Novel Global Harmony Search Algorithm for General Linear Complementarity Problem

Abstract

Linear complementarity problem (LCP) is studied. After reforming general LCP as the system of nonlinear equations by NCP-function, LCP is equivalent to solving an unconstrained optimization model, which can be solved by a recently proposed algorithm named novel global harmony search (NGHS). NGHS algorithm can overcome the disadvantage of interior-point methods. Numerical results show that the NGHS algorithm has a higher rate of convergence than the other HS variants. For LCP with a unique solution, NGHS converges to its unique solution. For LCP with multiple solutions, NGHS can find as many solutions as possible. Meanwhile, for unsolvable LCP, all algorithms are terminated on the solution with the minimum error.

1. Introduction

Consider the general linear complementarity problem (LCP):

$$\mathit{x}\ge \mathbf{0},\mathit{M}\mathit{x}+\mathit{q}\ge \mathbf{0},{\mathit{x}}^T\left(\mathit{M}\mathit{x}+\mathit{q}\right)=0,$$

(1)

where $\mathit{M}\in {\mathit{R}}^{n\times n},\mathit{q}\in {\mathit{R}}^n$ are given. LCP is a fundamental problem in mathematical programming. It is known that any differentiable linear and quadratic programming can be formulated into an LCP. LCP also has a wide range of applications in economics and engineering [1].

Many direct methods have been proposed for their solutions. The most famous among the pivotal methods for the LCP is Lemke’s method. The book by Cottle et al. [2] is a good reference for pivoting methods. Interior-point methods (IPMs) are an important method for LCP. Modern interior-point methods were introduced by Karmarkar in 1984 for linear programming [3]. Kojima et al. [4] proposed a polynomial-time algorithm for monotone LCP under the nonemptiness assumption of the set of feasible interior-point. Each algorithm in the class of interior-point methods for monotone LCP has a common feature that generates a sequence in the positive orthant of $R^n$ under the assumption of knowing a feasible initial point. However, it is a very difficult task to find a feasible initial point to start the interior-point methods. To overcome this difficulty, recent studies have focused on some new interior-point algorithms without the need to set a feasible initial point. In 1993, Kojima et al. presented the first infeasible interior-point algorithm with global convergence [5]; soon after, Zhang et al. [6,7] introduced this technique to the linear complementarity problem.

The study of LCP mainly concentrated on theory and algorithm: the former focused on the existence and uniqueness of solution [8,9,10,11], and the latter mainly designed efficient algorithms for LCP, such as interior-point algorithm, smooth function method, neural network method, matrix splitting iteration method, kernel function method, etc. [12,13,14,15,16,17]. In recent years, stochastic LCP [18], symmetry cone complementary [19], tensor complementary problem [20,21,22], parameter LCP [23,24], vertical linear complementary [25], and the sparse solution [26,27] of LCP have become research hotspots. Although most methods have polynomial-time complexity for monotone LCP with a unique solution after setting a feasible initial point properly, they cannot apply to nonmonotone LCP. Moreover, for the LCP with multiple solutions, how to obtain as many solutions as possible is rarely studied.

Based on the above reasons, in this paper, we use a recently proposed algorithm named novel global harmony search (NGHS) for solving general linear complementarity problems, including LCP with a unique solution or multiple solutions and the unsolvable LCP. We reform the LCP as an unconstrained optimization model, and then we use the NGHS algorithm to solve. If LCP is solvable (unique solution or multiple solutions), the objective function (merit function) converges to zero. If LCP is unsolvable, the objective function (merit function) converges to the solution with the minimum error. Thus, this method can also be used to test whether LCP has a solution. Numerical results, compared with the classical HS and HS variants, show that the NGHS method has good convergence property.

This paper is outlined as follows. In Section 2, we give some lemmas that ensure the solution to LCP (1) exists and reform the LCP as an unconstrained optimization model. In Section 3, the NGHS is discussed and compared with other HS variants. Numerical results and comparisons are provided in Section 4 by solving some given LCPs. Section 5 contains the concluding remarks.

2. Preliminaries and LCP as Optimization Model

We briefly summarize some of the major results in existence and uniqueness of the solution for standard LCP. This research has identified a wide variety of classes of square matrices that correspond to certain properties related to the LCP [28].

In 1958, Samelson, Thrall, and Wesler established that $\mathit{M}$ must be a P-matrix, i.e., a square matrix all of whose principal subdeterminants are greater than 0. For the LCP (1), this result leads to the theorem that (1) has a unique solution for all $\mathit{q}\in R^n$ iff $\mathit{M}$ is a P-matrix. For convenience, a positive (semi)definite matrix is also applied to the existence of a solution for standard LCP.

Definition 1.

The matrix $\mathit{M}$ is positive semidefinite, i.e., ${\mathit{d}}^T\mathit{M}\mathit{d}\ge 0$ for every $\mathit{d}\in R^n$.

Definition 2.

