Matrix Splitting Methods Research Articles

Fast Distributed Voltage Control for PV Generation Clusters Based on Approximate Newton Method

Solar power generation is exhibiting a distributed development trend, and has changed conventional characteristics of power supply and consumption. The integration of large-scale distributed photovoltaic (PV) generation forms high-penetration PV clusters in distribution networks, which aims to organize and control geographically scattered resources. However, due to the inherent uncertainty, high-penetration PV clusters are suffering from random disturbances and serious voltage risks. Besides, integrating system via power electronic interfaces, PV generation usually tracks system frequency without any backup and regulation capacity, which poses significant challenges to system operation, and requires greater flexibility of voltage control. In fact, by using appropriate methods, PV inverters can autonomously regulate reactive power output in a distributed manner to improve voltage profiles in clusters. In this article, a distributed voltage control method for PV generation clusters is presented to realize decentralized coordination of PV inverters. Based on matrix splitting and approximate Newton method, it can fast respond to reactive power mismatch and realize voltage profiles optimization. Exhibiting a more efficient, reliable and flexible performance than existing decentralized methods in case studies, the proposed method is demonstrated to effectively organize and control PV cluster to provide corresponding reactive power support and exert friendly effect on system operation.

Modulus-based matrix splitting methods for a class of horizontal nonlinear complementarity problems

In this paper, we generalize modulus-based matrix splitting methods to a class of horizontal nonlinear complementarity problems (HNCPs). First, we write the HNCP as an implicit fixed-point equation and we introduce the proposed solution procedures. We then prove the convergence of the methods under some assumptions. We also comment on how the proposed methods and convergence theorems generalize existing results on (standard) linear and nonlinear complementarity problems and on horizontal linear complementarity problems. Finally, numerical experiments are solved to demonstrate the efficiency of the procedures. In this context, the effects of the splitting, of the dimension of the matrices, and of the nonlinear term of the problem are analyzed.

On the convergence of modulus-based matrix splitting methods for horizontal linear complementarity problems in hydrodynamic lubrication

In this paper, we analyze the solution of horizontal linear complementarity problems arising from finite-difference discretizations of differential problems. In particular, we do so in the framework of the modeling of cavitation in lubricated contacts. We analyze the solution of such complementarity problems by recently introduced generalizations of projected and modulus-based matrix splitting methods. In this context, we extend the convergence analysis of some modulus-based matrix splitting methods to the problems of our concern. Finally, we analyze numerically the considered solution techniques by solving several test problems arising in hydrodynamic lubrication.

On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems

In this paper, we analyze the relationship between projected and (possibly accelerated) modulus-based matrix splitting methods for linear complementarity problems. In particular, first we show that some well-known projected splitting methods are equivalent, iteration by iteration, to some (accelerated) modulus-based matrix splitting methods with a specific choice of the parameter $${\varOmega }$$. We then generalize this result to any $${\varOmega }$$ by formulating new classes of projected splitting methods and also provide a formal projection-based formulation for general (accelerated) modulus-based matrix splitting methods. Finally, we introduce and solve several test problems to evaluate also numerically the equivalence between the analyzed methods.

Characterizations, approximation and perturbations of the core-EP inverse

Several characterizations and representations for the core-EP inverse are presented and investigated. To that goal, several explicit bordering technique for characterizing and defining the core-EP inverse are analyzed. Also the matrix splitting method and the iterative successive matrix squaring algorithm for computing the core-EP inverse are investigated. The perturbation bounds related with the core-EP inverse are also estimated.

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.

