Given a matrix and an n-dimensional vector q, the linear complementarity problem, denoted by LCP(q, M), is to find a vector such that

.

The set of solutions to this problem is denoted by. Throughout the paper, we always suppose. The LCP has many wide applications in physics, engineering, mechanics, and economics design [1] [2] . Numerical methods for solving LCPs, such as the Newton method, the interior point method and the non-smooth equation methods, have been extensively investigated in the literature. However, it seems that there are few methods to solve the sparse solutions for LCPs. In fact, it is very necessary to research the sparse solution of the LCPs such as portfolio selection [3] [4] and bimatrix games [5] in real applications.

In this paper, we consider the sparse solutions of the LCP. We call a sparse solution of LCP (q, M) if is a solution of the following optimization problem

To be more precise, we seek a vector by solving the l₀ norm minimization problem, where stands for the number of nonzero components of x. A solution of (1) is called the sparsest solution of the LCP.

Recently, Meijuan Shang, Chao Zhang and Naihua Xiu design a sequential smoothing gradient method to solve the sparse solution of LCP [6] . We inspire by the model and use the spectral method based on the generalized Fischer-Burmeister function to solve our new model (3). The spectral method is proposed by Barzilai and Borwein [7] and further analyzed by Raydan [8] [9] . The advantage of this method is that it requires little computational work and greatly speeds up the convergence of gradient methods. Therefore, this technique has received successful applications in unconstrained and constrained optimizations [10] -[13] .

In fact, the above minimization problem (1) is a sparse optimization with equilibrium constraints. From the problem of constraint conditions, as well as the non-smooth objective function, it is difficult to get solutions due to the equilibrium constraints to overcome the difficultly, and we use the NCP-functions to construct the penalty of violating the equilibrium constraints.

A function φ: R² → R¹ is called a NCP-function, if for any pair.

A popular NCP-functions is the Fischer-Burmeister (FB), which is defined as

The Fischer-Burmeister function has many interesting properties. However, it has limitations in dealing with monotone complementarity problems since it is too flat in the positive orthant, the region of main interest for a complementarity problem. In terms of the above disadvantage of the Fischer-Burmeister function, we consider the following generalized Fischer-Burmeister function [10] .

where p is any fixed real number from and denotes the p-norm, i.e.

In other words, in the function, we replace the 2-norm of in the FB function by a more general p-norm of. The function is still an NCP-function.

Define by

where with and. Obviously, if and only if. By further employing the l_p regularization term for seeking sparsity, we obtain the following unconstrained minimization problem to approximate

where is a given regularization parameter, and for any 0 < p < 1. We call (3) as l_p

regularized minimization problem.

Let us denote the first term of (3) by the function. That is

For any given, the function is shown to possess all favorable properties of the FB function; we can see [8] . It plays an important part in our study throughout the paper. We observe in [14] [15] that P has a great influence on the numerical performance of certain descent-type methods; a larger P yields a better convergence rate, whereas a small P often gives a better global convergence.

The paper is organized as follows: In Section 2, we present absolute lower bounds for nonzero entries in local solution of (3). In section 3, we approximate the minimal zero norm solutions of the LCP. In section 4, we give a sequential smoothing spectral gradient method to solve the model. In Section 5, numerical results are given to demonstrate the effectiveness of the sequential smoothing spectral gradient method.

In this section, we consider the minimizers of (3). We study the relation between the original model (1) and the l_p regularized model (3), which indicates the regularized model is a good approximation. We use a threshold lower bound L [6] for nonzero entries in local minimizers and the choice of the l_p minimization problem (3).

Most optimization algorithms are efficient only for convex and smooth problems. However, some algorithms for

Non-smooth and non-convex optimization problems have been developed recently. Note that the term (0 <

p < 1) in (3) is neither convex nor Lipschitz continuous in. Solving the non-convex, non-Lipschitz continuous minimization problem is not easy. We use some approximation methods to surmount the non-Lipschitz continuity problem in solving (3).

For, let

It is clear to see that, for any,

and is continuously differentiable with

We can construct a smoothing approximation of (2) as

by noting that is continuously differentiable, and since

for any.

Let denote the set of local minimizers of (10). We have the similar results as in Lemmma 2.1 and 2.2, corresponding to the smoothing counterpart (10).

Theorem 3.1. Let be a sequence of vectors being global minimizers of (10) with as.

Then, any accumulation point of is a global minimizer of (3).

Proof. Let be a global minimizer of (3) and be an accumulation point of.

We can deduce from (11) that

On the other hand, we have, and consequently

Which indicates is a global minimizer of (3).

Lemma 3.2. [6] for any, let be any local minimizer of (14) satisfying for an arbitrarily given initial points. Let L be defined in Lemma 2.2. Then, we have for any

We suggest a sequential Smoothing Spectral Gradient (SS-SG) Method to solve (3). With the SS-SG method, we need the Spectral Gradient method as the main step for decreasing the objective value. The smoothing method is very easy to implement and efficient to deal with optimization; see [15] .

We first introduce the spectral projected gradient method in [8] as follow.

Algorithm 1. Smoothing Spectral Gradient Method

Step 0: Choose an initial point, and parameters, ,. Let, ,.

Step1: Let, , where,. If

, then stop.

Step 2: Compute the step size by the Armijo line search, where satisfies

Set.

Step 3: If, then set; otherwise, choose.

Algorithm 2. Sequential Smoothing Spectral Gradient Method

Step 1: Find by using the algorithm 1 to solve

Step 2: Compute

Use the lower bound to set the entries of with small values to zeros and obtain the computed solution with

Step 3: Decrease the parameter and set.

In this section, we test some numerical experiments to demonstrate the effectiveness of our SG algorithm. In order to illustrating the effectiveness of the SS-SG algorithm we proposed, we introduce another algorithm of talking the LCPs. In [6] , the authors designed a sequential smoothing (SSG) method to solve the l_p regularized model and get a sparse solution of LCP(q, M). Numerical experiments show that our algorithm is more effective than (SSG) algorithm.

The program code was written in and run in MATLAB R2013 an environment. The parameters are chooses as, and. The maximum number of iterations in step 1 is set to be 2000. We end the SS-SG algorithm in Step 1, if and, or it reaches the maximum number of iterations.

In this paper, we have studied a l_p (0 < p < 1) model based on the generalized FB function defined as in (2) to find the sparsest solution of LCPs. Then, an l_p normregularized and unconstrained minimization model is proposed for relaxation, and we use a sequential smoothing spectral gradient method to solve the model. Numerical results demonstrate that the method can efficiently solve this regularized model and gets a sparsest solution of LCP with high quality.

This work is supported by Innovation Programming of Shanghai Municipal Education Commission (No. 14YZ094).

ChangGao,ZhenshengYu,FeiranWang, (2015) Spectral Gradient Algorithm Based on the Generalized Fiser-Burmeister Function for Sparse Solutions of LCPS. Open Journal of Statistics,05,543-551. doi: 10.4236/ojs.2015.56057