Formal Proofs for Nonlinear Optimization


  • Victor Magron LAAS-CNRS
  • Xavier Allamigeon INRIA and CMAP, Ecole Polytechnique, CNRS, Palaiseau
  • Stéphane Gaubert INRIA and CMAP, Ecole Polytechnique, CNRS, Palaiseau
  • Benjamin Werner LIX, Ecole Polytechnique, CNRS, Palaiseau



Polynomial Optimization Problems, Hybrid Symbolic-numeric Certification, Semidefinite Programming, Transcendental Functions, Semialgebraic Relaxations, Flyspeck Project, Maxplus Approximation, Templates Method, Proof Assistant


We present a formally verified global optimization framework. Given a semialgebraic or transcendental function f and a compact semialgebraic domain K, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of f over K.
This method allows to bound in a modular way some of the constituents of f by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations.  
Our implementation tool interleaves  semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent.
The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.


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How to Cite

Magron, V., Allamigeon, X., Gaubert, S., & Werner, B. (2015). Formal Proofs for Nonlinear Optimization. Journal of Formalized Reasoning, 8(1), 1–24.