Efficient solving of quantified inequality constraints over the real numbers

Abstract

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In this article, we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques. This is a revised and extended version of RATSCHAN, S., Continuous first-order constraint satisfaction, in

Robust Optimization Over Time: Problem Difficulties and Benchmark Problems

“…The difficulty in calculating the absolute best survival time in (7) originates from the difficulty associated with general nonlinear arithmetic constraint satisfaction problems (CSPs) [34]. Given a sequence of static fitness functions ( f 1 , f 2 , .…”

“…As stated in [34], solving arbitrary nonlinear arithmetic CSPs over the real numbers is undecidable. This means if f t+i , 0 ≤ i ≤ l, is an arbitrary nonlinear function of x, it is impossible to construct a single algorithm that will always lead to a yes/no answer as to whether there exists a solution x * simultaneously satisfying all the constraints in (10).…”