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A smoothing augmented Lagrangian method for solving simple bilevel programs

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  • Mengwei Xu
  • Jane Ye

Abstract

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program. Copyright Springer Science+Business Media New York 2014

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  • Mengwei Xu & Jane Ye, 2014. "A smoothing augmented Lagrangian method for solving simple bilevel programs," Computational Optimization and Applications, Springer, vol. 59(1), pages 353-377, October.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:1:p:353-377
    DOI: 10.1007/s10589-013-9627-7
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    References listed on IDEAS

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    1. 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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    1. Alain B. Zemkoho & Shenglong Zhou, 2021. "Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(2), pages 625-674, March.
    2. Xide Zhu & Peijun Guo, 2017. "Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 255-275, October.
    3. Yu-Wei Li & Gui-Hua Lin & Xide Zhu, 2024. "Solving Bilevel Programs Based on Lower-Level Mond-Weir Duality," INFORMS Journal on Computing, INFORMS, vol. 36(5), pages 1225-1241, September.
    4. Jörg Fliege & Andrey Tin & Alain Zemkoho, 2021. "Gauss–Newton-type methods for bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(3), pages 793-824, April.
    5. Lorenzo Lampariello & Simone Sagratella, 2017. "A Bridge Between Bilevel Programs and Nash Games," Journal of Optimization Theory and Applications, Springer, vol. 174(2), pages 613-635, August.
    6. M. Beatrice Lignola & Jacqueline Morgan, 2017. "Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 183-202, April.
    7. Gaoxi Li & Zhongping Wan, 2018. "On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 820-837, December.

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