The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady Navier–Stokes equations

In this paper, a generalized double shift-splitting (GDSS) preconditioner induced by a new matrix splitting method is proposed and implemented for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Detailed theoretical analysis of the iteration matrix is provided to show the GDSS method, which corresponds to the GDSS preconditioner, is unconditionally convergent. Additionally, a deteriorated GDSS (DGDSS) method is proposed. It is shown that, with suitable choice of parameter matrix, the DGDSS preconditioned matrix has an eigenvalue at 1 with multiplicity n, and the other m eigenvalues are of the form $$1-\lambda $$ with $$|\lambda |<1$$ , independently of the Schur complement matrix related. Finally, numerical experiments arising from a model Navier–Stokes problem are provided to validate and illustrate the effectiveness of the proposed preconditioner, with which a faster convergence for Krylov subspace iteration methods can be achieved.

Modeling transient churn-annular flows in a long vertical tube

This work investigates the transient behavior of high gas fraction gas-liquid flows in vertical pipes (annular and churn flows). Hyperbolic balance equations for mass, momentum and entropy are written for the gas and liquid, which is split between a continuous film and droplets entrained in the gas core. Closure relationships to calculate the wall and interfacial friction and the rates of droplet entrainment and deposition were obtained from the literature. A finite-difference solution algorithm based on a coefficient matrix splitting method was implemented to deal with sharp variations in the spatial and temporal domains, such as pressure and phase holdup waves. The model results were compared with steady-state experimental data from eight different sources, totaling more than 1500 data points for pressure gradient, liquid film flow rate and void/core fraction. The absolute average deviation between the model and the data was 17% for the pressure gradient and 5.8% for the void fraction. A comparison of the model results with fully transient air-water data generated in a 49-mm ID, 42-m long vertical pipe is also presented. The experimental results consist of two outlet pressure-induced and two inlet mass flow rate-induced transient tests. Two main transient parameters are compared, namely the local void fraction and the pressure difference between selected points along the test section and the outlet (taken as a reference). The comparisons between the experiments and the numerical model indicate that the model was capable of describing the transient annular to churn flow transition with absolute average deviations of 14.5% and 7.9% for the pressure difference and void fraction, respectively.

A modified general modulus-based matrix splitting method for linear complementarity problems of H-matrices

In this paper we establish a modified general modulus-based matrix splitting iteration method for solving the large sparse linear complementarity problems, and present the convergence analysis when the system matrix is an H+-matrix. Our result always improves some existing ones. In addition, numerical examples are given to show the effectiveness of the new method.

An Efficient Matrix Splitting Method for the Second-Order Cone Complementarity Problem

Given a symmetric and positive (semi)definite $n$-by-$n$ matrix $M$ and a vector, in this paper, we consider the matrix splitting method for solving the second-order cone linear complementarity problem (SOCLCP). The matrix splitting method is among the most widely used approaches for large scale and sparse classical linear complementarity problems, and its linear convergence is proved by [Luo and Tseng, SIAM J. Control Optim., 30 (1992), pp. 408--425]. Our first contribution is to prove that, when the general matrix splitting algorithm is applied to SOCLCP with $M$ symmetric and positive definite, it also converges at least linearly. Numerically, our second contribution is to propose a special and efficient matrix splitting algorithm, the block successive overrelaxation method, for solving the SOCLCP. The algorithm makes good use of the underlying geometry of the SOCLCP and each iteration only involves solving triangular linear systems and requires $O(n^2)$ flops; moreover, the algorithm does not destroy the sparse structure of $M$ and is able to exploit the sparsity effectively. Our code BSOR_BN_L based on this method is tested against four other state-of-the-art methods on problems with dense, ill-conditioned symmetric and positive definite matrices, as well as on problems with large scale sparse, symmetric and positive (semi)definite matrices. The numerical results are quite encouraging and BSOR_BN_L exhibits a very good performance in terms of its speed, accuracy, and efficiency in exploiting the sparsity of $M$.

Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an [formula omitted]-matrix

The theoretical analysis of the accelerated modulus-based matrix splitting iteration methods for the solution of the large sparse linear complementarity problem is further studied. The convergence conditions are presented by fully utilizing the H+-matrix property of the system matrix, and the optimal iteration parameters in accelerated modulus-based accelerated overrelaxation method are determined by minimizing the spectral radius of the iteration matrix. Numerical experiments further confirm the theoretical discussion, and show that the proposed methods accelerate the convergence of the modulus-based matrix splitting methods with less iteration steps and CPU time.

