Modulus-based Matrix Splitting Iteration Method Research Articles

The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

<abstract><p>In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.</p></abstract>

A two-step new modulus-based matrix splitting method for vertical linear complementarity problem

In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516).

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

In this paper, by applying the accelerated technique and relaxation two-sweep strategy to the modulus-based matrix splitting iteration method, we establish the accelerated relaxation two-sweep modulus-based matrix splitting method. The proposed method contains the accelerated modulus-based matrix splitting and the generalized accelerated modulus-based matrix splitting methods presented recently as special cases. Compared with some existing methods, the proposed method can use more information during each iteration, which may lead to higher computational efficiency. We investigate the convergence properties of the new method, and prove that it is convergent under certain assumptions. Also, for the accelerated relaxation two-sweep modulus-based accelerated overrelaxation method, we analyze the convergence regions of the parameters in detail. Numerical examples are offered to show that the proposed method is efficient and performs better than the generalized accelerated modulus-based matrix splitting one in actual implementation.

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

A Two-Step Modulus-Based Matrix Splitting Iteration Method Without Auxiliary Variables for Solving Vertical Linear Complementarity Problems

A Two-Step Modulus-Based Matrix Splitting Iteration Method Without Auxiliary Variables for Solving Vertical Linear Complementarity Problems

Modulus-based synchronous multisplitting iteration methods without auxiliary variable for solving vertical linear complementarity problems

In this work, by applying the synchronous multisplitting technique to the non-auxiliary variable modulus equation of the vertical linear complementarity problems, a new parallel method is constructed, which can generalize the existing modulus-based matrix splitting iteration method. The convergence conditions of the proposed method are discussed in the cases of more than two system matrices, and the existing results are improved. Numerical examples under OpenACC framework show that the proposed method can solve the large sparse vertical linear complementarity problems efficiently with high parallel efficiency.

Relaxation modulus-based matrix splitting iteration method for vertical linear complementarity problem

For solving vertical linear complementarity problems (VLCPs), we propose a relaxation modulus-based matrix splitting iteration method and a two-step relaxation modulus-based matrix splitting iteration method. When the system matrices are H+-matrices, we analyze the convergence theory of the proposed methods. Numerical experiments show that relaxation modulus-based matrix splitting iteration method and the two-step relaxation modulus-based matrix splitting iteration method are promising and have great performance in higher dimension.

Improved modulus-based matrix splitting iteration methods for quasi-complementarity problems

In this paper, the improved modulus-based matrix splitting iteration methods are proposed to solve the quasi-complementarity problems. It is proved that the proposed iteration methods are convergent under certain conditions by using some new arguments. Numerical experiments are given to illustrate the high efficiency of the new iteration methods. Numerical results show that the proposed methods have better performance than the existing methods.

An Improved Convergence Condition of the MMS Iteration Method for Horizontal LCP of H+-Matrices

In this paper, inspired by the previous work in (Appl. Math. Comput., 369 (2020) 124890), we focus on the convergence condition of the modulus-based matrix splitting (MMS) iteration method for solving the horizontal linear complementarity problem (HLCP) with H+-matrices. An improved convergence condition of the MMS iteration method is given to improve the range of its applications, in a way which is better than that in the above published article.

The double-relaxation modulus-based matrix splitting iteration method for linear complementarity problems

For the large sparse linear complementarity problems (LCP), by adopting the relaxation and relaxation two-sweep techniques for the modulus-based matrix splitting (MMS) iteration method, we establish a double-relaxation two-sweep modulus-based matrix splitting (DRTMMS) iteration method. This new method contains some known ones developed recently. Some sufficient conditions for guaranteeing the convergence of the DRTMMS method are presented when the system matrices are positive definite matrices and H+-matrices, which generalize some existing results. And the parameter selection strategy of the DRTMMS iteration method is given. Besides, the convergence analyses of the DRTM accelerated overrelaxation (DRTMAOR) method are given. Finally, numerical experiments illustrate that the DRTMMS method is efficient and has better performance than several existing MMS-like methods, and outperforms the projected splitting methods in some cases.

A new relaxed acceleration two-sweep modulus-based matrix splitting iteration method for solving linear complementarity problems

<abstract><p>A new relaxed acceleration two-sweep modulus-based matrix splitting (NRATMMS) iteration method is developed to solve linear complementarity problems. The convergence of the NRATMMS method is established with the system matrix $ A $ being an $ H_{+} $-matrix. Numerical experiments show that the proposed method is superior to some existing algorithms under appropriate conditions.</p></abstract>

A preconditioned new modulus-based matrix splitting method for solving linear complementarity problem of $ H_+ $-matrices

<abstract><p>For solving the linear complementarity problem (LCP), we propose a preconditioned new modulus-based matrix splitting (PNMMS) iteration method by extending the state-of-the-art new modulus-based matrix splitting (NMMS) iteration method to a more general framework with a customized preconditioner. We devise a generalized preconditioner that is associated with both $ H_+ $-matrix $ A $ and vector $ q $ of the LCP. The convergence analysis is conducted under some mild conditions. In particular, we provide a comparison theorem to theoretically show the PNMMS method accelerates the convergence rate. Numerical experiments further illustrate that the PNMMS method is efficient and has better performance for solving the large and sparse LCP.</p></abstract>

CONVERGENCE OF THE TWO-POINT MODULUS-BASED MATRIX SPLITTING ITERATION METHOD

In this paper, we discuss the convergence of the two-point modulus-based matrix splitting iteration method for solving the linear complementarity problem. Some convergence conditions are presented from the spectral radius and the matrix norm when the system matrix is a <i>P</i>-matrix. Besides, the quasi-optimal cases of the method are studied. Numerical experiments are provided to show the presented results.

