Numerical Constraint Satisfaction Problems with Non-isolated Solutions

Abstract

In recent years, interval constraint-based solvers have shown their ability to efficiently solve complex instances of non-linear numerical CSPs. However, most of the working systems are designed to deliver point-wise solutions with an arbitrary accuracy. This works generally well for systems with isolated solutions but less well when there is a continuum of feasible points (e.g. under-constrained problems, problems with inequalities). In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. In this paper, we address the issue of constructing concise inner and outer approximations of the complete solution set for non-linear CSPs. We propose a technique which combines the extreme vertex representation of orthogonal polyhedra 1,2,3, as defined in computational geometry, with adapted splitting strategies 4 to construct the approximations as unions of interval boxes. This allows for compacting the explicit representation of the complete solution set and improves efficiency.