^{1}

^{*}

^{1}

^{1}

This paper considers the computation of sparse solutions of the linear complementarity problems LCP(<i>q</i>, <i>M</i>). Mathematically, the underlying model is NP-hard in general. Thus an <i>lp</i>(0 <<i> p </i>< 1) regularized minimization model is proposed for relaxation. We establish the equivalent unconstrained minimization reformation of the NCP-function. Based on the generalized Fiser-Burmeister function, a sequential smoothing spectral gradient method is proposed to solve the equivalent problem. Numerical results are given to show the efficiency of the proposed method.

Given a matrix

The set of solutions to this problem is denoted by

In this paper, we consider the sparse solutions of the LCP. We call

To be more precise, we seek a vector _{0} norm minimization problem, where

Recently, Meijuan Shang, Chao Zhang and Naihua Xiu design a sequential smoothing gradient method to solve the sparse solution of LCP [

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: R^{2} → R^{1} 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 [

where p is any fixed real number from

In other words, in the function

Define

where _{p} regularization term for seeking sparsity, we obtain the following unconstrained minimization problem to approximate

where _{p} _{}

regularized minimization problem.

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

For any given

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 [_{p} minimization problem (3).

The following result is given in [

Lemma 2.1. [

of (3), and

In this section, we extend the above result to the l_{p} norm regularization model (2) for approximating minimal l_{0}_{ }norm solutions of the LCP. We provide a threshold lower bound L > 0 for any local minimizer, and show

that any nonzero entries of local minimizers must exceed L. Since

is bound below and

Lemma 2.2. [

Then we have: for any

Moreover, the number of nonzero entries in

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

First we present some properties of

Lemma 2.3. [

1)

2)

3) If

when

4) Given a point

sgn(.) represents the sign function;

Theorem 2.1. The function

where

en by

where

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

p < 1) in (3) is neither convex nor Lipschitz continuous in

For

It is clear to see that, for any

and

We can construct a smoothing approximation of (2) as

by noting that

for any

Let

Theorem 3.1. Let

Then, any accumulation point of

Proof. Let

We can deduce from (11) that

On the other hand, we have

Which indicates

Lemma 3.2. [

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 [

We first introduce the spectral projected gradient method in [

Algorithm 1. Smoothing Spectral Gradient Method

Step 0: Choose an initial point

Step1: Let

Step 2: Compute the step size

Set

Step 3: If

Algorithm 2. Sequential Smoothing Spectral Gradient Method

Step 1: Find

Step 2: Compute

Use the lower bound

Step 3: Decrease the parameter

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 [_{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

Example 1. We consider the LCP(q, M) with

The solution set is

When

Example 2. We consider the LCP(q, M) with

The solution set is

When

These examples show that, given the proper initial point, our algorithm can effectively find an approximate sparse solution.

Let us consider LCP(q, M) where

Here

Among all the solutions, the vector

In

In order to test the effectiveness of the SS-SG algorithm, we compare with the SSG algorithm of talking the LCPs. In [_{p} (0 < p < 1) regularized minimization model and designed a SSG method to solve the LCPs. The results are displayed in

100 | 789 | 2.71E−3 | 1 | 1 | 2.25 |

200 | 452 | 5.22E−3 | 1 | 1 | 2.27 |

500 | 14 | 3.91E−4 | 1 | 1 | 4.02 |

800 | 11 | 4.21E−4 | 1 | 1 | 4.73 |

1000 | 3 | 1.64E−5 | 1 | 1 | 4.73 |

1300 | 29 | 2.16E−5 | 1 | 1 | 25.41 |

100 | 2184 | 3.41E−3 | 1 | 1 | 9.66 |

200 | 549 | 4.20E−3 | 1 | 1 | 17.44 |

500 | 18 | 5.11E−3 | 1 | 1 | 28.77 |

800 | 9 | 2.23E−3 | 1 | 1 | 63.01 |

1000 | 3 | 1.24E−4 | 1 | 1 | 4.13 |

1300 | _ | _ | _ | _ | _ |

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