The matrix $\mathit{M}$ is positive definite, i.e., ${\mathit{d}}^T\mathit{M}\mathit{d}>0$ for every $\mathit{d}\in R^n,\mathit{d}\ne \mathbf{0}$

Let $\mathit{A}\in R^{n\times n}$. Splitting $\mathit{A}=\mathit{H}+\mathit{S}$, where $\mathit{H}=\frac{1}{2}(\mathit{A}+{\mathit{A}}^T)$, $\mathit{S}=\frac{1}{2}(\mathit{A}-{\mathit{A}}^T)$. Since ${\mathit{H}}^T=\mathit{H},{\mathit{S}}^T=-\mathit{S}$, we call $\mathit{A}=\mathit{H}+\mathit{S}$ Hermitian/skew-Hermitian splitting [29,30] and $\mathit{H}$ symmetry component matrix. Thus for any $\mathit{A}\in R^{n\times n}$, if ${\mathit{A}}^T=\mathit{A}$ and eigenvalues of matrix $\mathit{A}$ are all greater than (or equal to) 0, $\mathit{A}$ is positive (semi)definite matrix. If ${\mathit{A}}^T\ne \mathit{A}$, $\mathit{A}$ is positive (semi)definite if and only if $\mathit{H}$ is positive (semi)definite.

Lemma 1.

When $\mathit{M}$ is a positive definite matrix, LCP has a unique solution.

Lemma 2.

When $\mathit{M}$ is a positive semidefinite matrix, the solution set is nonempty and convex if the feasible set is nonempty.

Lemmas 1 and 2 are only sufficient conditions, but not necessary conditions. The LCP is called monotone if $\mathit{M}$ is a positive semidefinite.

Let $\varphi :R^2\to R^1$ be defined by

$$\varphi (a,b)=\sqrt{a^2+b^2}-(a+b)$$

(2)

Then

$$\varphi (a,b)=0\iff a\ge 0,b\ge 0,ab=0$$

(3)

From the characterization of NCP-function [31], LCP (1) can be recast as a system of nonlinear equations defined by

$$\mathsf{\Phi }(\mathit{x})\triangleq \left(\begin{array}{c}\begin{array}{c}\varphi (x_1,f_1(\mathit{x}))\\ \varphi (x_2,f_2(\mathit{x}))\end{array}\\ ⋮\\ \varphi (x_{n-1},f_{n-1}(\mathit{x}))\\ \varphi (x_n,f_n(\mathit{x}))\end{array}\right)=\mathbf{0}$$

where $f_i(\mathit{x})={(\mathit{M}\mathit{x}+\mathit{q})}_i,i=1,2,\cdots ,n.$ Define a merit function (or object function)

$${‖\mathsf{\Phi }(\mathit{x})‖}^2\triangleq {\displaystyle \sum _{i=1}^n{\left[\varphi (x_i,f_i(\mathit{x}))\right]}^2},$$

so (1) is equivalent to solving the unconstrained optimization model

$$\underset{x\in R^n}{{\displaystyle \min }}{‖\mathsf{\Phi }(\mathit{x})‖}^2\triangleq {\displaystyle \sum _{i=1}^n{\left[\varphi (x_i,f_i(\mathit{x}))\right]}^2}$$

(4)

If LCP is solvable (unique solution or multiple solutions), ${‖\mathsf{\Phi }(\mathit{x})‖}^2$ converges to zero. If LCP is unsolvable, ${‖\mathsf{\Phi }(\mathit{x})‖}^2$ converges to the solution with the minimum error.

Following, we use a recently proposed algorithm named novel global harmony search (NGHS) for solving some given LCPs.

3. NGHS Algorithm

3.1. HS Algorithm

Traditional optimization methods, such as interior-point methods, smooth function methods, neural network methods, and matrix splitting iteration methods, have taken on major roles in solving real world optimization problems. Nevertheless, their common drawbacks, requiring an initial setting in feasible region for decision variables, generate a demand for other types of algorithms, such as metaheuristic algorithms. These algorithms have been identified as efficient, intelligent techniques to solve complex real-world problems. In recent years, various metaheuristic algorithms have been proposed to solve a wide range of complex real-world problems.

The harmony search (HS) algorithm is one of the metaheuristic algorithms developed by Geem et al. [32,33]. Inspired by the improvisation of music players, they first play music randomly with existing instruments. This harmony is kept in the musician’s memory. In the next stage, according to the harmony in their memory, the musician plays new music that has changed from the previous one. The concept of HS is based on the idea that during the improvisation process, musicians try different combinations of memorized pitches, which is analogous to the optimization process applied to most engineering problems. Therefore, a feasible solution is called a harmony, and each decision variable corresponds to a note, which generates a value for finding the global optimum.

HS algorithm possesses several advantages over the traditional optimization techniques: (1) it is a simple metaheuristic algorithm that does not require an initial setting for decision variables; (2) it uses stochastic random searches, so derivative information is not necessary; and (3) it has a few parameters. For these reasons, HS has been demonstrated in several studies to be a useful optimization algorithm for various engineering applications owing to its convenient implementation, rapid convergence, and reasonable computational cost. This can be found in the literature [34,35,36,37,38,39,40].

