In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the s...In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at(0,0)and(2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.展开更多
An effective continuous algorithm is proposed to find approximate solutions of NP-hard max-cut problems. The algorithm relaxes the max-cut problem into a continuous nonlinear programming problem by replacing n discret...An effective continuous algorithm is proposed to find approximate solutions of NP-hard max-cut problems. The algorithm relaxes the max-cut problem into a continuous nonlinear programming problem by replacing n discrete constraints in the original problem with one single continuous constraint. A feasible direction method is designed to solve the resulting nonlinear programming problem. The method employs only the gradient evaluations of the objective function, and no any matrix calculations and no line searches are required. This greatly reduces the calculation cost of the method, and is suitable for the solution of large size max-cut problems. The convergence properties of the proposed method to KKT points of the nonlinear programming are analyzed. If the solution obtained by the proposed method is a global solution of the nonlinear programming problem, the solution will provide an upper bound on the max-cut value. Then an approximate solution to the max-cut problem is generated from the solution of the nonlinear programming and provides a lower bound on the max-cut value. Numerical experiments and comparisons on some max-cut test problems （small and large size） show that the proposed algorithm is efficient to get the exact solutions for all small test problems andwell satisfied solutions for most of the large size test problems with less calculation costs.展开更多
基金The work is partially supported by the National Natural Science Foundation of China (No. 11801362).
文摘In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at(0,0)and(2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.
基金This work is supported by National Natural Science Foundation of China at 10231060.
文摘An effective continuous algorithm is proposed to find approximate solutions of NP-hard max-cut problems. The algorithm relaxes the max-cut problem into a continuous nonlinear programming problem by replacing n discrete constraints in the original problem with one single continuous constraint. A feasible direction method is designed to solve the resulting nonlinear programming problem. The method employs only the gradient evaluations of the objective function, and no any matrix calculations and no line searches are required. This greatly reduces the calculation cost of the method, and is suitable for the solution of large size max-cut problems. The convergence properties of the proposed method to KKT points of the nonlinear programming are analyzed. If the solution obtained by the proposed method is a global solution of the nonlinear programming problem, the solution will provide an upper bound on the max-cut value. Then an approximate solution to the max-cut problem is generated from the solution of the nonlinear programming and provides a lower bound on the max-cut value. Numerical experiments and comparisons on some max-cut test problems （small and large size） show that the proposed algorithm is efficient to get the exact solutions for all small test problems andwell satisfied solutions for most of the large size test problems with less calculation costs.
