# American Institute of Mathematical Sciences

September  2020, 16(5): 2267-2281. doi: 10.3934/jimo.2019053

## Differential equation method based on approximate augmented Lagrangian for nonlinear programming

 School of Mathematics, Zhejiang Ocean University, Key Laboratory of Oceanographic Big Data Mining & Application of Zhejiang Province, Zhoushan, Zhejiang 316022, China

* Corresponding author: Hongying Huang

Received  May 2018 Revised  February 2019 Published  May 2019

Fund Project: The research is supported by the National Natural Science Foundation of China under Grant Nos.61673352 and 11771398

This paper analyzes the approximate augmented Lagrangian dynamical systems for constrained optimization. We formulate the differential systems based on first derivatives and second derivatives of the approximate augmented Lagrangian. The solution of the original optimization problems can be obtained at the equilibrium point of the differential equation systems, which lead the dynamic trajectory into the feasible region. Under suitable conditions, the asymptotic stability of the differential systems and local convergence properties of their Euler discrete schemes are analyzed, including the locally quadratic convergence rate of the discrete sequence for the second derivatives based differential system. The transient behavior of the differential equation systems is simulated and the validity of the approach is verified with numerical experiments.

Citation: Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2267-2281. doi: 10.3934/jimo.2019053
##### References:

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##### References:
Performances of the variable $x$ and $z$ in Problem 71
Performances of the variable $x$ and $z$ in Problem 53
Performances of the variable $x$ and $z$ in Problem 100
Performances of the variable $x$ and $z$ in Problem 113
Performances of the variable x and z in Problem 100
Performances of the cost function and the objective function in Problem 100
Performances of the variable x and z in Problem 113
Performances of the cost function and the objective function in Problem 113
numerical results
 Test n p q IT $S(z)$ $f(x^*)$ $F(x^*)$ P.71 4 10 1 349 8.125604 $\times10^{-10}$ 17.014 17.0140173 P.53 5 13 3 127 1.175666 $\times10^{-11}$ 4.0930 4.093023 P.100 7 4 0 967 3.829630$\times10^{-12}$ 678.6796 680.6300573 P.113 10 8 0 991 2.452665$\times10^{-12}$ 24.3062 24.306291
 Test n p q IT $S(z)$ $f(x^*)$ $F(x^*)$ P.71 4 10 1 349 8.125604 $\times10^{-10}$ 17.014 17.0140173 P.53 5 13 3 127 1.175666 $\times10^{-11}$ 4.0930 4.093023 P.100 7 4 0 967 3.829630$\times10^{-12}$ 678.6796 680.6300573 P.113 10 8 0 991 2.452665$\times10^{-12}$ 24.3062 24.306291
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