Modification of iterative methods for solving linear complementarity problems

Purpose – The purpose of this paper is to present the efficient iterative methods for solving linear complementarity problems (LCP), using a class of pre-conditioners. Design/methodology/approach – By using the concept of solving the fixed-point system of equations associated to the LCP, pre-conditioning techniques and Krylov subspace methods the authors design some projected methods to solve LCP. Furthermore, within the computational framework, some models of pre-conditioners candidates are investigated and evaluated. Findings – The proposed algorithms have a simple and graceful structure and can be applied to other complementarity problems. Asymptotic convergence of the sequence generated by the method to the unique solution of LCP is proved, along with a result regarding the convergence rate of the pre-conditioned methods. Finally, a computational comparison of the standard methods against pre-conditioned methods based on Example 1 is presented which illustrate the merits of simplicity, power and effectiveness of the proposed algorithms. Research limitations/implications – Comparison between the authors' methods and other similar methods for the studied problem shows a remarkable agreement and reveals that their models are superior in point of view rate of convergence and computing efficiency. Originality/value – For solving LCP more attention has recently been paid on a class of iterative methods called the matrix-splitting such as AOR, MAOR, GAOR, SSOR, etc. But up to now, no paper has discussed the effect of pre-conditioning technique for matrix-splitting methods in LCP. So, this paper is planning to fill in this gap and the authors use a class of pre-conditioners with iterative methods and analyze the convergence of these methods for LCP.

Iterative methods with analytical preconditioning technique to linear complementarity problems: application to obstacle problems

For solving linear complementarity problems LCP more attention has recently been paid on a class of iterative methods called the matrix-splitting. But up to now, no paper has discussed the effect of preconditioning technique for matrix-splitting methods in LCP . So, this paper is planning to fill in this gap and we use a class of preconditioners with generalized Accelerated Overrelaxation (GAOR ) methods and analyze the convergence of these methods for LCP . Furthermore, Comparison between our methods and other non-preconditioned methods for the studied problem shows a remarkable agreement and reveals that our models are superior in point of view of convergence rate and computing efficiency. Besides, by choosing the appropriate parameters of these methods, we derive same results as the other iterative methods such as AOR , JOR , SOR etc. Finally the method is tested by some numerical experiments.

Modified matrix splitting method for the support vector machine and its application to the credit classification of companies in Korea

This research proposes a solving approach for the ν-support vector machine (SVM) for classification problems using the modified matrix splitting method and incomplete Cholesky decomposition. With a minor modification, the dual formulation of the ν-SVM classification becomes a singly linearly constrained convex quadratic program with box constraints. The Kernel Hessian matrix of the SVM problem is dense and large. The matrix splitting method combined with the projection gradient method solves the subproblem with a diagonal Hessian matrix iteratively until the solution reaches the optimum. The method can use one of several line search and updating alpha methods in the projection gradient method. The incomplete Cholesky decomposition is used for the calculation of the large scale Hessian and vectors. The newly proposed method applies for a real world classification problem of the credit prediction for small-sized Korean companies.

