In this article, we consider the  path-following method based on the perturbed Newton flow with the new trust-region time-stepping scheme for the standard linear programming problem. For the problem with deficient rank matrix and the noise right-hand-side vector , we also give the pre-processing method based on the singular value decomposition . Then, we analyze the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, we test the new method for some problems with deficient rank matrices, and compare it with other  popular interior-point methods such as the path-following method (the subroutine pathfollow.m coded by M.~C. Ferris \mbox{%DIFAUXCMD \cite{Gonzaga1992,FMW2007}\hspace{0pt}%DIFAUXCMD ) and Mehrotra's predictor-corrector algorithm (}  the built-in subroutine linprog.m of the MATLAB environment , which was implemented by S. Mehrotra and Y. Zhang \mbox{%DIFAUXCMD \cite{MATLAB,Mehrotra1992,Zhang1998}\hspace{0pt}%DIFAUXCMD ). }  Numerical results show that the new method is more robust than those methods for the large-scale deficient-rank problems without sacrificing its computational efficiency .
In this article, we consider the primal-dual path-following method and the trust-region strategy for the standard linear programming problem. For the deficient rank problem with the small noisy data , we also give the pre-processing 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 deficient rank 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 }\hspace{0pt}%DIFAUXCMD ) and Mehrotra's predictor-corrector algorithm (} and the built-in subroutine linprog.m of the MATLAB environment }\hspace{0pt}%DIFAUXCMD ). } . Numerical results show that the new method is more robust than the other two methods for the deficient rank problem with the small noise data .
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primal-dual
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based on the perturbed Newton flow with the new
and the
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time-stepping scheme
strategy
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problem with deficient rank matrix and the noise right-hand-side vector
deficient rank problem with the small noisy data
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singular value decomposition
QR decomposition with column pivoting
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analyze
prove
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we test the new method for some problems with deficient rank matrices, and
for some deficient rank problems with or without the small noisy data from the NETLIB collection, we
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two
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such as the path-following method (the
, i.e. the
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coded by M.~C. Ferris \mbox{%DIFAUXCMD \cite{Gonzaga1992,FMW2007
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and
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, which was implemented by S. Mehrotra and Y. Zhang \mbox{%DIFAUXCMD \cite{MATLAB,Mehrotra1992,Zhang1998
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.
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those
the other two
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large-scale deficient-rank problems without sacrificing its computational efficiency
deficient rank problem with the small noise data