New Accelerated Modulus-Based Iteration Method for Solving Large and Sparse Linear Complementarity Problem

In this article, we establish a class of new accelerated modulus-based iteration methods for solving the linear complementarity problem. When the system matrix is an $H_+$-matrix, we present appropriate criteria for the convergence analysis. Also, we demonstrate the effectiveness of our proposed method and reduce the number of iterations and CPU time to accelerate the convergence performance by providing two numerical examples for various parameters. Keywords. Linear complementarity problem, Iteration method, $P$-matrix, $H_{+}$-matrix, Convergence analysis, Matrix splitting.


Introduction
The large and sparse matrices are matrices that have a large number of rows and columns but a small number of non-zero elements.In other words, they are matrices where the majority of the elements are zero.Sparse matrices are commonly used to represent complex systems or large datasets in fields such as computer science, mathematics, physics and engineering.The sparsity of the matrix means that it is not practical to store each element individually and specialized data structures and algorithms must be used to efficiently store and manipulate the matrix.
Given A 1 ∈ R n×n and a vector q ∈ R n .The linear complementarity problem, represented as LCP(q, A 1 ), is to find the solution z ∈ R n to the following system: The free boundary problem, the Nash equilibrium point of the bimatrix game, operations research, control theory, mathematical economics, optimization theory, stochastic optimal control, the American option pricing problem, and elasticity theory are among the applications of the linear complementarity problem that are extensively studied in the literature on mathematical programming.
The methods available for solving the linear complementarity problems are into two groups namely the pivotal method [6], [8] and the iterative method [31], [15], [9], [18], [22], [24] and [26].The basic idea behind the pivotal method is to get a basic feasible complementary vector through a series of pivot steps, while the iterative method creates a series of iterates that lead to a solution .Reformulating the LCP(q, A 1 ) as an equation whose solution must be the same as the LCP (q, A 1 ) is one of the most well-known and highly sought-after techniques for creating fast and economical iteration methods.As a result, some useful LCP(q, A 1 ) equivalent forms have emerged.Mangasarian [29] presented three methods: projected Jacobi over-relaxation, projected SOR, and projected symmetric SOR.For more information on designing iteration methods using the idea of Mangasarian, see also [1], [4] and [39].Bai in [3] given the following general equivalent form: with r > 0, where Ω 1 ∈ R n×n is a positive diagonal matrix, and initially, a class of modulus-based matrix splitting iteration algorithms was developed.
Using the ideas of Shilang [35] and Bai [3], we present a class of new accelerated modulus-based iteration methods for solving the large and sparse LCP(q, A).
Also, we show that the linear complementarity problem and fixed point equation are equivalent and provide some convergence domains for our proposed method.
The following is the structure of the article: Some required definitions, notations and well-known lemmas are provided in section 2, all of which will be used for the discussions in the remaining sections of this work.In section 3, a new accelerated modulus-based iteration method with the help of the new equivalent fixed point form of the LCP(q, A) is provided.In section 4, we establish some convergence domains for the proposed method.A numerical comparison between the proposed methods and the modulus-based matrix splitting methods, introduced by Bai [3], is illustrated in section 5. Section 6 contains the conclusion of the article.

Preliminaries
In this part, we briefly discuss the basic results, definitions, and notations, most of which may be found in [12], [13] and [33].
The matrices A 1 and B 1 are denoted by and matrix if all of its non-diagonal elements are less than equal to zero; an M -matrix if A −1 1 ≥ 0 as well as Z-matrix; an Hmatrix, if A 1 is an M -matrix and an H + -matrix if A 1 is an H-matrix as well as āii > 0 ∀ i ∈ {1, 2, . . ., n}; a P -matrix if all its principle minors are positive such

