On stopping criteria in verified nonlinear systems or optimization algorithms

Traditionally, iterative methods for nonlinear systems use heuristic domain and range stopping criteria to determine when accuracy tolerances have been met. However, such heuristics can cause stopping at points far from actual solutions, and can be unreliable due to the effects of round-off error or inaccuracies in data. In verified computations, rigorous determination of when a set of bounds has met a tolerance can be done analogously to the traditional approximate setting. Nonetheless, the range tolerance possibly cannot be met. If the criteria are used to determine when to stop subdivision of n-dimensional bounds into subregions, then failure of a range tolerance results in excessive, unnecessary subdivision, and could make the algorithm impractical. On the other hand, interval techniques can detect when inaccuracies or round-off will not permit residual bounds to be narrowed. These techniques can be incorporated into range thickness stopping criteria that complement the range stopping criteria. In this note, the issue is first introduced and illustrated with a simple example. The thickness stopping criterion is then formally introduced and analyzed. Third, inclusion of the criterion within a general verified global optimization algorithm is studied. An industrial example is presented. Finally, consequences and implications are discussed.