et at. [8] established a potential reduction method for the same case.

In this paper, we give a new smoothing Newton method for solving the GCP based on a smoothing NCP-function. This method is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the method needs only to solve one linear system of equations and perform one line search per iteration.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and results of a smoothing NCP-function based on the Fischer-Burmeister function. In Section 3, we present a one-step smoothing Newton method for the GCP and state some preliminary results. In Section 4, we establish the global and superlinear convergence of the proposed method. Conclusions are given in Section 5.

To end this section, we will give some notations used in this paper. All vectors are column ones. The inner product of vectors 𝑥,𝑦∈𝑅^𝑛 is denoted by 𝑥^𝑇𝑦. For convenience, vector (𝑥^𝑇,𝑦^𝑇)^𝑇 is denoted by (𝑥,𝑦), where 𝑥,𝑦 are both column vectors. Let ‖⋅‖ denote the Euclidean norm of a vector or a matrix. For vector 𝑢∈𝑅^𝑛, 𝐷_𝑢=diag(𝑢) denotes the diagonal matrix with the 𝑖th diagonal element being 𝑢_𝑖. Vector 𝑒 denotes the vector of all ones whose dimension is defined by the context of its use.

2. Preliminaries

To establish our method for the solution of the GCP, we now formulate the GCP as a system of equations via the following smoothing NCP-function based on the Fischer-Burmeister function [9]: 𝜙𝜇,𝑎,𝑏=𝑎+𝑏−𝑎²+𝑏²,if𝑎²+𝑏²𝑎≥𝜇,𝑎+𝑏−²+𝑏²−𝜇2𝜇2,if𝑎²+𝑏²<𝜇, (4) where 𝜇>0 is a smoothing parameter. By simple calculation, we have 𝜙^′_𝜇𝜇,𝑎,𝑏=0if𝑎²+𝑏²𝑎≥𝜇,₂+𝑏²2𝜇²−12if𝑎²+𝑏²𝜙<𝜇,_′𝑎=𝑎𝜇,𝑎,𝑏1−𝑎²+𝑏²if𝑎²+𝑏²𝑎≥𝜇,1−𝜇if𝑎²+𝑏²𝜙<𝜇,_′𝑏=𝑏𝜇,𝑎,𝑏1−𝑎²+𝑏²if𝑎²+𝑏²𝑏≥𝜇,1−𝜇if𝑎²+𝑏²<𝜇. (5)

The smoothing NCP-function possesses a few nice properties [10].

Lemma 1. The NCP-function 𝜙(𝜇,𝑎,𝑏) defined by (4) has the following properties:(i)for any (𝜇,𝑎,𝑏)∈𝑅₊₊×𝑅², one has lim_𝜇→0𝜙𝜇,𝑎,𝑏=𝜙_𝐹𝐵(𝑎,𝑏), (6) where 𝜙_𝐹𝐵(𝑎,𝑏)=𝑎+𝑏−𝑎²+𝑏², a basic property of which is that 𝜙_𝐹𝐵(𝑎,𝑏)=0⇔𝑎≥0,𝑏≥0,𝑎𝑏=0;(ii)for any 𝜇₁,𝜇₂∈𝑅₊₊, one has |𝜙(𝜇₁,𝑎,𝑏)−𝜙(𝜇₂,𝑎,𝑏)|<|𝜇₁−𝜇₂|;(iii)for any (𝜇,𝑎,𝑏)∈𝑅₊₊×𝑅², one has 0<𝜙^′_𝑎(𝜇,𝑎,𝑏)<2,0<𝜙^′_𝑏(𝜇,𝑎,𝑏)<2;(iv)let {𝑎_𝑘},{𝑏_𝑘} be any two sequences such that 𝑎_𝑘,𝑏_𝑘→+∞ or 𝑎_𝑘→−∞ or 𝑏_𝑘→−∞. Then, for any 𝜇∈𝑅₊₊, one has lim_𝑘→∞|𝜙(𝜇,𝑎_𝑘,𝑏_𝑘)|=+∞.

From Lemma 1, we can find that the function 𝜙(𝜇,𝑎,𝑏) is indeed a smoothing approximation function of the Fischer-Burmeister function 𝜙_FB(𝑎,𝑏).