The steps in the procedure of the classical harmony search algorithm are as follows:

Step 1. Initialize the problem and algorithm parameters. The optimization problem is specified as follows:

$$\mathrm{Minimize} f(\mathit{x}) \mathrm{subject} \mathrm{to} x_i\in X_i,i=1,2,\cdots ,n,$$

where $f(\mathit{x})$ is an objective function; $\mathit{x}$ is the set of each decision variable $x_i$; $n$ is the number of decision variables; $X_i$ is the set of the possible range of values (named search space) for each decision variable; and $X_i:x_i{}^L\le X_i\le x_i{}^U$. The HS algorithm parameters are also specified in this step. These are the harmony memory size (HMS), or the number of solution vectors in the harmony memory; harmony memory considering rate (HMCR); pitch adjusting rate (PAR); and the number of improvisations (Tmax), or stopping criterion.

Step 2. Initialize the harmony memory. The HM matrix is filled with as many randomly generated solution vectors as the HMS:

$$\mathrm{HM}=\left[\left.\begin{array}{c}{\mathit{x}}^1\\ {\mathit{x}}^2\\ ⋮\\ {\mathit{x}}^{\mathrm{HMS}}\end{array}\right|\begin{array}{c}f({\mathit{x}}^1)\\ f({\mathit{x}}^2)\\ ⋮\\ f({\mathit{x}}^{\mathrm{HMS}})\end{array}\right]=\left[\left.\begin{array}{cccc}{x}_1^1& x_2^1& \cdots & x_n^1\\ x_1^2& x_2^2& \cdots & x_n^2\\ ⋮& ⋮& \cdots & ⋮\\ x_1^{\mathrm{HMS}}& x_2^{\mathrm{HMS}}& \cdots & x_n^{\mathrm{HMS}}\end{array}\right|\begin{array}{c}f({\mathit{x}}^1)\\ f({\mathit{x}}^2)\\ ⋮\\ f({\mathit{x}}^{\mathrm{HMS}})\end{array}\right]$$

Step 3. Improvise a new harmony. Generating a new harmony is called ‘improvisation’. A new harmony vector, ${\mathit{x}}^{\prime }=({x^{\prime }}_1,{x^{\prime }}_2,\cdots ,{x^{\prime }}_n)$, is generated based on three rules: (1) memory consideration, (2) pitch adjustment, and (3) random selection. The procedure works as shown in Figure 1.

Here, ${x^{\prime }}_i(i=1,2,\cdots ,n)$ is the ith component of ${\mathit{x}}^{\prime }$, and $x_i^j(j=1,2,\cdots ,\mathrm{HMS})$ is the ith component of the jth candidate solution vector in HM. Both r and rand() are uniformly generated random numbers in the region of (0, 1), and bw is an arbitrary distance bandwidth.

Step 4. Update harmony memory. If the new harmony vector ${\mathit{x}}^{\prime }=(x_1^{\prime },x_2^{\prime },\cdots ,x_n^{\prime })$ is better than the worst harmony in the HM, judged in terms of the objective function value, the new harmony is included in the HM, and the existing worst harmony is excluded from the HM.

Step 5. Check the stopping criterion. If the stopping criterion (Tmax) is satisfied, computation is terminated. Otherwise, Steps 3 and 4 are repeated.

3.2. NGHS Algorithm

Experiments with the classical HS algorithm over the benchmark problems show that the algorithm suffers from the problem of premature and/or false convergence and slow convergence especially over multimodal fitness landscape. To enrich the searching behavior and to avoid being trapped into the local optimum, more improved HS algorithms were presented. The NGHS algorithm modifies the improvisation step of the HS such that the new harmony can mimic the global-best harmony in the HM [41,42,43,44]. In step 3 it works as shown in Figure 2.

Here, ‘best’ and ‘worst’ are the indexes of the best harmony and the worst harmony in HM, respectively. r and rand() are all uniformly generated random number in (0, 1).

The NGHS method has strong global search ability in the early stage of the optimization process and has strong local search ability in the late stage of optimization process. In the early stage of the optimization process, all solution vectors are sporadic in feasible space, so $x_s$ is large, which is beneficial to the global search, while in the late stage of the optimization process, all nonbest solution vectors are inclined to move to the global-best solution vector, so most solution vectors are close to each other. In this case, most $x_s$ is small and most trust regions are narrow, which is beneficial to the local search of the NGHS.

The NGHS and the HS are different in the following:

(1) In Step 1, harmony memory considering rate (HMCR), pitch adjusting rate (PAR), and adjusting step (bw) are excluded from the NGHS, and genetic mutation probability ($p_m$) is included in the NGHS;

(2) The HS carries out mutation with the probability HMCR $\times$ PAR and carries out random selection with the probability 1-HMCR, while the NGHS carries out genetic mutation with the probability $p_m$. In fact, both operations are exactly the same, and they are used to keep the individual variety better, which can effectively prevent both algorithms from being trapped into the local optimum.

