Modulus-based Matrix Splitting Iteration Method Research Articles

An Efficient Class of Modulus-Based Matrix Splitting Methods for Nonlinear Complementarity Problems

In this paper, we present an efficient method for finding a numerical solution for nonlinear complementarity problems (NCPs). We first reformulate an NCP as an equivalent system of fixed-point equations and then present a modulus-based matrix splitting iteration method. We prove the convergence of the proposed method with theorems with the relevant conditions. Our preliminary numerical results show that the method is feasible and effective.

Convergence of modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

Convergence of modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

A two-step iteration method for the horizontal nonlinear complementarity problem

In this paper, for solving horizontal nonlinear complementarity problem (HNCP), a two-step modulus-based matrix splitting iteration method is established. The convergence analysis of the proposed method is presented. Numerical examples are reported to show the efficiency of the proposed method.

Two-step modulus-based matrix splitting iteration methods for retinex problem

Retinex theory was proposed by Land in the 1970s. Its theoretical basis is that the color of an object is determined by the reflection ability of the object to the light of long wave, medium wave, and short wave, rather than by the absolute value of the reflected light intensity. There is color sense consistency in Retinex theory. The main purpose of Retinex problem is to recover the reflection properties of objects from the images obtained under a certain illumination intensity, so as to obtain the real colors of objects. There are many methods for solving the Retinex problem. In this paper, based on a variational minimization model with physical constraints of reflectance value, we first transform the Retinex problem into a linear complementarity problem, and then propose a class of two-step modulus-based matrix splitting iteration methods to solve this problem. We also prove the convergence of the two-step modulus-based matrix splitting iteration methods for solving the linear complementarity problem. The experimental results show that the convergence speeds of the proposed methods are much faster than the existing methods for solving the Retinex problem, and that the advantages of the new methods over the existing ones could be significant in the quality of the reflectance recovery.

Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem

In this paper, we obtain a new equivalent fixed-point form of the linear complementarity problem by introducing a relaxed matrix and establish a class of relaxed modulus-based matrix splitting iteration methods for solving the linear complementarity problem. Some sufficient conditions for guaranteeing the convergence of relaxed modulus-based matrix splitting iteration methods are presented. Numerical examples are offered to show the efficacy of the proposed methods.

Accelerated Relaxation Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

For solving large sparse linear complementarity problems effectively, by utilizing the parametric method, the acceleration strategy and the relaxation technique to the modulus-based matrix splitting (MMS) iteration method, we develop the accelerated relaxation MMS (ARMMS) iteration method, which generalizes the generalized accelerated MMS (GAMMS) and the relaxation MMS (RMMS) ones proposed recently. Meanwhile, it is proved that the ARMMS iteration method is convergent under proper restrictions when the system matrix is a positive definite matrix or an $$H_{+}$$ -matrix. In addition, we also study the convergence properties of the ARM accelerated overrelaxation (ARMAOR) method. Numerical experiments show that the proposed iteration method is efficient, and it outperforms some existing ones with suitable choice of the parameter and matrix splitting.

A Robust Modulus-Based Matrix Splitting Iteration Method for Mixed-Cell-Height Circuit Legalization

Modern circuits often contain standard cells of different row heights to meet various design requirements. Taller cells give larger drive strengths and higher speed at the cost of larger areas and power. Multi-row height standard cells incur challenging issues for layout designs, especially the mixed-cell-height legalization problem with heterogeneous cell structures. Honoring the good cell positions from global placement, we present in this article a robust modulus-based matrix splitting iteration method (RMMSIM) to solve the mixed-cell-height legalization problem. Fixing the cell ordering from global placement and relaxing the right-boundary constraints, our proposed method first converts the problem into an equivalent linear complementarity problem (LCP), and then properly splits the matrices in the LCP so that the RMMSIM can solve the LCP optimally. The RMMSIM effectively explores the sparse characteristic of a circuit, and takes only linear time per iteration; as a result, it can solve the QP very efficiently. Finally, an allocation scheme for illegal cells is used to align such cells to placement sites on rows and fix the placement of out-of-right-boundary cells, if any. Experimental results show the effectiveness and efficiency of our proposed algorithm. In addition, the RMMSIM convergence and optimality are theoretically proved and empirically validated. In particular, this article provides a new RMMSIM formulation for various optimization problems that require solving large-scale convex quadratic programming problems efficiently.

A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems

In this paper, for solving horizontal linear complementarity problems, a two-step modulus-based matrix splitting iteration method is established. The convergence analysis of the proposed method is presented, including the case of accelerated overrelaxation splitting. Numerical examples are reported to show the efficiency of the proposed method.

Convergence analysis on matrix splitting iteration algorithm for semidefinite linear complementarity problems

In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the modulus-based matrix splitting iteration methods, which are obtained by reformulating equivalently SDLCP as an implicit fixed-point matrix equation. The convergence of the proposed modulus-based matrix splitting iteration methods has been analyzed. Numerical experiments have shown that the modulus-based iteration methods are effective for solving SDLCPs.