For the GCP, we have the following equivalent statements: 𝑥is a solution of the GCP, ⇔𝐹(𝑥)∈𝒦,𝐺(𝑥)∈𝒦^∘, and 𝐹(𝑥)^𝑇𝐺(𝑥)=0, ⇔ there exist𝜆₁∈𝑅^𝑠,𝜆₂∈𝑅^𝑡, such that 𝐴𝐹(𝑥)≥0,𝐵𝐹(𝑥)=0, 𝐺(𝑥)=𝐴^𝑇𝜆₁+𝐵^𝑇𝜆₂,𝜆₁≥0, 𝐹(𝑥)^𝑇𝐺(𝑥)=0, ⇔ there exist𝜆₁∈𝑅^𝑠,𝜆₂∈𝑅^𝑡, such that 𝐴𝐹(𝑥)≥0,𝜆₁≥0,(𝜆₁)^𝑇𝐴𝐹(𝑥)=0, 𝐵𝐹(𝑥)=0, 𝐺(𝑥)−𝐴^𝑇𝜆₁−𝐵^𝑇𝜆₂=0.

Let 𝑧∶=(𝜇,𝑥,𝜆₁,𝜆₂)∈𝑅₊₊×𝑅^𝑛×𝑅^𝑠×𝑅^𝑡 and define a vector-valued function 𝐻∶𝑅₊₊×𝑅^{𝑛+𝑠+𝑡}→𝑅^{𝑛+𝑠+𝑡+1} as follows: 𝐻(𝑧)∶=𝐻𝜇,𝑥,𝜆₁,𝜆₂𝑒∶=^𝜇Φ−1𝜇,𝜆₁(𝑥)(𝑥)𝐺(𝑥),𝐴𝐹𝐵𝐹−𝐴^𝑇𝜆₁−𝐵^𝑇𝜆₂, (7) where Φ𝜇,𝜆₁(𝑥)=𝜙𝜆,𝐴𝐹𝜇,₁₁,[𝐴𝐹(𝑥)]₁𝜙𝜇,𝜆₁₂,[𝐴𝐹(𝑥)]₂⋮𝜙𝜇,𝜆_1𝑠,[𝐴𝐹(𝑥)]_𝑠. (8)

Then, by Lemma 1, the following result is straightforward.

Theorem 2. 𝑥 is a solution of the 𝐺𝐶𝑃 if and only if there exist 𝜆₁∈𝑅^𝑠,𝜆₂∈𝑅^𝑡 such that 𝐻(𝑧)=0. (9)

Theorem 2 means that the GCP is equivalent to an equation system in the sense that their solution sets are coincident.

By (5)–(7), it is not difficult to see that the Jacobian of 𝐻(𝑧) has the following form: 𝐻^′(𝑧)=𝑒^𝜇{𝑣}𝐷000vec_𝑎𝐴𝐹^′(𝑥)𝐷_𝑏00𝐵𝐹^′(𝑥)000𝐺^′(𝑥)−𝐴^𝑇−𝐵^𝑇, (10) where vec{𝑣}∶=𝜙^′_𝜇𝜇,𝜆₁₁,[𝐴𝐹(𝑥)]₁,𝜙^′_𝜇𝜇,𝜆₁₂,[𝐴𝐹(𝑥)]₂𝜙,…,_′𝜇𝜇,𝜆_1𝑠,[𝐴𝐹(𝑥)]_𝑠^𝑇,𝐷_𝑎∶=diag𝑎₁(𝑧),𝑎₂(𝑧),…,𝑎_𝑠(𝑧),𝑎_𝑖(𝑧)=[(𝑥1−𝐴𝐹)]_𝑖𝜆₁²_𝑖+[𝐴𝐹(𝑥)]²_𝑖(𝑧),[]𝑖∉𝛼1−𝐴𝐹(𝑥)_𝑖𝜇(𝑧),𝐷𝑖∈𝛼_𝑏∶=diag𝑏₁(𝑧),𝑏₂(𝑧),…,𝑏_𝑠(𝑧),𝑏_𝑖(𝑧)=𝜆1−^1𝑖𝜆₁²_𝑖+[𝐴𝐹(𝑥)]²_𝑖(𝑧),𝜆𝑖∉𝛼1−^1𝑖𝜇(𝑧),𝛼(𝑧)={}∣𝑖∈𝛼𝑖∈1,2,…,𝑠𝜆₁²_𝑖+[𝐴𝐹(𝑥)]²_𝑖<𝜇. (11)

And for the functions 𝐻 and Φ, they have the following nice properties.

