Quantified Constraints Under Perturbation

Quantified constraints (i.e. first-order formulae over the real numbers) are often exposed to perturbations: usually constants that come from measurements are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the behavior of quantified constraints under perturbation by showing that one can formulate the problem of solving quantified constraints as a nested parametric optimization problem followed by one sign computation. Using the fact that minima and maxima are stable under perturbation, but the sign of a real number is stable only for non-zero inputs, we derive practically useful conditions for the stability of quantified constraints under perturbation.