(3) In Step 4, the NGHS replaces the worst harmony ${\mathit{x}}^{worst}$ in HM with the new harmony ${\mathit{x}}^{\prime }$ even if ${\mathit{x}}^{\prime }$ is worse than ${\mathit{x}}^{worst}$.

4. Computational Results

In this section, we perform some numerical tests in order to illustrate the implementation and efficiency of the NGHS method for some linear complementary problems. All the experiments were performed on MatlabR2009a system with Intel(R) Core (TM) 4 × 3.3 GHz and 2 GB RAM.

4.1. Problems

4.1.1. LCP with a Unique Solution

LCP1. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccc}3& -2& -1\\ -2& 2& 1\\ -1& 1& 1\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}14\\ -11\\ -7\end{array}\right].$$

Eigenvalues of the symmetric matrix $\mathit{M}$ are ${\lambda }_1=0.3080,{\lambda }_2=0.6431,{\lambda }_3=5.0489$; thus, $\mathit{M}$ is a (symmetric) positive definite matrix, and LCP1 has a unique solution ${\mathit{x}}^{*}={(0,4,3)}^T$.

LCP2. Consider the following LCP, where

$$\mathit{M}=\left[\begin{array}{ccccc}4& -1& & & \\ -1& 4& -1& & \\ & -1& 4& -1& \\ & & \ddots & \ddots & -1\\ & & & -1& 4\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}-1\\ -1\\ -1\\ ⋮\\ -1\end{array}\right].$$

Here $\mathit{M}$ is a tridiagonal matrix, whose eigenvalues are all greater than 0; thus, $\mathit{M}$ is a (symmetric) positive definite matrix, and LCP2 has a unique solution ${\mathit{x}}^{*}={(0.3660,0.4641,\cdots ,0.4641,0.3660)}^T$.

LCP3. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccccc}1& 2& 2& \cdots & 2\\ 2& 5& 6& \cdots & 6\\ 2& 6& 9& \cdots & 10\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 6& 10& \cdots & 4n-3\end{array}\right]\begin{array}{cccc},& & & \mathit{q}=\left[\begin{array}{c}\begin{array}{c}-1\\ -1\end{array}\\ -1\\ ⋮\\ -1\end{array}\right]\end{array}.$$

Since eigenvalues of the matrix $\mathit{M}$ are all greater than 0, $\mathit{M}$ is a (symmetric) positive definite matrix, and LCP3 has a unique solution ${\mathit{x}}^{*}={(1,0,\cdots ,0)}^T$.

LCP 4. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccccc}1/n& 0& 0& \cdots & 0\\ 0& 2/n& 0& \cdots & 0\\ 0& 0& 3/n& \cdots & 0\\ ⋮ & ⋮& ⋮ & \ddots & ⋮\hfill \\ 0& 0& 0& \cdots & n/n\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}-1\\ -1\\ -1\\ ⋮\\ -1\end{array}\right].$$

Since $\mathit{M}$ is a (symmetric) positive definite, LCP4 has a unique solution ${\mathit{x}}^{*}={(n,n/2,\cdots ,1)}^T$.

LCP 5. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{cccc}3& 0& -1& 0\\ -1& 3& -1& 0\\ 0& -1& 4& -2\\ -1& -1& -1& 5\end{array}\right]\begin{array}{cccc},& & & \mathit{q}=\left[\begin{array}{c}-2\\ 3\\ -4\\ 5\end{array}\right]\end{array}.$$

Based on the Hermitian/skew-Hermitian splitting, eigenvalues of symmetry component matrix $\frac{1}{2}(\mathit{M}+{\mathit{M}}^T)$ are all greater than 0; thus, $\mathit{M}$ is a (nonsymmetric) positive definite matrix, and LCP5 has a unique solution ${\mathit{x}}^{*}={(1,0,1,0)}^T$.

LCP 6. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{c}\begin{array}{c}\begin{array}{cccc}1& 2& 2& \begin{array}{cc}\cdots & 2\end{array}\end{array}\\ \begin{array}{cccc}0& 1& 2& \begin{array}{cc}\cdots & 2\end{array}\end{array}\end{array}\\ \begin{array}{cccc}0& 0& 1& \begin{array}{cc}\cdots & 2\end{array}\end{array}\\ \begin{array}{cccc}⋮& ⋮& ⋮& \begin{array}{cc}\ddots & ⋮\end{array}\end{array}\\ \begin{array}{cccc}0& 0& 0& \begin{array}{cc}\cdots & 1\end{array}\end{array}\end{array}\right]\begin{array}{cccc},& & & \mathit{q}=\left[\begin{array}{c}\begin{array}{c}-1\\ -1\end{array}\\ -1\\ ⋮\\ -1\end{array}\right]\end{array}.$$

Since eigenvalues of symmetry component matrix $\frac{1}{2}(\mathit{M}+{\mathit{M}}^T)$ are all greater than or equal to 0, $\mathit{M}$ is a (nonsymmetric) positive semidefinite matrix, and LCP6 has a unique solution ${\mathit{x}}^{*}={(0,0,\cdots ,0,1)}^T$.

