# American Institute of Mathematical Sciences

October  2019, 15(4): 1517-1534. doi: 10.3934/jimo.2018107

## An interior point continuous path-following trajectory for linear programming

 1 School of Science, Nanjing Audit University, Nanjing 211815, Jiangsu Province, China 2 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China

* Corresponding author: Li-Zhi Liao

Received  January 2018 Revised  March 2018 Published  October 2019 Early access  July 2018

Fund Project: The work of Liming Sun was supported in part by the National Natural Science Foundation of China (Grant No. 11701287) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20171071). The work of Li-Zhi Liao was supported in part by grants from the General Research Fund (GRF) of Hong Kong and FRG of Hong Kong Baptist University.

In this paper, an interior point continuous path-following trajectory is proposed for linear programming. The descent direction in our continuous trajectory can be viewed as some combination of the affine scaling direction and the centering direction for linear programming. A key component in our interior point continuous path-following trajectory is an ordinary differential equation (ODE) system. Various properties including the convergence in the limit for the solution of this ODE system are analyzed and discussed in detail. Several illustrative examples are also provided to demonstrate the numerical behavior of this continuous trajectory.

Citation: Liming Sun, Li-Zhi Liao. An interior point continuous path-following trajectory for linear programming. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1517-1534. doi: 10.3934/jimo.2018107
##### References:

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##### References:
Transient behaviors of the continuous path of $x(t)$ and the objective function $c^Tx$ in Example 4.1 with starting point $x_0$
Transient behaviors of the continuous path of $x(t)$ and the objective function $c^Tx$ in Example 4.1 with starting point $x_0^{'}$
Transient behaviors of the continuous path of $x(t)$ and the objective function $c^Tx$ in Example 4.2 with starting point $x_0$
Transient behaviors of the continuous path of $x(t)$ and the objective function $c^Tx$ in Example 4.2 with starting point $x_0^{'}$
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