Changfeng Ma

Iterative method to solve the generalized coupled Sylvester-transpose linear matrix equations over reflexive or anti-reflexive matrix ☆

Abstract The iterative method of generalized coupled Sylvester-transpose linear matrix equations A X B + C Y T D = S 1 , E X T F + G Y H = S 2 over reflexive or anti-reflexive matrix pair ( X , Y ) is presented. On the condition that the coupled matrix equations are consistent, we show that the solution pair ( X ∗ , Y ∗ ) proposed by the iterative method can be obtained within finite iterative steps in the absence of roundoff-error for any initial value given a reflexive or anti-reflexive matrix. Moreover, the optimal approximation reflexive or anti-reflexive matrix solution pair to an arbitrary given reflexive or anti-reflexive matrix pair can be derived by searching the least Frobenius norm solution pair of the new generalized coupled Sylvester-transpose linear matrix equations. Finally, some numerical examples are given which illustrate that the introduced iterative algorithm is quite efficient.

A generalized shift-splitting preconditioner for singular saddle point problems

Recently, some authors (Cao et?al., 2014; Chen and Ma, 2015; Salkuyeh et?al., 2014) discussed the (generalized) shift-splitting preconditioner for nonsingular saddle point problems. In this paper, we further study the generalized shift-splitting preconditioner for solving singular saddle point problems with symmetric positive definite?(1, 1)-block. Theoretical analysis shows that the generalized shift-splitting iteration method is unconditionally semi-convergent. Numerical experiments are given to illustrate the efficiency of the proposed preconditioner with appropriate parameters.

A new SOR-Like method for the saddle point problems

Abstract In previous years, Golub, Wu and Yuan presented a generalized successive over-relaxation (SOR-Like) method for solving the saddle point problems. In this paper, we present a new SOR-Like (NSOR-Like) method which has three parameters. Our new method can be applied to the nonsingular saddle point problems as well as the singular cases. The characteristic of eigenvalues of the iteration matrix of this NSOR-Like method is analyzed. Then we give the convergence (semi-convergence) theorem of the new iterative method by giving the restrictions imposed on the parameter. Moreover, that convergence (semi-convergence) theorem is applied to some special cases to give the convergence region for the parameters. We can see that NSOR-Like method has a wider convergence (semi-convergence) region for ω and τ than the Parameterized Uzawa method which covers Preconditioned Uzawa method and Uzawa method. In addition, the optimal iteration parameters and the corresponding convergence (semi-convergence) factor for the Uzawa method are presented.

A globally convergent Levenberg-Marquardt method for solving nonlinear complementarity problem

The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the least l2-norm solution of a optimization problem. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a smoothing Levenberg–Marquardt method is applied to solve the parameterized optimization problems. The global convergence of the proposed method is proved under an assumption that the level set of the problem is compact. 2007 Elsevier Inc. All rights reserved.

A new smoothing and regularization Newton method for P0-NCP

The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization Newton method for solving nonlinear complementarity problem with P0-function (P0-NCP). Without requiring strict complementarity assumption at the P0-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented compared to many previous methods.

The quadratic convergence of a smoothing Levenberg–Marquardt method for nonlinear complementarity problem

Abstract The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a possibly inconsistent nonsmooth system of equations. Based on the ideas developed in smoothing Newton methods, we approximated the problem of the least l2-norm solution of the equivalent nonsmooth equations of NCP(F) with a family of parameterized optimization problem with twice continuously differentiable objective functions by making use of a new smoothing function. Then we presented a smoothing Levenberg–Marquardt method to solve the parameterized smooth optimization problem. By using the smooth and semismooth technique, the local quadratic convergence of the proposed method is proved under some suitable assumptions.

A globally and superlinearly convergent smoothing Broyden-like method for solving nonlinear complementarity problem

Abstract The nonlinear complementarity problem (denoted by NCP( F )) has attracted much attention due to its various applications in economics, engineering and management science. In this paper, we propose a smoothing Broyden-like method for solving nonlinear complementarity problem. The algorithm considered here is based on the smooth approximation Fischer–Burmeister function and makes use of the derivative-free line search rule of Li in [D.H. Li, M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optim. Meth. Software 13(3) (2000) 181–201]. We show that, under suitable conditions, the iterates generated by the proposed method converge to a solution of the nonlinear complementarity problem globally and superlinearly.

A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation

Abstract A nested splitting conjugate gradient (NSCG) iterative method and a preconditioned NSCG (PNSCG) iterative method are presented for solving the generalized Sylvester equation with large sparse coefficient matrices, respectively. Both methods are actually inner/outer iterations, which employ the CG-like method as inner iteration to approximate each outer iteration, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrices. Convergence conditions of both methods are studied in depth and numerical experiments demonstrate the efficiency of the proposed methods. Moreover, experimental results show that the PNSCG method is more accurate, robust and effective than the NSCG method.

A parameterized SHSS iteration method for a class of complex symmetric system of linear equations

In this paper, we present a parameterized variant of the single-step Hermitian and skew-Hermitian (SHSS) iteration method for solving a class of complex symmetric system of linear equations. We study the convergence properties of the parameterized SHSS (P-SHSS) iteration method, the choices of the parameters and the quasi-optimal parameters. Numerical experiments are given to verify the effectiveness of the P-SHSS iteration method.

An inexact relaxed DPSS preconditioner for saddle point problem

Based on the relaxed deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner, in this paper, we proposed a class of relaxed deteriorated positive-definite and skew-Hermitian splitting (RDPSS) preconditioner for solving the saddle point problem. The proposed RDPSS preconditioner is a technical modification of the deteriorated positive-definite and skew-Hermitian splitting (DPSS) preconditioner 36. The PSS preconditioner is a straightforward application of the positive-definite and skew-Hermitian splitting (PSS) iteration method for solving non-Hermitian positive definite linear systems initially established by Bai et?al. 37. Numerical results have shown that the proposed RDPSS preconditioner is advantageous over the existing DPSS preconditioner.