Lemma 3. Let 𝐻 and Φ be defined by (7) and (8), respectively. Then(i)Φ is continuously differentiable at any 𝑧∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡},(ii)𝐻 is continuously differentiable at any 𝑧∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡} with its Jacobian 𝐻^′(𝑧) being defined by (10). If matrix 𝐵 has full row rank and 𝐺^′(𝑥)[𝐹^′(𝑥)]⁻¹ is positive definite, then 𝐻^′(𝑧) is nonsingular on 𝑅₊₊×𝑅^{𝑛+𝑠+𝑡}.

Proof. From Lemma 1, it is not difficult to see that Φ is continuously differentiable at any 𝑧∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡}.
Next we prove (ii). It follows from the definition of 𝐻 and (i) that 𝐻 is continuously differentiable on 𝑅₊₊×𝑅^{𝑛+𝑠+𝑡}. And by simple calculation, we obtain the Jacobian 𝐻^′(𝑧) defined by (10). Now we prove the nonsingularity of 𝐻^′(𝑧).
Since 𝜇>0, it follows that 𝑒^𝜇>1. And thus we know that to prove the nonsingularity of 𝐻^′(𝑧), it is sufficient to prove the nonsingularity of 𝐷∶=^{𝐷𝑎𝐴𝐹′(𝑥)𝐷𝑏0}_𝐵𝐹^′_{𝐺(𝑥)00′(𝑥)−𝐴𝑇−𝐵𝑇}.
Since 𝐹^′(𝑥) is nonsingular, we make the following transformation to 𝐷 without changing its rank: 𝐷𝐹^′(𝑥)⁻¹00−𝐷_𝑏⁻¹𝐷_𝑎𝐴𝐼_𝑠000𝐼_𝑡=0𝐷_𝑏0𝐵𝐺00_′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴−𝐴^𝑇−𝐵^𝑇. (12) Since 𝐷_𝑏 is nonsingular, to prove the nonsingularity of 𝐷, it suffices to show the nonsingularity of the following submatrix of it: 𝐺𝐵0_′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴−𝐵^𝑇. (13) Let 𝐺𝐵0_′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴−𝐵^𝑇𝑑₁𝑑₂=0, (14) where 𝑑₁∈𝑅^𝑛,𝑑₂∈𝑅^𝑡. Then we obtain, 𝐵𝑑₁𝐺=0_′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴𝑑₁−𝐵^𝑇𝑑₂=0. (15) That is, 𝑑₁=𝐺^′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴⁻¹𝐵^𝑇𝑑₂,𝐵𝐺^′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴⁻¹𝐵^𝑇𝑑₂=0. (16) Since 𝐺^′(𝑥)𝐹^′(𝑥)⁻¹ is positive definite and 𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴 is semipositive definite, it holds that (𝐺^′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴)⁻¹ is also positive definite. Premultiplying (16) by 𝑑^𝑇₂ yields 𝐵^𝑇𝑑₂^𝑇𝐺^′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴⁻¹𝐵^𝑇𝑑₂=0. (17) Noting that (𝐺^′(𝑥)𝐹^′(𝑥)⁻¹+𝐴^𝑇𝐷_𝑏⁻¹𝐷_𝑎𝐴)⁻¹ is positive definite and 𝐵 has full row rank, we obtain 𝑑₂=0, and thus 𝑑₁=0. Hence, we complete the proof.

3. Algorithm and Preliminaries

In this section we propose our smoothing Newton method for the GCP.

Let 𝛾∈(0,1). Define the following real-valued functions 𝜃,𝜌∶𝑅₊₊×𝑅^{𝑛+𝑠+𝑡}→𝑅₊ by 𝜃(𝑧)=‖𝐻(𝑧)‖²,𝜌(𝑧)=𝛾min{1,‖𝐻(𝑧)‖,𝜃(𝑧)}. (18) And for convenience, let 𝑧^𝑘=(𝜇_𝑘,𝑥^𝑘,𝜆^𝑘₁,𝜆^𝑘₂) in what follows.

Algorithm 4. Consider the following.

Step 1. Choose parameters 𝛿∈(0,1),𝜎∈(0,1/2),𝜇₀>0, and 𝛾∈(0,1) such that 𝛾𝜇₀<1/2. Let 𝜇∶=(𝜇₀,0)∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡} and let 𝑧⁰ be an arbitrary initial point. Set 𝑘∶=0.