THE MODULUS-BASED MATRIX SPLITTING METHOD WITH INNER ITERATION FOR A CLASS OF NONLINEAR COMPLEMENTARITY PROBLEMS

In this paper, we propose a modulus-based matrix splitting iteration method with inner iteration for a class of nonlinear complementarity problems. Convergence conditions of the iteration method are analyzed carefully, which shows that the iteration sequence generated by this method converges to a solution of the NCP under certain conditions. Moreover, the convergence conditions of the proposed method are studied when the system matrix is symmetric positive definite or is an <i>H</i>₊-matrix. Theoretical results are supported by the numerical experiments, which implies that the iteration method with inner iteration is more effective and feasible for solving certain nonlinear complementarity problems.

A Robust Two-Step Modulus-Based Matrix Splitting Iteration Method for Mixed-Size Cell Circuit Legalization Problem

As semiconductor manufacturing moves into the nanotechnology era, mixed-size standard cell circuit designs have become mainstream, but they also pose significant challenges to placement. To address the mixed-size standard cell circuit legalization problem, in this paper, we first formulate it as a quadratic programming problem with nonnegative constraints, resulting in an equivalent linear complementarity problem (LCP). In particular, the system matrix involved in the LCP is positive semi-definite, so we perturb it to a positive-definite matrix. Then, we cast the linear complementarity problem as an equivalent implicit fixed-point equation. To improve the convergence rate of the class of modulus-based matrix splitting iteration method (MMS), we develop a robust two-step MMS to solve the fixed-point equation. Finally, we analyze the convergence property and illustrate the approximation of the solution between the perturbed LCP and the original LCP. Numerical experiments for the legalization problem are provided, and the results reveal that our approach is competitive with the existing state-of-the-art methods for solving mixed-size cell circuit legalization problems.

A preconditioned general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

A preconditioned general modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

Mixed-Cell-Height Placement With Complex Minimum-Implant-Area Constraints

Mixed-cell-height standard cells are prevailingly used in advanced technologies to achieve better design tradeoffs among timing, power, and routability. As feature size decreases, the placement of cells with multiple threshold voltages may violate the complex minimum-implant-area (MIA) layer rule arising from the limitations of patterning technologies. Existing works consider the mixed-cell-height placement problem only during legalization or handle the MIA constraints during detailed placement. In this article, we address the mixed-cell-height placement problem with MIA constraints in two major stages: 1) post-global placement (Post-GP) and 2) MIA-aware legalization. In the Post-GP stage, we first present a continuous and differentiable cost function to address the Vdd/Vss alignment constraints and add weighted pseudonets to MIA-violation cells dynamically. Then, we propose a proximal optimization method based on the given global placement result to simultaneously consider Vdd/Vss alignment constraints, MIA constraints, cell distribution, cell displacement, and total wirelength. In the MIA-aware legalization stage, we develop a graph-based method to cluster cells of specific threshold voltages and apply a strip-packing-based binary linear programming to reshape cells. Then, we propose a matching-based technique to resolve intrarow MIA violations and reduce filler insertion. Furthermore, we formulate inter-row MIA-aware legalization as a quadratic programming problem, which is efficiently solved by a modulus-based matrix splitting iteration method. Finally, MIA-aware cell allocation and refinement are performed to further improve the result. Experimental results show that without any extra area overhead, our algorithm still can achieve 5.4% shorter final total wirelength than the state-of-the-art work.

New convergence results of the modulus-based methods for vertical linear complementarity problems

In this work, for vertical linear complementarity problems, new convergence results of the modulus-based matrix splitting iteration method are given for the general cases, which can also extend and improve the existing ones proposed by “F. Mezzadri, A modulus-based formulation for the vertical linear complementarity problems, Numer. Algorithms 90 (2022) 1547–1568”.

An efficient modulus-based matrix splitting iteration method for quasi-complementarity problems

An efficient modulus-based matrix splitting iteration method for quasi-complementarity problems

An accelerated modulus-based matrix splitting iteration method for mixed-size cell circuits legalization

Mixed-size cell circuits dominate in advanced technology node designs, with attendant increases in layout complexity. The introduction of multi-row-height cells requires additional constraints, such as power line alignment, to be considered in the legalization stage, in addition to eliminating overlap while maintaining the results of the global placement stage as much as possible. In this paper, a three-stage legalization methodology is presented. The first is cell preprocessing, namely cell diffusion based on the network flow algorithm. Then the legalization problem is modeled as a quadratic programming problem, which is converted into a linear complementarity problem and solved by an accelerated modulus-based matrix splitting iteration method. Finally, the illegal cells are reshaped. Experimental results reveal that our method achieves 11.8% and 0.4% reduction in total displacement, respectively, over existing two state-of-the-art works, and is 1.407x faster than the modulus-based matrix splitting iteration method. In particular, the proposed approach provides a reference for solving large-scale quadratic programming problems.