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

TL;DR

This paper presents a new approximation algorithm combining max-plus basis and linear templates methods to certify lower bounds of complex multivariate functions, enabling formal proofs in systems like Coq.

Contribution

It introduces a novel algorithm that bounds function components using quadratic forms, leading to semialgebraic problems solvable by sum-of-squares relaxations, with applications in formal proof certification.

Findings

Efficiently certifies lower bounds for complex functions.

Integrates max-plus basis and linear templates methods.

Demonstrates effectiveness on literature examples.

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.