Step 2. If ||𝐻(𝑧^𝑘)||=0, stop; otherwise, compute 𝜌_𝑘∶=𝜌(𝑧^𝑘).

Step 3. Compute Δ𝑧^𝑘∶=(Δ𝜇_𝑘,Δ𝑥^𝑘,Δ𝜆^𝑘₁,Δ𝜆^𝑘₂)∈𝑅^{𝑛+𝑠+𝑡+1}by 𝐻𝑧^𝑘+𝐻^′𝑧^𝑘Δ𝑧^𝑘=𝜌_𝑘𝜇𝑒^𝜇𝑘. (19)

Step 4. Let 𝜏_𝑘=𝛿^𝑚𝑘, where 𝑚_𝑘 is the smallest nonnegative integer 𝑚 such that 𝜃𝑧^𝑘+𝛿^𝑚Δ𝑧^𝑘−𝜃𝑧^𝑘≤−2𝜎1−2𝛾𝜇₀𝛿^𝑚𝜃𝑧^𝑘. (20)

Step 5. Let 𝑧^𝑘+1∶=𝑧^𝑘+𝜏_𝑘Δ𝑧^𝑘,𝑘∶=𝑘+1; go to Step 2.

The following theorem shows that Algorithm 4 is well defined and generates an infinite sequence with some good features. Define the set Ω∶=𝑧=𝜇,𝑥,𝜆₁,𝜆₂∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡}∣𝜇≥𝜌(𝑧)𝜇₀. (21)

Theorem 5. Let 𝑧⁰=(𝜇₀,𝑥⁰,𝜆⁰₁,𝜆⁰₂)∈𝑅₊₊×𝑅^{𝑛+𝑠+𝑡} be given in Algorithm 4. Then Algorithm 4 is well defined and generates an infinite sequence {𝑧^𝑘} with 𝜇_𝑘∈𝑅₊₊ and 𝑧^𝑘∈Ω for all 𝑘.

