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A smoothing augmented Lagrangian method for nonconvex, nonsmooth constrained programs and its applications to bilevel problems

  • * Corresponding author: Mengwei Xu

    * Corresponding author: Mengwei Xu 
The second author is supported by NSFC grant 11601376. The third author is supported by NSFC grant 11601389, the Doctoral Foundation of Tianjin Normal University grant 52XB1513 and and 2017- Outstanding Young Innovation Team Cultivation Program of Tianjin Normal University grant 135202TD1703. The fourth author is supported by NSFC grant 11571059 and 11731013.
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  • In this paper, we consider a class of nonsmooth and nonconvex optimization problem with an abstract constraint. We propose an augmented Lagrangian method for solving the problem and construct global convergence under a weakly nonsmooth Mangasarian-Fromovitz constraint qualification. We show that any accumulation point of the iteration sequence generated by the algorithm is a feasible point which satisfies the first order necessary optimality condition provided that the penalty parameters are bounded and the upper bound of the augmented Lagrangian functions along the approximated solution sequence exists. Numerical experiments show that the algorithm is efficient for obtaining stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the nonsmooth Mangasarian-Fromovitz constraint qualification.

    Mathematics Subject Classification: 65K10, 90C26.

    Citation:

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  • Table 1.  Mirrlees' problem

    (x*; y*) d(x*; y*)
    Algorithm 3.1 (1, 0.957504) 5.73e-006
    SQP algorithm (1.000002, 0.957598) 9.79e-005
    SAL algorithm (1.000905, 0.957459) 9.06e-004
     | Show Table
    DownLoad: CSV

    Table 2.  Example 4.4

    (x*; y*) d(x*; y*)
    Algorithm 3.1 (0.500003, 0.500003) 4.08e-006
    SQP algorithm (0.499996, 0.499996) 5.85e-006
    SAL algorithm (0.500000, 0.499995) 2.89e-005
     | Show Table
    DownLoad: CSV
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