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A smoothing-regularization approach to mathematical programs with vanishing constraints

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  • Wolfgang Achtziger
  • Tim Hoheisel
  • Christian Kanzow

Abstract

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothing- and regularization parameter t>0 such that the resulting problem is more tractable and coincides with the original program for t=0. We investigate the convergence behavior of a sequence of stationary points of the smooth and regularized problems by letting t tend to zero. Numerical results illustrating the performance of the approach are given. In particular, a large-scale example from topology optimization of mechanical structures with local stress constraints is investigated. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Wolfgang Achtziger & Tim Hoheisel & Christian Kanzow, 2013. "A smoothing-regularization approach to mathematical programs with vanishing constraints," Computational Optimization and Applications, Springer, vol. 55(3), pages 733-767, July.
    • Handle: RePEc:spr:coopap:v:55:y:2013:i:3:p:733-767
    DOI: 10.1007/s10589-013-9539-6
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    References listed on IDEAS

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    1. X. M. Hu & D. Ralph, 2004. "Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 365-390, November.
    2. A. F. Izmailov & M. V. Solodov, 2009. "Mathematical Programs with Vanishing Constraints: Optimality Conditions, Sensitivity, and a Relaxation Method," Journal of Optimization Theory and Applications, Springer, vol. 142(3), pages 501-532, September.
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    Cited by:

    1. Qingjie Hu & Jiguang Wang & Yu Chen, 2020. "New dualities for mathematical programs with vanishing constraints," Annals of Operations Research, Springer, vol. 287(1), pages 233-255, April.
    2. Sajjad Kazemi & Nader Kanzi, 2018. "Constraint Qualifications and Stationary Conditions for Mathematical Programming with Non-differentiable Vanishing Constraints," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 800-819, December.
    3. Tadeusz Antczak, 2022. "Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints," 4OR, Springer, vol. 20(3), pages 417-442, September.
    4. Tadeusz Antczak, 2023. "On directionally differentiable multiobjective programming problems with vanishing constraints," Annals of Operations Research, Springer, vol. 328(2), pages 1181-1212, September.
    5. Matúš Benko & Helmut Gfrerer, 2017. "An SQP method for mathematical programs with vanishing constraints with strong convergence properties," Computational Optimization and Applications, Springer, vol. 67(2), pages 361-399, June.

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