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Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems

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  • Mengwei Xu
  • Jane Ye
  • Liwei Zhang

Abstract

In this paper, we propose a smoothing augmented Lagrangian method for finding a stationary point of a nonsmooth and nonconvex optimization problem. We show that any accumulation point of the iteration sequence generated by the algorithm is a stationary point provided that the penalty parameters are bounded. Furthermore, we show that a weak version of the generalized Mangasarian Fromovitz constraint qualification (GMFCQ) at the accumulation point is a sufficient condition for the boundedness of the penalty parameters. Since the weak GMFCQ may be strictly weaker than the GMFCQ, our algorithm is applicable for an optimization problem for which the GMFCQ does not hold. Numerical experiments show that the algorithm is efficient for finding stationary points of general nonsmooth and nonconvex optimization problems, including the bilevel program which will never satisfy the GMFCQ. Copyright Springer Science+Business Media New York 2015

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  • Mengwei Xu & Jane Ye & Liwei Zhang, 2015. "Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems," Journal of Global Optimization, Springer, vol. 62(4), pages 675-694, August.
    • Handle: RePEc:spr:jglopt:v:62:y:2015:i:4:p:675-694
    DOI: 10.1007/s10898-014-0242-7
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    References listed on IDEAS

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    1. Jane J. Ye, 2011. "Necessary Optimality Conditions for Multiobjective Bilevel Programs," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 165-184, February.
    2. J. A. Mirrlees, 1999. "The Theory of Moral Hazard and Unobservable Behaviour: Part I," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(1), pages 3-21.
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    Cited by:

    1. Qian Liu & Yuqing Xu & Yang Zhou, 2020. "A class of exact penalty functions and penalty algorithms for nonsmooth constrained optimization problems," Journal of Global Optimization, Springer, vol. 76(4), pages 745-768, April.

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