Main results
For a given vector s ∈ R n , we indicate the vectors s + = max{0, s} and , where I is the identity matrix of order n and L 1 is the strictly lower triangular matrix of A 1 .In the following result, we convert the LCP(q, A 1 ) into a fixed point formulation.
be a nonsingular, then the equivalent formulation of the LCP(q, A 1 ) in form of fixed point equation is Proof.We have z = τ (|s| + s) and ω = Ω 1 (|s| − s), from Equation (1) we obtain Let τ = 1 r , the above equation can be rewritten as, In the following, Based on Equation (3), we propose an iteration method which is known as Method 3.1 to solve the LCP(q, A 1 ).
Then we use the following equation for Method 3.1 is Let Residual be the Euclidean norm of the error vector, which is defined as follows: Consider a nonnegative initial vector z (0) ∈ R n .The iteration process continues until the iteration sequence {z (k) } +∞ k=0 ⊂ R n converges.For k = 0, 1, 2, . .., the iterative process continues until the iterative sequence z (k+1) ∈ R n converges.
The iteration process stops if Res(z (k) ) < ǫ.For computing z (k+1) we use the following steps.
Step 2: Using the following scheme, create the sequence z (k) : and set z (k+1) = 1 r (|s (k+1) | + s (k+1) ), where z (k) is a k th approximate solution of LCP(q, A 1 ) and s (k is a k th approximate solution of Equation (3).
Furthermore, the proposed Method 3.1 offers a generic framework for solving LCP(q, A 1 ).We created a new family of accelerated modulus-based relaxation methods using matrix splitting.In particular, we express the system matrix 4) gives the new accelerated modulus iteration method is 4) gives the new accelerated modified modulus-based iteration method is

Convergence analysis
In the following result, we prove the convergence conditions when the system matrix A 1 is a P -matrix.Theorem 4.1.Let A 1 ∈ R n×n be a P -matrix and s * be the solution of Equation k=1 generated by Method 3.1 converges to the solution s * for any initial vector s (0) ∈ R n .Proof.Let s * be the solution of Equation ( 3), then error is Therefore, the sequence {s (k) } +∞ k=1 converges to the solution s * .
When the system matrix A 1 is an H + -matrix, the following result discusses the convergence domain of Ω 1 for a new accelerated modulus-based iteration method.
Theorem 4.2.Let A 1 be an H + -matrix and

and either one of the following conditions holds:
(1) Then the sequence {s (k) } +∞ k=1 generated by Method 3.1 converges to the solution s * for any initial vector s

and it holds that
and it holds that Since A 1 is an M -matrix, then there exists a positive vector v > 0 such that By using the Lemma 2.4, we are able to determine that ρ(T )<1.
] is an M -matrix.Then there exists a positive vector v > 0 such that By using the Lemma 2.4, we are able to determine that ρ(T )<1.
Because of this, according to Theorem 4.1, the iteration sequence {s (k) } +∞ k=1 generated by Method 3.1 converges to s * for any initial vector s (0) .
Matlab version 2021a on an Acer Desktop (Intel(R) Core(TM) i7-8700 CPU @ 3.2 GHz 3.19GHz, 16.00GB RAM) is used for all calculations.The numerical results for the new accelerated modulus-based iteration method and modulusbased matrix splitting method in [3] are listed in Tables 1 and 2.
Example 5.1.The system matrix A 1 ∈ R n×n is generated by where δ 1 is nonnegative real parameter and , where P 1 ∈ R n×n , L 1 ∈ R m×m and I 2 is the identity matrix of order m.
where δ 1 are nonnegative real parameters and  From Tables 1 and 2, we can observe that the iteration steps required by our proposed NAMGS and NAMSOR methods have lesser number of iteration steps, faster processing (CPU time) and a greater computational efficiency than the MGS and MSOR methods in [3] respectively.

Conclusion
In this article, we present a class of new accelerated modulus-based iteration methods for solving the LCP(q, A 1 ) based on matrix splitting.The large and sparse structure of A 1 is preserved throughout the iteration process by this iteration form.Additionally, when system matrix A 1 is an H + -matrix, we demonstrate some convergence conditions.Finally, two numerical examples are provided to illustrate the effectiveness of the proposed methods.

P 1 ∈
R n×n , L 1 ∈ R m×m and I 2 is the identity matrix of order m.