LCP 7. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{cccc}2& 1& 1& 1\\ 1& 2& 0& 1\\ 1& 0& 1& 2\\ -1& -1& -2& 0\end{array}\right]\begin{array}{cccc},& & & \mathit{q}=\left[\begin{array}{c}-8\\ -6\\ -4\\ 3\end{array}\right]\end{array}.$$

Since eigenvalues of symmetry component matrix $\frac{1}{2}(\mathit{M}+{\mathit{M}}^T)$ are all greater than or equal to 0, $\mathit{M}$ is a (nonsymmetric) positive semidefinite matrix, and LCP7 has a unique solution ${\mathit{x}}^{*}={(2.5,0.5,0,2.5)}^T$.

In fact, $\forall \mathit{x}\ne \mathbf{0},$ $\mathit{x}={(x_1,x_2,x_3,x_4)}^T$,

$${\mathit{x}}^T\mathit{M}\mathit{x}=2x_1{}^2+2x_1{x}_2+2x_1{x}_3+2x_2{}^2+x_3{}^2={(x_1+x_2)}^2+x_2{}^2+{(x_1+x_3)}^2\ge 0$$

so $\mathit{M}$ is a positive semidefinite matrix, too.

LCP8. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccccccc}1& 0& -0.5& 0& 1& 3& 0\\ 0& 0.5& 0& 0& 2& 1& -1\\ -0.5& 0& 1& 0.5& 1& 2& -4\\ 0& 0& 0.5& 0.5& 1& -1& 0\\ -1& -2& -1& -1& 0& 0& 0\\ -3& -1& -2& 1& 0& 0& 0\\ 0& 1& 4& 0& 0& 0& 0\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}\begin{array}{c}-1\\ -3\\ 1\\ -1\end{array}\\ 5\\ 4\\ -1.5\end{array}\right].$$

Since eigenvalues of symmetry component matrix $\frac{1}{2}(\mathit{M}+{\mathit{M}}^T)$ are all greater than or equal to 0, $\mathit{M}$ is a (nonsymmetric) positive semidefinite matrix, and LCP7 has a unique solution ${\mathit{x}}^{*}={(0.0909,2.3636,0,0.1818,0.9091,0,0)}^T$.

In fact, $\forall \mathit{x}\ne \mathbf{0},$ $\mathit{x}={(x_1,x_2,x_3,x_4,x_5,x_6,x_7)}^T$,

$${\mathit{x}}^T\mathit{M}\mathit{x}=x_1{}^2-x_1{x}_3+\frac{1}{2}{x}_2{}^2+x_3{}^2+x_3{x}_4+\frac{1}{2}{x}_4{}^2={\left(x_1-\frac{1}{2}{x}_3\right)}^2+\frac{1}{2}{x}_2{}^2+\frac{3}{4}{\left(x_2+\frac{2}{3}{x}_3\right)}^2+\frac{1}{6}{x}_4{}^2\ge 0,$$

so $\mathit{M}$ is a positive semidefinite matrix, too.

LCP 9. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{cccc}1& -4& 1& 0\\ 0& 1& 0& 1\\ -1& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right]\begin{array}{cccc},& & & \mathit{q}=\left[\begin{array}{c}-5\\ -5\\ 1\\ 1\end{array}\right]\end{array}.$$

For arbitrary $\mathit{x}={(x_1,x_2,x_3,x_4)}^T$,

$${\mathit{x}}^T\mathit{M}\mathit{x}={(x_1-2x_2)}^2-3x_2{}^2.$$

If $\tilde{\mathit{x}}={(2,1,0,0)}^T$, then ${\tilde{\mathit{x}}}^T\mathit{M}\tilde{\mathit{x}}=-3<0$, which demonstrates $\mathit{M}$ is not a positive semidefinite matrix. LCP9 has a unique solution ${\mathit{x}}^{*}={(1,1,8,4)}^T$.

LCP 10. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{c}\begin{array}{cccc}7& \begin{array}{cc}\begin{array}{cc}0& 4\end{array}& 2\end{array}& 2& 4\end{array}\\ \begin{array}{cccc}2& \begin{array}{cc}\begin{array}{cc}8& 1\end{array}& 5\end{array}& 9& 0\end{array}\\ \begin{array}{cccc}1& \begin{array}{cc}\begin{array}{cc}0& 3\end{array}& 0\end{array}& 3& 2\end{array}\\ \begin{array}{cccc}0& \begin{array}{cc}\begin{array}{cc}1& 4\end{array}& 6\end{array}& 1& 0\end{array}\\ \begin{array}{cccc}8& \begin{array}{cc}\begin{array}{cc}0& 1\end{array}& 2\end{array}& 0& 3\end{array}\\ \begin{array}{cccc}5& \begin{array}{cc}\begin{array}{cc}0& 9\end{array}& 2\end{array}& 6& 8\end{array}\end{array}\right],\begin{array}{cc}& \begin{array}{cc}& \mathit{q}=\left[\begin{array}{c}-7.3\\ \begin{array}{c}7.9\\ -3.4\\ 10.2\end{array}\\ -5.9\\ 7.2\end{array}\right]\end{array}\end{array}.$$

