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Certification of Bounds of Non-linear Functions: The Templates Method

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Book cover Intelligent Computer Mathematics (CICM 2013)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7961))

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Abstract

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales’ proof of Kepler’s conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.

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Author information

Authors and Affiliations

  1. INRIA and CMAP, École Polytechnique, Palaiseau, France

    Xavier Allamigeon & Stéphane Gaubert

  2. INRIA and LIX, École Polytechnique, Palaiseau, France

    Victor Magron & Benjamin Werner

Authors
  1. Xavier Allamigeon
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  2. Stéphane Gaubert
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  3. Victor Magron
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  4. Benjamin Werner
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Editor information

Editors and Affiliations

  1. Computing and Software, McMaster University, 1280 Main Street West, L8S 4K1, Hamilton, ON, Canada

    Jacques Carette

  2. School of Informatics, University of Edinburgh, 10 Crichton Street, EH8 9AB, Edinburgh, Scotland, UK

    David Aspinall

  3. School of Computer Science, University of Birmingham, Edgbaston, B15 2TT, Birmingham, UK

    Christoph Lange

  4. Faculty of Informatics, Department of Computer Graphics and Design, Masaryk University, Botanická 68a, 60200, Brno, Czech Republic

    Petr Sojka

  5. Johannes Kepler University Linz (JKU), RISC, Altenberger Strasse 69, 4040, Linz, Austria

    Wolfgang Windsteiger

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Allamigeon, X., Gaubert, S., Magron, V., Werner, B. (2013). Certification of Bounds of Non-linear Functions: The Templates Method. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds) Intelligent Computer Mathematics. CICM 2013. Lecture Notes in Computer Science(), vol 7961. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39320-4_4

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  • DOI: https://doi.org/10.1007/978-3-642-39320-4_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39319-8

  • Online ISBN: 978-3-642-39320-4

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