Inner and Outer Approximations of Existentially Quantified Equality Constraints

Abstract

Abstract. We propose a branch and prune algorithm that is able to compute inner and outer approximations of the solution set of an existentially quantified constraint where existential parameters are shared between several equations. While other techniques that handle such constraints need some preliminary formal simplification of the problem or only work on simpler special cases, our algorithm is the first pure numerical algorithm that can approximate the solution set of such constraints in the general case. Hence this new algorithm allows computing inner approximations that were out of reach until today.

Presentation of a highly tuned multithreaded interval solver for underdetermined and well-determined nonlinear systems

“…The first of the underdetermined ones is a set of two equations -a quadratic one and a linear one -in five variables [13]. It is called the Academic problem.…”

Set-membership state estimation with fleeting data

“…This operation requires a tube forv (t) which can be obtained thanks to the relationv (t) = ∂v ∂x (x (t)) * ẋ (t) . Note that an extension of Theorem 1 to the case where v is a vector function can be obtained by using the mathematical tools presented in [9]. Theorem 2.…”

Solving set-valued constraint satisfaction problems

“…The idea that is developed in this paper follows the foundations of interval analysis that has been built to handle uncertain real numbers [13], [18], to solve realvalued nonlinear problems (see e.g. [5], [14], [8]), or to provide mathematical proofs (see, e.g., [17], [6], [15]). …”