Approximation Techniques for Non-linear Problems with Continuum of Solutions

Abstract

Most of the working solvers for numerical constraint satisfaction problems (NCSPs) are designed to delivering point-wise solutions with an arbitrary accuracy. When there is a continuum of feasible points this might lead to prohibitively verbose representations of the output. In many practical applications, such large sets of solutions express equally relevant alternatives which need to be identified as completely as possible. The goal of this paper is to show that by using appropriate approximation techniques, explicit representations of the solution sets, preserving both accuracy and completeness, can still be proposed for NCSPs with continuum of solutions. We present a technique for constructing concise inner and outer approximations as unions of interval boxes. The proposed technique combines a new splitting strategy with the extreme vertex representation of orthogonal polyhedra [1,2,3], as defined in computational geometry. This allows for compacting the representation of the approximations and improves efficiency.