Validating numerical semidefinite programming solvers for polynomial invariants

  • Pierre Roux
  • Yuen-Lam Voronin
  • Sriram Sankaranarayanan
Article
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Abstract

Semidefinite programming (SDP) solvers are increasingly used as primitives in many program verification tasks to synthesize and verify polynomial invariants for a variety of systems including programs, hybrid systems and stochastic models. On one hand, they provide a tractable alternative to reasoning about semi-algebraic constraints. However, the results are often unreliable due to “numerical issues” that include a large number of reasons such as floating-point errors, ill-conditioned problems, failure of strict feasibility, and more generally, the specifics of the algorithms used to solve SDPs. These issues influence whether the final numerical results are trustworthy or not. In this paper, we briefly survey the emerging use of SDP solvers in the static analysis community. We report on the perils of using SDP solvers for common invariant synthesis tasks, characterizing the common failures that can lead to unreliable answers. Next, we demonstrate existing tools for guaranteed semidefinite programming that often prove inadequate to our needs. Finally, we present a solution for verified semidefinite programming that can be used to check the reliability of the solution output by the solver and a padding procedure that can check the presence of a feasible nearby solution to the one output by the solver. We report on some successful preliminary experiments involving our padding procedure.

Keywords

Static analysis Polynomial invariants Numerical optimization Sum of squares relaxation Semidefinite programming Floating-point arithmetic Formal proofs 

Notes

Acknowledgements

The authors would like to thank Didier Henrion, Pierre-Loïc Garoche and Assalé Adjé for interesting discussions on this subject.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Pierre Roux
    • 1
  • Yuen-Lam Voronin
    • 2
  • Sriram Sankaranarayanan
    • 2
  1. 1.ONERA – The French Aerospace LabToulouseFrance
  2. 2.University of ColoradoBoulderUSA

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