# On smooth relaxations of obstacle sets

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## Abstract

We consider the problem of avoiding obstacle sets described by finitely many smooth convex inequality constraints, as it frequently occurs in, for example, trajectory or location planning. We present and discuss a general method to relax such sets by the upper level set of a single smooth convex function, covering different smoothing approaches like hyperbolic and entropic smoothing.

Based on error bounds and Lipschitz continuity, special attention is paid to the computations of the maximal geometric approximation error and of a guaranteed safety margin. Our results thus allow to safely avoid the obstacle by obeying a single nonconvex smooth constraint. Numerical results indicate that our technique gives rise to smoothing methods which perform well even for smoothing parameters very close to zero.

### Keywords

Relaxation Error bound Lipschitz continuity Hyperbolic smoothing Entropic smoothing Obstacle problem## Notes

### Acknowledgements

We thank two anonymous referees for their precise and substantial remarks which helped to significantly improve this paper. The first author also wishes to thank Adil Bagirov and Diethard Klatte for bringing the references (Li and Fang 1997) and (Klatte 1998; Klatte and Thiere 1996), respectively, to his attention.

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