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

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.


Introduction
The generalized complementarity problem, denoted by GCP, is to find a vector  * ∈   such that where  and  are continuous functions from   to   , K is a nonempty closed convex cone in   , 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][2][3] and references therein.In particular, if K =   + and () = , 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  * is a solution of GCP if and only if ( * ) is a solution of ( ∘  −1 , K) if  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 =   + .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  = ,  and  are both continuously differentiable on   , and K is a polyhedral cone in   ; that is, there exist  ∈  × ,  ∈  × such that where  and  are both positive integers.It is easy to verify that its polar cone K ∘ has the following representation: Obviously, if  is an identity matrix and  = 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  and  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 ,  ∈   is denoted by   .For convenience, vector (  ,   )  is denoted by (, ), where ,  are both column vectors.Let ‖ ⋅ ‖ denote the Euclidean norm of a vector or a matrix.For vector  ∈   ,   = diag() denotes the diagonal matrix with the th diagonal element being   .Vector  denotes the vector of all ones whose dimension is defined by the context of its use.

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]: where  > 0 is a smoothing parameter.By simple calculation, we have The smoothing NCP-function possesses a few nice properties [10].

Theorem 2. 𝑥 is a solution of the 𝐺𝐶𝑃 if and only if there exist
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 () has the following form: where And for the functions  and Φ, they have the following nice properties.Lemma 3. Let  and Φ be defined by (7) and (8), respectively.Then with its Jacobian   () being defined by (10).If matrix  has full row rank and Proof.From Lemma 1, it is not difficult to see that Φ is continuously differentiable at any  ∈  ++ ×  ++ .Next we prove (ii).It follows from the definition of  and (i) that  is continuously differentiable on  ++ ×  ++ .And by simple calculation, we obtain the Jacobian   () defined by (10).Now we prove the nonsingularity of   ().
Since  > 0, it follows that   > 1.And thus we know that to prove the nonsingularity of   (), it is sufficient to prove the nonsingularity of  := ( Since   () is nonsingular, we make the following transformation to  without changing its rank: ) . ( Since   is nonsingular, to prove the nonsingularity of , it suffices to show the nonsingularity of the following submatrix of it: Let where  1 ∈   ,  2 ∈   .Then we obtain, That is, Since   ()  () −1 is positive definite and Noting that (  ()  () −1 +   −1    ) −1 is positive definite and  has full row rank, we obtain  2 = 0, and thus  1 = 0. Hence, we complete the proof.

Algorithm and Preliminaries
In this section we propose our smoothing Newton method for the GCP.
Algorithm 4. Consider the following.
The following theorem shows that Algorithm 4 is well defined and generates an infinite sequence with some good features.Define the set Theorem 5. Let  0 = ( 0 ,  0 ,  0 1 ,  0 2 ) ∈  ++ ×  ++ be given in Algorithm 4. Then Algorithm 4 is well defined and generates an infinite sequence {  } with   ∈  ++ and   ∈ Ω for all .
Proof .We firstly prove that Algorithm 4 is well defined.Obviously, if   > 0 for all , it follows from Lemma 3(ii) that the matrix   (  ) is nonsingular.So, Step 3 of Algorithm 4 is well defined at the th iteration.Now we prove that   > 0 by mathematical induction on .
By Step 3, we have where the last inequality follows from  > 1− − for all  > 0.
Therefore, for any  ∈ (0, 1] we obtain that which means that, for   ∈ (0, 1),  +1 =   +   Δ  > 0. And thus we prove the desired result.Next we prove that Step 4 is well defined.Let From Lemma 3 we can know that (⋅) is continuously differentiable around   .Hence, (24) implies that

𝜔 (𝛼) = 𝑜 (𝛼) . (25)
So, for any  ∈ (0, 1], we obtain from ( 19), (24), and (25) that where the first inequality follows from the fact that    − 1 ≤ ‖(  )‖,   ≤ ‖(  )‖ and the second inequality follows from the fact that   ≤ .This indicates that Step 4 of Algorithm 4 is well defined at the th iteration.Now we prove the second part of conclusion; that is,   ∈ Ω for all .We prove this result also by mathematical induction on .

Global and Superlinear Convergence
Theorem 6. Assume that the sequence {  } is generated by Algorithm 4; then any accumulation point of {  } is a solution of (9).
where  > 0 is some constant.By Theorem 19 in [12], it is easy to see from Lemma where the third equality follows from the fact  − − 1 +  = ( 2 ).Therefore, for all   sufficiently close to  * , we obtain which completes the whole proof.

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.