Proof . We firstly prove that Algorithm 4 is well defined. Obviously, if 𝜇_𝑘>0 for all 𝑘, it follows from Lemma 3(ii) that the matrix 𝐻^′(𝑧^𝑘) is nonsingular. So, Step 3 of Algorithm 4 is well defined at the 𝑘th iteration. Now we prove that 𝜇_𝑘>0 by mathematical induction on 𝑘.
Obviously, 𝜇₀>0. Assume 𝜇_𝑘>0 for 𝑘; next we prove 𝜇_𝑘+1>0.
By Step 3, we have Δ𝜇_𝑘=𝑒^−𝜇𝑘𝑒^𝜇𝑘𝜌_𝑘𝜇₀−𝑒^𝜇𝑘+1=𝑒^−𝜇𝑘−1+𝜌_𝑘𝜇₀>−𝜇_𝑘+𝜌_𝑘𝜇₀, (22) where the last inequality follows from 𝜇>1−𝑒^−𝜇 for all 𝜇>0. Therefore, for any 𝛼∈(0,1] we obtain that 𝜇_𝑘+𝛼Δ𝜇_𝑘>𝜇_𝑘+𝛼−𝜇_𝑘+𝜌_𝑘𝜇₀=(1−𝛼)𝜇_𝑘+𝛼𝜌_𝑘𝜇₀>0, (23) which means that, for 𝜏_𝑘∈(0,1), 𝜇_𝑘+1=𝜇_𝑘+𝜏_𝑘Δ𝜇_𝑘>0. And thus we prove the desired result.
Next we prove that Step 4 is well defined. Let 𝜔(𝛼)∶=𝜃𝑧^𝑘+𝛼Δ𝑧^𝑘−𝜃𝑧^𝑘−2𝛼𝐻𝑧^𝑘𝑇𝐻^′𝑧^𝑘Δ𝑧^𝑘. (24) From Lemma 3 we can know that 𝜃(⋅) is continuously differentiable around 𝑧^𝑘. Hence, (24) implies that 𝜔(𝛼)=𝑜(𝛼). (25) So, for any 𝛼∈(0,1], we obtain from (19), (24), and (25) that 𝜃𝑧^𝑘+𝛼Δ𝑧^𝑘−𝜃𝑧^𝑘=2𝛼𝐻𝑧^𝑘𝑇𝐻^′𝑧^𝑘Δ𝑧^𝑘(𝛼)𝑧+𝜔=2𝛼𝐻^𝑘𝑇𝜌_𝑘𝜇𝑒^𝜇𝑘−𝐻𝑧^𝑘+𝑜(𝛼)=2𝛼𝑒^𝜇𝑘𝜌_𝑘𝐻𝑧^𝑘𝑇𝜇−2𝛼𝐻𝑧^𝑘𝑇𝐻𝑧^𝑘+𝑜(𝛼)=2𝛼𝑒^𝜇𝑘𝜌_𝑘⋅𝜇₀𝑒^𝜇𝑘−1−2𝛼𝜃𝑧^𝑘+𝑜(𝛼)=2𝛼𝜇₀𝜌_𝑘𝑒^𝜇𝑘−1²+2𝛼𝜇₀𝜌_𝑘𝑒^𝜇𝑘𝑧−1−2𝛼𝜃^𝑘+𝑜(𝛼)≤2𝛼𝜇₀𝜌_𝑘𝜃𝑧^𝑘+2𝛼𝛾𝜇₀𝜃𝑧^𝑘−2𝛼𝜃𝑧^𝑘(𝛼)≤+𝑜4𝛼𝛾𝜇₀−2𝛼𝜃𝑧^𝑘+𝑜(𝛼)=−21−2𝛾𝜇₀𝛼𝜃𝑧^𝑘+𝑜(𝛼), (26) where the first inequality follows from the fact that 𝑒^𝜇𝑘−1≤‖𝐻(𝑧^𝑘)‖,𝜌_𝑘≤𝛾‖𝐻(𝑧^𝑘)‖ and the second inequality follows from the fact that 𝜌_𝑘≤𝛾. This indicates that Step 4 of Algorithm 4 is well defined at the 𝑘th iteration.
Now we prove the second part of conclusion; that is, 𝑧^𝑘∈Ω for all 𝑘. We prove this result also by mathematical induction on 𝑘.
Obviously, from the definition of 𝜌(⋅) and the choice of 𝛾, it holds that 𝜌(𝑧⁰)≤𝛾<1; that is, 𝜇₀≥𝜌₀𝜇₀. So, 𝑧⁰∈Ω. Suppose that 𝑧^𝑘∈Ω; that is, 𝜇_𝑘≥𝜌_𝑘𝜇₀. Then, by (22), we have 𝜇_𝑘+1−𝜌_𝑘+1𝜇₀=𝜇_𝑘+𝜏_𝑘Δ𝜇_𝑘−𝜌_𝑘+1𝜇₀>1−𝜏_𝑘𝜇_𝑘+𝜏_𝑘𝜌_𝑘𝜇₀−𝜌_𝑘+1𝜇₀≥1−𝜏_𝑘𝜌_𝑘𝜇₀+𝜏_𝑘𝜌_𝑘𝜇₀−𝜌_𝑘+1𝜇₀=𝜇₀𝜌_𝑘−𝜌_𝑘+1. (27) By Step 4 and the definition of 𝜌(⋅), we obtain 𝜃𝑧^𝑘+1≤𝜃𝑧^𝑘,𝜌_𝑘+1≤𝛾,𝜌_𝑘+1≤𝛾𝜃𝑧^𝑘+1. (28) So, if ‖𝐻(𝑧^𝑘)‖≤1, then ‖𝐻(𝑧^𝑘+1)‖≤‖𝐻(𝑧^𝑘)‖≤1, and hence 𝜌_𝑘+1=𝛾𝜃𝑧^𝑘+1≤𝛾𝜃𝑧^𝑘=𝜌_𝑘. (29) On the other hand, if ‖𝐻(𝑧^𝑘)‖>1, then 𝜌_𝑘=𝛾≥𝜌_𝑘+1. (30) Thus, from (27), we complete the proof.

4. Global and Superlinear Convergence

Theorem 6. Assume that the sequence {𝑧^𝑘} is generated by Algorithm 4; then any accumulation point of {𝑧^𝑘} is a solution of (9).

