17 - Approximation of sets defined with quantifiers

Summary

Introduction

In many applications of interest one has to handle a class of sets R ⊂ ℝⁿ whose definition contains a set K ⊂ ℝ^n+p and some quantifier ∃ or ∀. In other words, R is described with the help of additional variables y ∈ ℝ^p and constraints linking x ∈ ℝⁿ and y ∈ ℝ^p. To obtain an exact explicit description of R ⊂ ℝⁿ solely in terms of the x-variables is a difficult and challenging problem in general. If K ⊂ ℝ^n+p is semi-algebraic then it is certainly possible and this is part of Elimination Theory in Algebraic Geometry. Most of the associate dalgorithms are based on Gröbner basis methods (with exact arithmetic) and in practice quantifier elimination is very costly and is limited to small size problems or particular cases.

The goal of this chapter is less ambitious as we only want to obtain approximations of R with some convergence properties. We provide a nested sequence of outer (respectively inner) approximations R^k, k ∈ ℕ, such that the Lebesgue volume of R^k converges to that of R as k → ∞. Moreover the outer (respectively inner) approximations R^k are of the form {x ∈ ℝⁿ : p_k(x) ≤ 0} for some polynomial p_k, k ∈ ℕ, of increasing degree. Therefore approximations (possibly crude if k is small) can be obtained for cases where exact elimination is out of reach.

Problem statement

Consider two sets of variables x ∈ ℝⁿ and y ∈ ℝ^m coupled with a constraint (x, y) ∈ K, where K is the basic semi-algebraic set:

K ≔ {(x, y) ∈ ℝⁿ × ℝ^m : x ∈ B; g_j (x, y) ≥ 0, j = 1, …, s}

for some polynomials g_j, j = 1, …, s, and B ⊂ ℝⁿ is a simple set (e.g. some box or ellipsoid).