For arbitrary $\mathit{x}={(x_1,x_2,x_3,x_4,x_5,x_6)}^T$,

$$\begin{gathered} {\mathit{x}}^T\mathit{M}\mathit{x}=7x_1{}^2+(9x_6+2x_2+5x_3+10x_5+2x_4)x_1+(2x_6+5x_2+6x_4+2x_5)x_4+(8x_2+x_4)x_2 \\ +(x_5+x_2+3x_3+4x_4+9x_6)x_3+(2x_3+3x_5+8x_6)x_6+(6x_6+9x_2+3x_3+x_4)x_5. \end{gathered}$$

If $\tilde{\mathit{x}}={(0,0,0,0,-8,3)}^T$, then ${\tilde{\mathit{x}}}^T\mathit{M}\tilde{\mathit{x}}=-144<0$, which demonstrates $\mathit{M}$ is not a positive semidefinite matrix. LCP10 has a unique solution ${\mathit{x}}^{*}={(0.7,0,0.3,0,0.6,0)}^T$.

4.1.2. LCP with Multiple Solutions

LCP 11. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccccc}0& 0& 2& 2& 1\\ 0& 0& 1& 2& 2\\ 1& 2& 0& 0& 0\\ 3& 1& 0& 0& 0\\ 2& 3& 0& 0& 0\end{array}\right],\mathit{q}=\left[\begin{array}{c}\begin{array}{c}-1\\ -1\end{array}\\ -1\\ -1\\ -1\end{array}\right].$$

For arbitrary $\mathit{x}={(x_1,x_2,x_3,x_4,x_5)}^T$,

$${\mathit{x}}^T\mathit{M}x=3x_1{x}_3+5x_1{x}_4+3x_1{x}_5+3x_2{x}_3+3x_2{x}_4+5x_2{x}_5.$$

If $\tilde{\mathit{x}}={(0,2,-1,-1,-1)}^T$, then ${\tilde{\mathit{x}}}^T\mathit{M}\tilde{\mathit{x}}=-22<0$, which demonstrates $\mathit{M}$ is not a positive semidefinite matrix. The solution set of LCP11′ is

$${\mathit{x}}^{*}={(\lambda ,1-3\lambda ,0,0.5,0)}^T,0\le \lambda \le 0.2$$

LCP 12. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}-1\\ -1\\ -1\end{array}\right].$$

$\forall \mathit{x}={(x_1,x_2,x_3)}^T$, and ${\mathit{x}}^T\mathit{M}\mathit{x}={(x_1+x_1+x_3)}^2\ge 0$, so $\mathit{M}$ is a (symmetric) positive semidefinite matrix. The solution set of LCP12 is

$${\mathit{x}}^{*}={({\lambda }_1,{\lambda }_2,{\lambda }_3)}^T,{\lambda }_1+{\lambda }_2+{\lambda }_3=1,{\lambda }_1\ge 0,{\lambda }_2\ge 0,{\lambda }_3\ge 0.$$

LCP 13. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccc}1& 0& -1\\ 0& 2& 0\\ -1& 0& 1\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}-1\\ 0\\ 1\end{array}\right].$$

$\forall \mathit{x}={(x_1,x_2,x_3)}^T$, and ${\mathit{x}}^T\mathit{M}\mathit{x}={(x_1-x_3)}^2+2x_2^2\ge 0$, so $\mathit{M}$ is a (symmetric) positive semidefinite matrix. The solution set of LCP13 is ${\mathit{x}}^{*}={(\lambda +1,0,\lambda )}^T,\lambda \ge 0$.

LCP 14. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccccc}1& 0& 1& 1& -1\\ 0& 0& 0& 0& 0\\ 1& 0& 1& 1& -1\\ 1& 0& 1& 1& -1\\ -1& 0& -1& -1& 1\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}-5\\ 3\\ -3\\ -5\\ 10\end{array}\right].$$

$\forall \mathit{x}={(x_1,x_2,x_3,x_4,x_5)}^T$, and ${\mathit{x}}^T\mathit{M}\mathit{x}={(x_1+x_3+x_4-x_5)}^2\ge 0$, so $\mathit{M}$ is a (symmetric) positive semidefinite matrix. The solution set of LCP14 is ${\mathit{x}}^{*}={(\lambda ,0,0,5-\lambda ,0)}^T,0\le \lambda \le 5$.

4.1.3. LCP without Solution

LCP 15. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right],\begin{array}{ccc}& & \mathit{q}=\left[\begin{array}{c}-2\\ -1\end{array}\right]\end{array}.$$

LCP 15 is feasible but not solvable [45,46,47].

LCP 16. Consider the following linear complementary problem, where

$$\mathit{M}=\left[\begin{array}{ccc}1& 1& 0\\ -5& 1& 0\\ -1& 1& 0\end{array}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathit{q}=\left[\begin{array}{c}-2\\ -1\\ -2\end{array}\right].$$

LCP 16 is feasible but not solvable [45,46,47].