Proof. Without loss of generality, we assume that {𝑧^𝑘} converges to 𝑧^∗=(𝜇^∗,𝑥^∗,𝜆^∗₁,𝜆^∗₂). It follows from (20) that {𝜃(𝑧^𝑘)} is monotonically decreasing. We assume that {𝜃(𝑧^𝑘)} converges to 𝜃(𝑧^∗). Then, to prove the result, it is sufficient to prove that 𝜃(𝑧^∗)=0, which will be proved by reduction to absurdity. Assume that 𝜃(𝑧^∗)>0; then we obtain 𝜌^∗∶=𝜌𝑧^∗=𝛾min1,𝐻𝑧^∗,𝜃𝑧^∗>0,0<𝜌^∗𝜇₀≤𝜇^∗≤𝜇₀,lim_𝑘→∞𝜆_𝑘=0. (31) On the one hand, from Step 4 of Algorithm 4, it holds that 𝜃𝑧^𝑘+𝛿^𝑚𝑘−1Δ𝑧^𝑘−𝜃𝑧^𝑘>−2𝜎1−2𝛾𝜇₀𝛿^𝑚𝑘−1𝜃𝑧^𝑘. (32) That is, 𝜃𝑧^𝑘+𝛿^𝑚𝑘−1Δ𝑧^𝑘−𝜃𝑧^𝑘2𝛿^𝑚𝑘−1>−𝜎1−2𝛾𝜇₀𝜃𝑧^𝑘. (33) Let 𝑘→∞; we obtain 𝐻𝑧^∗𝑇𝐻^′𝑧^∗Δ𝑧^∗≥−𝜎1−2𝛾𝜇₀𝜃𝑧^∗. (34) On the other hand, by (19) and the definitions of 𝜌, 𝐻, and 𝜃, we have that 𝐻𝑧^∗𝑇𝐻^′𝑧^∗Δ𝑧^∗=−𝐻𝑧^∗𝑇𝐻𝑧^∗+𝜌^∗𝑒^𝜇∗𝐻𝑧^∗𝑇𝜇=−𝜃𝑧^∗+𝜌^∗𝑒^𝜇∗𝑒^𝜇∗−1𝜇₀=−𝜃𝑧^∗+𝜌^∗𝜇₀𝑒^𝜇∗−1²+𝜌^∗𝜇₀𝑒^𝜇∗𝑧−1≤−𝜃^∗+𝜌^∗𝜇₀𝐻𝑧^∗2+𝜌^∗𝜇₀𝐻𝑧^∗≤−𝜃𝑧^∗+2𝛾𝜇₀𝜃𝑧^∗=−1−2𝛾𝜇₀𝜃𝑧^∗. (35) Combining (34) and (35), we deduced that −1−2𝛾𝜇₀𝜃𝑧^∗≥−𝜎1−2𝛾𝜇₀𝜃𝑧^∗. (36) From the assumption that 𝜃(𝑧^∗)>0, it is implied that (1−𝜎)1−2𝛾𝜇₀≤0, (37) which contradicts the fact that 𝜎<1,2𝛾𝜇₀<1. Thus, we have 𝜃(𝑧^∗)=0; that is, 𝐻(𝑧^∗)=0. And this completes the proof.

Assumption 7. The solution set 𝑆={𝑥∈𝑅^𝑛∣𝐹(𝑥)∈𝒦,𝐺(𝑥)∈𝒦^∘,𝐹(𝑥)^⊤𝐺(𝑥)=0} of GCP is nonempty and bounded.

Theorem 8. Suppose that Assumption 7 is satisfied and 𝑧^∗=(𝜇^∗,𝑥^∗,𝜆^∗₁,𝜆^∗₂) is an accumulation point of the iteration sequence {𝑧^𝑘} generated by Algorithm 4. If all 𝑉∈𝜕𝐻(𝑧^∗) are nonsingular, then(i)𝜏_𝑘≡1, for all 𝑧^𝑘 sufficiently close to 𝑧^∗;(ii)the whole sequence {𝑧^𝑘} converges to 𝑧^∗; that is, lim_𝑘→∞𝑧^𝑘=𝑧^∗;(iii){𝑧^𝑘} converges to 𝑧^∗ superlinearly; that is, ‖𝑧^𝑘+1−𝑧^∗‖=𝑜‖𝑧^𝑘−𝑧^∗‖. Moreover, 𝜇_𝑘+1=𝑜(𝜇_𝑘).

