On smooth relaxations of obstacle sets

We present and discuss a method to relax sets described by finitely many smooth convex inequality constraints by the level set of a single smooth convex inequality constraint. Based on error bounds and Lipschitz continuity, special attention is paid to the maximal approximation error and a guaranteed safety margin. Our results allow to safely avoid the obstacle by obeying a single smooth constraint. Numerical results indicate that our technique gives rise to a smoothing method which performs well even for smoothing parameters very close to zero.