4.2. Parameters Setting

Simulations were carried out to compare the optimization capabilities of the NGHS method with respect to: (1) classical HS (HS) [33], (2) HSCH (Harmony Search Algorithm with Chaos [48]), and (3) HSWB (Harmony Search with Worst and Best [49]). To make the comparison fair, the populations for all HS variants were initialized using the same random seeds. All HS-variants algorithms set the same parameters: HMS = 10, HMCR = 0.85, PAR = 0.35, Tmax = 50,000. In NGHS, we set $p_m$ = 0.005. In LCP4 and LCP9, we set search space in the region ${[-n,n]}^n$ and ${[-10,10]}^n$, respectively. In other LCPs, we set search space in the region ${[-5,5]}^n$.

4.3. Results and Analysis

To judge the accuracy of different algorithms, ten independent runs of four algorithms (HS, HSCH, HSWB, NGHS) were carried out. The best, the mean, the worst fitness values, the standard deviation (Std), and the mean time are recorded in

Table 1. Results for 10 runs on given LCPs.

LCPs	Algorithm	Best	Mean	Worst	Std	Meantime (s)
LCP1 n = 3	HS	0.00012	0.002018	0.006212	0.002172	0.7930772
	HSCH	9.53 × 10−8	3.17 × 10−7	6.51 × 10−7	1.89 × 10−7	0.8098424
	HSWB	6.24 × 10−8	7.69 × 10−7	3.53 × 10−6	1.03 × 10−6	0.8240383
	NGHS	4.93 × 10−30	7.66 × 10−29	2.33 × 10−28	7.28 × 10−29	0.8852437
LCP2 n = 10	HS	0.049171	0.077259	0.092279	0.012563	1.4272887
	HSCH	1.08 × 10−7	5 × 10−7	9.78 × 10−7	2.53 × 10−7	1.4395747
	HSWB	2.93 × 10−8	1.57 × 10−7	5.17 × 10−7	1.46 × 10−7	1.4575034
	NGHS	1.05 × 10−27	7.91 × 10−26	3.6 × 10−25	1.09 × 10−25	1.5603642
LCP3 n = 10	HS	0.000879	0.007951	0.018899	0.006224	1.2304132
	HSCH	2.21 × 10−6	7.45 × 10−6	1.53 × 10−5	4.52 × 10−6	1.2584141
	HSWB	2.48 × 10−6	6.4 × 10−6	1.45 × 10−5	4.03 × 10−6	1.2722705
	NGHS	4.44 × 10−22	7.55 × 10−19	5.88 × 10−18	1.81 × 10−18	1.3203617
LCP4 n = 10	HS	8.8 × 10−5	0.000458	0.000849	0.000281	1.2091804
	HSCH	1.46 × 10−9	3.52 × 10−9	8.17 × 10−9	2.06 × 10−9	1.228747
	HSWB	5.36 × 10−10	1.78 × 10−9	3.94 × 10−9	1.24 × 10−9	1.2424723
	NGHS	0	0	0	0	1.3108552
LCP5 n = 4	HS	6.71 × 10−5	0.014856	0.03293	0.010225	0.8205359
	HSCH	1.78 × 10−7	5.49 × 10−7	1.35 × 10−6	3.8 × 10−7	0.8371677
	HSWB	1.95 × 10−8	1.79 × 10−7	6.17 × 10−7	1.74 × 10−7	0.8490459
	NGHS	0	7.44 × 10−31	9.86 × 10−31	3.55 × 10−31	0.9181975
LCP6 n = 10	HS	0.000197	0.000531	0.001297	0.000323	1.2196362
	HSCH	4.01 × 10−10	2.39 × 10−9	4.95 × 10−9	1.37 × 10−9	1.2251887
	HSWB	2.91 × 10−10	1.3 × 10−9	2.42 × 10−9	6.24 × 10−10	1.2407602
	NGHS	0	2.47 × 10−33	2.47 × 10−32	7.8 × 10−33	1.3161821
LCP7 n = 4	HS	0.005125	0.015587	0.030891	0.007932	0.8171942
	HSCH	1.26 × 10−5	2.73 × 10−5	4.65 × 10−5	1.4 × 10−5	0.8416201
	HSWB	2.94 × 10−6	6 × 10−5	0.000436	0.000133	0.8573362
	NGHS	2.78 × 10−21	7.55 × 10−19	6.11 × 10−18	1.89 × 10−18	0.9224063
LCP8 n = 7	HS	0.003457	0.011776	0.046185	0.012459	0.9173754
	HSCH	6.31 × 10−6	0.109186	1.091539	0.345164	0.9359316
	HSWB	8.64 × 10−6	0.065945	0.49082	0.158359	0.9496413
	NGHS	3.01 × 10−12	8.63 × 10−11	6.03 × 10−10	1.84 × 10−10	1.0178789
LCP9 n = 4	HS	1.38 × 10−5	0.002363	0.022127	0.006945	0.8277304
	HSCH	9 × 10−7	3.13 × 10−5	0.000128	3.81 × 10−5	0.8443199
	HSWB	3.46 × 10−6	4.87 × 10−5	0.000227	7.57 × 10−5	0.8582979
	NGHS	4.19 × 10−28	2.24 × 10−27	6.97 × 10−27	2.14 × 10−27	0.9241014
LCP10 n = 6	HS	0.012854	0.099804	0.257314	0.088732	0.8767348
	HSCH	4.5 × 10−5	0.000248	0.00062	0.000173	0.8921561
	HSWB	1.41 × 10−5	0.000227	0.000747	0.000251	0.9055236
	NGHS	4.6 × 10−23	5.11 × 10−14	2.94 × 10−13	1.07 × 10−13	0.9837116
LCP11 n = 5	HS	4.63 × 10−5	0.006906	0.025728	0.009857	0.8943217
	HSCH	1.55 × 10−7	1.8 × 10−5	6.46 × 10−5	2.4 × 10−5	0.9109492
	HSWB	7.3 × 10−8	6.71 × 10−5	0.000493	0.000151	0.9244671
	NGHS	8.71 × 10−32	3.15 × 10−27	1 × 10−26	3.81 × 10−27	0.9889576
LCP12 n = 3	HS	4.88 × 10−14	1.37 × 10−6	9 × 10−6	2.98 × 10−6	0.8374631
	HSCH	2.17 × 10−14	1.01 × 10−9	5.51 × 10−9	1.82 × 10−9	0.8581188
	HSWB	3.09 × 10−13	9.33 × 10−10	7.38 × 10−9	2.29 × 10−9	0.871607
	NGHS	0	0	0	0	0.9422746
LCP13 n = 3	HS	2.22 × 10−9	3.06 × 10−6	1.44 × 10−5	5.13 × 10−6	0.7916857
	HSCH	3.92 × 10−11	6.18 × 10−10	1.26 × 10−9	4.53 × 10−10	0.8085951
	HSWB	1 × 10−12	4.56 × 10−10	2.16 × 10−9	6.96 × 10−10	0.8221383
	NGHS	0	0	0	0	0.9289744
LCP14 n = 5	HS	3.56 × 10−5	0.00053	0.001597	0.000515	0.848102
	HSCH	3.57 × 10−9	2.33 × 10−8	5.41 × 10−8	1.85 × 10−8	0.8672321
	HSWB	3.04 × 10−10	6.88 × 10−9	1.67 × 10−8	6.24 × 10−9	0.8818419
	NGHS	0	6.51 × 10−31	1.23 × 10−30	4.8 × 10−31	0.9494152
LCP15 n = 2	HS	0.41112	0.411127	0.411169	1.49 × 10−5	0.7566656
	HSCH	0.41112	0.41112	0.41112	4.29 × 10−10	0.7321389
	HSWB	0.41112	0.41112	0.41112	4.43 × 10−10	0.7490452
	NGHS	0.41112	0.41112	0.41112	6.14 × 10−17	0.8155493
LCP16 n = 3	HS	0.032256	0.040354	0.063529	0.01081	0.7415792
	HSCH	0.032256	0.032256	0.032256	4.11 × 10−8	0.7572208
	HSWB	0.032256	0.032256	0.032256	1.1 × 10−7	0.7765369
	NGHS	0.032256	0.032256	0.032256	6.04 × 10−17	0.8379764

