The generalized least squares (LS) problem ... (Ax - b)[sup TW[sup -1](Ax - b) appears in, many application areas. Here W is an m × m symmetric positive definite matrix and A is an m × n matrix with m ≥ n. ...The generalized least squares (LS) problem ... (Ax - b)[sup TW[sup -1](Ax - b) appears in, many application areas. Here W is an m × m symmetric positive definite matrix and A is an m × n matrix with m ≥ n. Since the problem has many solutions in rank deficient case, some special preconditioned techniques are adapted to obtain the minimum 2-norm solution. A block SOR method and the preconditioned conjugate gradient (PCG) method are proposed here. Convergence and optimal relaxation parameter for the block SOR method are studied. An error bound for the PCG method is given. The comparison of these methods is investigated. Some remarks on the implementation of the methods and the operation cost are given as well. [ABSTRACT FROM AUTHOR]展开更多
In this article,we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem.For the rank-deficient problem with the small noisy data,we also giv...In this article,we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem.For the rank-deficient problem with the small noisy data,we also give the preprocessing method based on the QR decomposition with column pivoting.Then,we prove the global convergence of the new method when the initial point is strictly primal-dual feasible.Finally,for some rankdeficient problems with or without the small noisy data from the NETLIB collection,we compare it with other two popular interior-point methods,i.e.the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment.Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.展开更多
基金CNPq, Brazil!301035/93-8University of Macao!RG010/ 99- 00S / JXQ / FST
文摘The generalized least squares (LS) problem ... (Ax - b)[sup TW[sup -1](Ax - b) appears in, many application areas. Here W is an m × m symmetric positive definite matrix and A is an m × n matrix with m ≥ n. Since the problem has many solutions in rank deficient case, some special preconditioned techniques are adapted to obtain the minimum 2-norm solution. A block SOR method and the preconditioned conjugate gradient (PCG) method are proposed here. Convergence and optimal relaxation parameter for the block SOR method are studied. An error bound for the PCG method is given. The comparison of these methods is investigated. Some remarks on the implementation of the methods and the operation cost are given as well. [ABSTRACT FROM AUTHOR]
基金This work was supported in part by Grant 61876199 from National Natural Science Foundation of China,Grant YBWL2011085 from Huawei Technologies Co.,Ltd.,and Grant YJCB2011003HIInnovation Research Program of Huawei Technologies Co.,Ltd..The first author is grateful to professor Li-Zhi Liao for introducing him the interiorpoint methods when he visited Hong Kong Baptist University in July,2012.
文摘In this article,we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem.For the rank-deficient problem with the small noisy data,we also give the preprocessing method based on the QR decomposition with column pivoting.Then,we prove the global convergence of the new method when the initial point is strictly primal-dual feasible.Finally,for some rankdeficient problems with or without the small noisy data from the NETLIB collection,we compare it with other two popular interior-point methods,i.e.the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment.Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.
