An interior point continuous path-following trajectory for linear programming

Liming Sun; Li-Zhi Liao

doi:10.3934/jimo.2018107

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2019, Volume 15, Issue 4: 1517-1534. Doi: 10.3934/jimo.2018107

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An interior point continuous path-following trajectory for linear programming

Liming Sun^1, and
Li-Zhi Liao^2, ,

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
* Corresponding author: Li-Zhi Liao

Received: December 31, 2017

Revised: February 28, 2018

Early access: June 2018

Published: July 2019

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.

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C05; Secondary: 35A24.

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Figure 1. Transient behaviors of the continuous path of $x(t)$ and the objective function $c^Tx$ in Example 4.1 with starting point $x_0$

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Figure 2. Transient behaviors of the continuous path of $x(t)$ and the objective function $c^Tx$ in Example 4.1 with starting point $x_0^{'}$

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Figure 3. 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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Figure 4. 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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