Proof. It follows from Theorem 6 that 𝐻(𝑧^∗)=0 and 𝑥^∗∈𝑆. Because all 𝑉∈𝜕𝐻(𝑧^∗) are nonsingular, it follows from Proposition  3.1 in [11] that, for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we have 𝐻^′𝑧^𝑘−1≤𝐶, (38) where 𝐶>0 is some constant. By Theorem  19 in [12], it is easy to see from Lemma 3 that 𝐻(⋅) is semismooth. Hence, for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we get 𝐻𝑧^𝑘−𝐻𝑧^∗−𝐻^′𝑧^𝑘𝑧^𝑘−𝑧^∗=𝑜𝑧^𝑘−𝑧^∗. (39) On the other hand, the fact that 𝐻(⋅) is semismooth implies that 𝐻(⋅) is locally Lipschitz continuous near 𝑧^∗. Therefore, for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we have 𝐻𝑧^𝑘2=𝐻𝑧^𝑘−𝐻𝑧^∗2=𝑂𝑧^𝑘−𝑧^∗2. (40) Thus, we obtain from (40) and the definition of 𝜌(⋅) that 𝜌_𝑘𝜇₀≤𝛾𝜇₀𝜃𝑧^𝑘=𝑂𝐻𝑧^𝑘2=𝑂𝑧^𝑘−𝑧^∗2. (41) Then, by (38), (39), and (41), we have 𝑧^𝑘+Δ𝑧^𝑘−𝑧^∗=𝑧^𝑘+𝐻^′𝑧^𝑘−1−𝐻𝑧^𝑘+𝜌_𝑘𝜇𝑒^𝜇𝑘−𝑧^∗≤𝐻^′𝑧^𝑘−1𝐻𝑧^𝑘−𝐻𝑧^∗−𝐻^′𝑧^𝑘𝑧^𝑘−𝑧^∗+𝜌_𝑘𝜇₀𝑒^𝜇𝑘=𝑜𝑧^𝑘−𝑧^∗. (42) Similar to the proof of Theorem  3.1 in [13], for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we get 𝑧^𝑘−𝑧^∗=𝑂𝐻𝑧^𝑘−𝐻𝑧^∗. (43) Then, combining this with (42), since 𝐻(⋅) is semismooth at 𝑧^∗, for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we have 𝐻𝑧^𝑘+Δ𝑧^𝑘=𝑂𝑧^𝑘+Δ𝑧^𝑘−𝑧^∗=𝑜𝑧^𝑘−𝑧^∗=𝑜𝐻𝑧^𝑘−𝐻𝑧^∗=𝑜𝐻𝑧^𝑘. (44) By Theorem 6, lim_𝑘→∞‖𝐻(𝑧^𝑘)‖=0. Hence, (44) implies that when 𝑧^𝑘 is sufficiently close to 𝑧^∗, 𝜏_𝑘=1 can satisfy (20), which proves (i). Thus, for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we have 𝑧^𝑘+1=𝑧^𝑘+Δ𝑧^𝑘, (45) which together with (42) proves (ii) and 𝑧^𝑘+1−𝑧^∗=𝑜𝑧^𝑘−𝑧^∗. (46) Next, from (i), (ii), and (41), we obtain for all sufficiently large 𝑘 that 𝜇_𝑘+1=𝜇_𝑘+Δ𝜇_𝑘=𝜇_𝑘+𝑒^−𝜇𝑘−1+𝜌_𝑘𝜇₀=𝑂𝜇²_𝑘+𝑂𝐻𝑧^𝑘2, (47) where the third equality follows from the fact 𝑒^−𝜇−1+𝜇=𝑜(𝜇²). Therefore, for all 𝑧^𝑘 sufficiently close to 𝑧^∗, we obtain 𝜇_𝑘+1=𝑜𝜇_𝑘, (48) which completes the whole proof.

5. Conclusion

For the generalized complementarity problem over a polyhedral cone, we formulated it as a system of equations via modified smoothing NCP-function based on the Fischer-Burmeister function. Then we presented a Newton-type method to solve the equivalent system of equations. The proposed method is shown to be globally and superlinearly convergent under mild assumptions. Furthermore, our method solves only one linear system of equations and performs only one line search at each iteration. Compared to the previous literatures, our method has stronger convergence results under weaker conditions.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 11171180, 11271226, and 61100028, the Natural Science Foundation of Shandong Province under Grant ZR2012AL07, and by the Fundamental Research Funds for the Central Universities under Grant N110404017.