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An approach to solve local and global optimization problems based on exact objective filled penalty functions

  • *Corresponding author: Jiahui Tang

    *Corresponding author: Jiahui Tang 

This work is supported by National Natural Science Foundation of China(No.72072036)

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  • In this work, an exact objective penalty function and an exact objective filled penalty function are proposed to solve constrained optimization problems. There are two main innovations of these two functions: using the exact objective penalty function to find a locally optimal point; a better locally optimal point than the current one can be found by exact objective filled penalty function. A local search method and a global search method for constrained optimization problems are proposed. We can use this new global search method to compute an approximately globally optimal point for constrained optimization problems. The convergence of the sequences obtained by the new algorithms are also analyzed respectively. Finally, the rationality of these two search methods are illustrated by numerical experiments.

    Mathematics Subject Classification: Primary: 90C30.

    Citation:

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  • Table 1.  Numerical Results of the EOP Algorithm

    $ k $ $ \sigma_{k} $ $ \varepsilon_{k} $ $ M_{k} $ $ x_{k} $ $ f(x_{k}) $
    1 $ 10 $ 0.01 -25 $ (0.5045, -1.21, -1.12, -1.99)^{T} $
    2 $ 50 $ 0.001 -25 $ ( 0.7231, -1.4788, 2.6252, -5.0909 )^{T} $ $ -44.5773 $
    3 $ 500 $ 0.0001 -25 $ ( 0.7273, -1.4895, 2.6217, -5.1149 )^{T} $ $ -44.3926 $
    4 $ 5000 $ $ 1e-05 $ -25 $ ( 0.7300, -1.4963, 2.6195, -5.1301 )^{T} $ $ -44.2751 $
    5 $ 50000 $ $ 1e-06 $ -25 $ ( 0.7306, -1.4978, 2.6190, -5.1333 )^{T} $ $ -44.2502 $
    6 $ 500000 $ $ 1e-07 $ -25 $ ( 0.7309, -1.4987, 2.6187, -5.1355 )^{T} $ $ -44.2331 $
     | Show Table
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    Table 2.  Numerical Results of the EOP Algorithm and EOFP Algorithm

    $ n $ 10 15
    local optimal point of the EOP Algorithm (-0.0074, -0.0005, -0.0020, -0.0000, 0.0439, 0.0045, -0.0016, 0.0035, -0.0020, 0.0014)T (0.0000, 0.0000, -0.0000, 0.0000, -0.0000, 1.0001, 0.9974, 0.0000, -0.0000, 0.0000, -0.0000, -0.0000, 0.0000, 0.0000, -0.0000, )T
    globla optimal point of the EOFP Algorithm (0.0007, 0.0003, 0.0003, -0.0000, 0.0018, 0.0008, 0.0006, 0.0004, 0.0003, 0.0002)T (0.1986, 0.3573, -0.0269, 0.1132, -0.0393, -0.3979, -0.2196, 0.0445, -0.0046, 0.4476, -0.0678, -0.1024, 0.0134, 0.0345, -0.3531)T
    local optimal value of the EOP Algorithm $ -0.97385 $ $ 0.89497 $
    gobal optimal value of the EOFP Algorithm $ -0.99993 $ $ -1.50000 $
     | Show Table
    DownLoad: CSV
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