A Complementarity Framework for Forward Contracting Under Uncertainty

We consider a particular instance of a stochastic multi-leader multi-follower equilibrium problem in which players compete in the forward and spot markets in successive periods. Proving the existence of such equilibria has proved difficult, as has the construction of globally convergent algorithms for obtaining such points. By conjecturing a relationship between forward and spot decisions, we consider a variant of the original game and relate the equilibria of this game to a related simultaneous stochastic Nash game where forward and spot decisions are made simultaneously. We characterize the complementarity problem corresponding to the simultaneous Nash game and prove that it is indeed solvable. Moreover, we show that an equilibrium to this Nash game is a local Nash equilibrium of the conjectured variant of the multi-leader multi-follower game of interest. Numerical tests reveal that the difference between equilibrium profits between the original and constrained games are small. Under uncertainty, the equilibrium point of interest is obtainable as the solution to a stochastic mixed-complementarity problem. Based on matrix-splitting methods, a globally convergent decomposition method is suggested for such a class of problems. Computational tests show that the effort grows linearly with the number of scenarios. Further tests show that the method can address larger networks as well. Finally, some policy-based insights are drawn from utilizing the framework to model a two-settlement six-node electricity market.

Inexact multisplitting methods for linear complementarity problems

We present an inexact multisplitting method for solving the linear complementarity problems, which is based on the inexact splitting method and the multisplitting method. This new method provides a specific realization for the multisplitting method and generalizes many existing matrix splitting methods for linear complementarity problems. Convergence for this new method is proved when the coefficient matrix is an H + -matrix. Then, two specific iteration forms for this inexact multisplitting method are presented, where the inner iterations are implemented either through a matrix splitting method or through a damped Newton method. Convergence properties for both these specific forms are analyzed, where the system matrix is either an H + -matrix or a symmetric matrix.

Qualitative analysis of one-step iterative methods and consistent matrix splittings

A qualitative analysis of one-step iterative methods is presented with special regard to the connection between concavity preservation and time-monotonicity. We also analyze the relation of one-step iterative methods to matrix splitting methods.

A matrix-splitting method for symmetric affine second-order cone complementarity problems

The affine second-order cone complementarity problem (SOCCP) is a wide class of problems that contains the linear complementarity problem (LCP) as a special case. The purpose of this paper is to propose an iterative method for the symmetric affine SOCCP that is based on the idea of matrix splitting. Matrix-splitting methods have originally been developed for the solution of the system of linear equations and have subsequently been extended to the LCP and the affine variational inequality problem. In this paper, we first give conditions under which the matrix-splitting method converges to a solution of the affine SOCCP. We then present, as a particular realization of the matrix-splitting method, the block successive overrelaxation (SOR) method for the affine SOCCP involving a positive definite matrix, and propose an efficient method for solving subproblems. Finally, we report some numerical results with the proposed algorithm, where promising results are obtained especially for problems with sparse matrices.

Convergence rate analysis of iteractive algorithms for solving variational inequality problems

We present a unified convergence rate analysis of iterative methods for solving the variational inequality problem. Our results are based on certain error bounds; they subsume and extend the linear and sublinear rates of convergence established in several previous studies. We also derive a new error bound for $\gamma$-strictly monotone variational inequalities. The class of algorithms covered by our analysis in fairly broad. It includes some classical methods for variational inequalities, e.g., the extragradient, matrix splitting, and proximal point methods. For these methods, our analysis gives estimates not only for linear convergence (which had been studied extensively), but also sublinear, depending on the properties of the solution. In addition, our framework includes a number of algorithms to which previous studies are not applicable, such as the infeasible projection methods, a separation-projection method, (inexact) hybrid proximal point methods, and some splitting techniques. Finally, our analysis covers certain feasible descent methods of optimization, for which similar convergence rate estimates have been recently obtained by Luo [14].

Qualitative analysis of matrix splitting methods

Qualitative properties of matrix splitting methods for linear systems with tridiagonal and block tridiagonal Stieltjes-Toeplitz matrices are studied. Two particular splittings, the so-called symmetric tridiagonal splittings and the bidiagonal splittings, are considered, and conditions for qualitative properties like nonnegativity and shape preservation are shown for them. Special attention is paid to their close relation to the well-known splitting techniques like regular and weak regular splitting methods. Extensions to block tridiagonal matrices are given, and their relation to algebraic representations of domain decomposition methods is discussed. The paper is concluded with illustrative numerical experiments.