Abstract
This article suggests two new modified iteration methods called the modified Gauss-Seidel (MGS) method and the modified fixed point (MFP) method to solve the absolute value equation. Using appropriate assumptions, we examine the convergence of the given methods. Lastly, numerical examples illustrate the usefulness of the new strategies.
1. Introduction
Let the absolute value equation (AVE) be
In this case, the order of the matrix is , , and describes the component-wise absolute value of vector. The AVE is an important nondifferentiable and nonlinear problem in optimization, such as convex quadratic programming, linear complementarity problems (LCPs), and linear programming (see [1–12]).
The development of numerical procedures for AVE has been extensively studied recently, and several techniques have been presented. For example, Fakharzadeh and Shams [13] introduced the mixed-type splitting approach to determine AVE (1) and showed the new convergence properties under appropriate conditions. Edalatpour et al. [14] introduced the generalized form of the Gauss-Seidel approach for determining (1). The following summarizes this approach: where and are expressed in Equation (4), respectively. Ke and Ma [15] offered an SOR-like strategy to obtain the AVE (1). This method is defined as follows: where , , , and . Chen et al. [16] showed the SOR-like strategy with optimal parameters and examined some novel convergence situations different from [15]. Zamani and Hladík [17] offered a new concave minimization approach for AVE (1), which addresses the deficiency of the system proposed in [18] and others (see [19–23]).
The remainder of this study is organized in the following manner. Section 2 presents various notations and definitions. Section 3 discusses the proposed methods as well as their convergence for AVE (1). We demonstrate the efficiency of the new techniques in Section 4 by providing numerical examples. In the last section, we make concluding remarks.
2. Preliminaries
The purpose of this section is to briefly review some of the symbols and concepts used in this article.
Suppose ; we describe the spectral radius and absolute value of as and , respectively.
Lemma 1 (see [24]). Let be an invertible matrix. If , then for any vector , Equation (1) has a unique solution.
3. Modified Iteration Methods
Here, we present a detailed discussion of the proposed modified methods for solving AVEs.
3.1. MGS Method for AVE
Let us divide matrix as follows:
where , , and are strictly upper-triangular, strictly lower-triangular, and diagonal matrices of , respectively. Based on (4), the MGS approach to solving (1) can be expressed as follows:
In the MGS method, let with and . Based on an starting vector and for until the iterative sequence is convergent, calculate
The next step is to examine the convergence of the MGS approach by utilizing the subsequent theorem.
Theorem 2. Assume that AVE (1) is solvable, and matrix satisfies Lemma 1. If where Then, for any starting vector , the sequence created by the MGS approach converges to the unique solution of the Equation (1).
Proof. Suppose that is the solution of Equation (1); then we obtain Using absolute values for each side of the first equation of (8), we get or equivalently Similarly, the second part of (8) indicates From (10) and (11), we obtain So, is nonnegative. Note that if then the sequence of MGS approach converges to the unique solution of AVE.
3.2. MFP Method for AVE
First, we will briefly discuss the fixed point method of determining the AVE. The AVE (1) is equivalent to the fixed point problem of solving such that where and is a positive diagonal matrix. If we take , then, the fixed point method for solving AVE (1) is defined as follows:
The purpose of this article is to discuss the MFP method. Then, the offered method is expressed as follows:
In the MFP method, let with and . Based on an starting vector and for until the iterative sequence is convergent, calculate
The next step is to examine the convergence of the MFP method by utilizing the subsequent theorem.
Theorem 3. Let matrix satisfy Lemma 1. If then the sequence derived from (17) converges to the unique solution .
Proof. Suppose represents the solution to AVE (1); then we obtain Using absolute values for each side of the first equation of (20), we get or equivalently Similarly, the second part of (20) indicates From (22) and (23), we obtain So, Evidently, if , the iteration sequence created by the MFP method is convergent (Table 1).
4. Numerical Tests
In this section, we investigate experimentally the performance of newly developed approaches for determining the AVEs. The computations have been performed on an Intel Core (TM) i7-10875H, with 32 GB memory, and a 5.1 GHz CPU, as well as using MATLAB (2017a). The initial guess is a zero vector, and the current iteration ends when it meets the requirement
Example 1 (see [16]). Let and vector . Table 2 summarizes the outcomes.
Listed in Table 2 are the iteration steps (Iter), CPU time in seconds (Time), and residuals (RES) for each method. Table 2 provides numerical results that show the suggested methods to be more efficient in terms of time and iteration steps than all other compared methods. Additionally, we compare all approaches graphically for and . The graphical outcomes are shown in Figure 1.
Example 2 (see [16]). Let and using where is the identity matrix, , and is a tridiagonal matrix. Table 3 outlines the results.
(a)
(b)
From Table 3, we can notice that the MGS method has higher accuracy for different values of . The MGS method needs less time and a number of iterations as compared with other methods. Furthermore, we observe that the MFP method converges to the solution rapidly as compared with the SORLaopt method, SORLo method, and GGS method. However, the time of the MFP approach is better than the SORLopt strategy. Furthermore, we take and for Example 2 and compare all approaches in a graph form. The graphical outcomes are shown in Figure 2.
(a)
(b)
The graphical forms in Figures 1 and 2 show the implementation of the offered methods. Graphically, representation displays that the convergence of the presented strategies is faster than other methods.
5. Conclusion
We examined two new iteration approaches for determining AVEs called the MGS method as well as the MFP method, and we discussed the necessary circumstances for convergence between both methods. To ensure the efficacy of the defined approaches, numerical examples have also been conducted. According to the theoretical analyses and numerical investigations, the presented strategies seem to be vowing to determine the AVEs.
Data Availability
The data utilized to support the findings of this analysis are available from the corresponding author upon request.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This research was supported by JC2B6AYP.