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Differential equation method based on approximate augmented Lagrangian for nonlinear programming

  • * Corresponding author: Hongying Huang

    * Corresponding author: Hongying Huang

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

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  • 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.

    Mathematics Subject Classification: Primary: 90C30, 90C48; Secondary: 90C59.


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  • Figure 1.  Performances of the variable $ x $ and $ z $ in Problem 71

    Figure 2.  Performances of the variable $ x $ and $ z $ in Problem 53

    Figure 3.  Performances of the variable $ x $ and $ z $ in Problem 100

    Figure 4.  Performances of the variable $ x $ and $ z $ in Problem 113

    Figure 5.  Performances of the variable x and z in Problem 100

    Figure 6.  Performances of the cost function and the objective function in Problem 100

    Figure 7.  Performances of the variable x and z in Problem 113

    Figure 8.  Performances of the cost function and the objective function in Problem 113

    Table 1.  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
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