Two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems

The modulus-based matrix splitting iteration has received substantial attention as a momentous tool for complementarity problems. For the purpose of solving the horizontal linear complementarity problem, we introduce the two-step modulus-based matrix splitting iteration method. We also show the theoretical analysis of the convergence. Numerical experiments illustrate the effectiveness of the proposed approach.

Mixed-cell-height legalization considering complex minimum width constraints and half-row fragmentation effect

With the increasing complexity in modern circuit designs, the 6T&6TPPNN circuits have become popular in advanced technologies for better trade-offs among routability, timing, power and performance. The 6TPPNN cells incur great challenges to layout designs, especially the legalization problem due to their complex minimum width (MW) constraints and the fragmentation effect (FE). In this paper, we address the mixed 6T&6TPPNN cell legalization problem considering both MW constraints and FE issues. Given a global placement result, we first align all cells to the correct rows/half-rows that satisfy the VDD/VSS alignment constraints, and then propose a clustering-based MW constraints solving method to eliminate all the MW violations. After that, we formulate the legalization problem with MW constraints as a quadratic programming (QP) problem, which not only satisfies the MW constraints, but also avoid causing excessive dead spaces. Finally, we integrate the FE issues into the formulated MW-aware QP model, and adopt the modulus-based matrix splitting iteration method (MMSIM) to solve the mathematical model effectively and efficiently. Experimental results show that our algorithm can resolve all MW constraints and mitigate the half-row fragmentation effect without any extra area overhead in a reasonable time.

On convergence of the modulus-based matrix splitting iteration method for horizontal linear complementarity problems of [formula omitted]-matrices

Horizontal linear complementarity problem has wide applications, such as in mechanical and electrical engineering, structural mechanics, piecewise linear system, telecommunication systems and so on. In this paper, we focus on the convergence conditions of the modulus-based matrix splitting iteration method proposed recently for solving horizontal linear complementarity problems. By the proposed theorems, the assumptions on the matrix splitting and the system matrices are weakened, and the convergence domain is enlarged. Numerical examples are presented to show the improvement.

A class of two-step modulus-based matrix splitting iteration methods for quasi-complementarity problems

In this paper, a class of two-step modulus-based matrix splitting (TMMS) iteration methods are proposed to solve the quasi-complementarity problems. In the new TMMS iteration method, we first use the modulus technique to equivalently transform the quasi-complementarity problem into a fixed point equation, and then construct a class of two-step matrix splitting iteration methods to further accelerate the convergence rates. Theoretical results show that the proposed TMMS iteration methods are global convergent under certain conditions when the system matrix is either a positive definite matrix or an $$H_+$$-matrix. Finally, two numerical examples are presented to demonstrate that the TMMS iteration methods are superior to some existing modulus-based iteration methods studied recently for solving the quasi-complementarity problems.

The Nonlinear Lopsided HSS-Like Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems with Positive-Definite Matrices

In this paper, by means of constructing the linear complementarity problems into the corresponding absolute value equation, we raise an iteration method, called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method, for solving the linear complementarity problems whose coefficient matrix in $$\mathbb {R}^{n\times n}$$ is large sparse and positive definite. From the convergence analysis, it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions. Numerical examples demonstrate that the presented method precede to other methods in practical implementation.

Two-Step Modulus-Based Matrix Splitting Iteration Methods for a Class of Implicit Complementarity Problems

Two-Step Modulus-Based Matrix Splitting Iteration Methods for a Class of Implicit Complementarity Problems

A Relaxation Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

A Relaxation Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Newton-type methods for solving quasi-complementarity problems via sign-based equation

In this paper, for solving quasi-complementarity problems, the discussion of the sign patterns of the solution is presented. Under some assumptions, the proposed theorem shows that the sign patterns of the solution of the quasi-complementarity problem can be obtained by solving a linear system. Then, a Newton’s iteration can be applied to a differentiable nonlinear sign-based equation. The quadratic convergence conditions of the Newton iteration are given by discussing the singularity of the Jacobian. Moreover, a hybrid method is established by the existing modulus-based matrix splitting iteration method to obtain global convergence. By numerical examples, the proposed methods are shown to have higher precision and faster convergence rate than some existing methods. Meanwhile, the hybrid method is a practical efficient method in application.

Modulus-based matrix splitting methods for horizontal linear complementarity problems

In this paper, we extend modulus-based matrix splitting iteration methods to horizontal linear complementarity problems. We consider both standard and accelerated methods and analyze their convergence. In this context, we also generalize existing results on modulus-based matrix splitting iteration methods for (non-horizontal) linear complementarity problems. Lastly, we analyze the proposed methods by numerical experiments involving both symmetric and non-symmetric matrices.

Two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

In this paper, a class of two-step modulus-based matrix splitting (TMMS) iteration methods are proposed to solve the implicit complementarity problems. It is proved that the TMMS iteration methods are convergent under certain conditions when the system matrix is either a positive definite matrix or an H+-matrix. Two numerical examples are given to illustrate the effectiveness of the new proposed iteration methods. Numerical results show that the new proposed TMMS iteration methods have better performance than the existing modulus-based relaxation iteration methods for solving the implicit complementarity problems.

Modulus-Based Matrix Splitting Iteration Methods for a Class of Stochastic Linear Complementarity Problem

For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.