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A smoothing-regularization approach to mathematical programs with vanishing constraints

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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.

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Acknowledgements

The authors would like to thank two anonymous referees for their very constructive comments on an early version of this paper. Some preliminary works on this paper were finished while the first author was guest at the Department of Mathematics at the Technical University of Denmark (DTU), Lyngby/Copenhagen, Denmark. W. Achtziger is indebted to the Otto-Mønsted-Fonds making this stay possible. Moreover, W. Achtziger thanks M. Stolpe (DTU Wind Energy) for his very useful hints during the work on the numerical experiments.

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Authors and Affiliations

  1. Department of Mathematics, AM2, University of Erlangen-Nuremberg, Cauerstraße 11, 91058, Erlangen, Germany

    Wolfgang Achtziger

  2. Institute of Mathematics, University of Würzburg, Am Hubland, 97074, Würzburg, Germany

    Tim Hoheisel & Christian Kanzow

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  1. Wolfgang Achtziger
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  2. Tim Hoheisel
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  3. Christian Kanzow
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Correspondence to Wolfgang Achtziger.

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Achtziger, W., Hoheisel, T. & Kanzow, C. A smoothing-regularization approach to mathematical programs with vanishing constraints. Comput Optim Appl 55, 733–767 (2013). https://doi.org/10.1007/s10589-013-9539-6

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