Robust Satisfiability of Systems of Equations

We study the problem of \emph{robust satisfiability} of systems of nonlinear equations, namely, whether for a given continuous function \(f:\,K\to\mathbb{R}^n\) on a~finite simplicial complex \(K\) and \(\alpha>0\), it holds that each function \(g:\,K\to\mathbb{R}^n\) such that \(\|g-f\|_\infty \...

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Bibliographic Details
Published in:arXiv.org 2014-02
Main Authors: Franek, Peter, Krcal, Marek
Format: Article
Language:English
Subjects:
Continuity (mathematics)
Homotopy theory
Mathematical analysis
Nonlinear equations
Nonlinear systems
Polynomials
Reduction
Robustness (mathematics)
Online Access:Get full text
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Summary:We study the problem of \emph{robust satisfiability} of systems of nonlinear equations, namely, whether for a given continuous function \(f:\,K\to\mathbb{R}^n\) on a~finite simplicial complex \(K\) and \(\alpha>0\), it holds that each function \(g:\,K\to\mathbb{R}^n\) such that \(\|g-f\|_\infty \leq \alpha\), has a root in \(K\). Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed \(n\), assuming \(\dim K \le 2n-3\). This is a substantial extension of previous computational applications of \emph{topological degree} and related concepts in numerical and interval analysis. Via a reverse reduction we prove that the problem is undecidable when \(\dim K\ge 2n-2\), where the threshold comes from the \emph{stable range} in homotopy theory. For the lucidity of our exposition, we focus on the setting when \(f\) is piecewise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.
ISSN:2331-8422