. Figure 3 shows the convergence of the mean fitness value and boxplot of the final best fitness value for given LCPs. The values plotted for every generation are averaged over ten independent runs. The boxplot is the best fitness value in the final population for the different algorithms.

From Table 1, for solvable LCPs (LCP1~LCP14), we can see that all of the best fitness, mean fitness, worst fitness, and standard deviation (Std) obtained by NGHS are better than those obtained by HS, HSCH, and HSWB. So NGHS has a very fast convergence speed when compared with HS, HSCH, and HSWB, which is indicated in the convergence plot of the mean fitness value of Figure 3. From these figures, we can see that almost all of NGHS find the solutions with the unchangeable convergence speed for solving LCPs. Meanwhile, almost all of mean time of NGHS are very similar to that of HS, HSCH, and HSWB.

After ten independent runs, for LCPs with unique solution (LCP1~LCP10), NGHS converges to its unique solution respectively; and for LCPs with multiple solutions (LCP11~LCP14), NGHS can find as many solutions as possible. Figure 4 shows the computation results of NGHS (10 independent runs) for LCP11~LCP14. Meanwhile, for unsolvable LCPs (LCP15~LCP16), four algorithms are terminated on the solution with the minimum error, and NGHS can more early terminate than other HS variants.

5. Conclusions

General linear complementarity problem is studied in this paper. After reforming the LCP as an unconstrained optimization model, the LCP can be solved by the NGHS algorithm. For the solvable LCP, NGHS terminated on the solution. For the unsolvable LCP, NGHS terminated on the solution with the minimum error.

Thus, we can use this method to decide whether the solution of LCP is existent or not. That is, if objective function (merit function) cannot converge to the zeros anyway, the LCP is maybe unsolvable.