NDLI: Certification of inequalities involving transcendental functions: Combining SDP and max-plus approximation

Content Provider	IEEE Xplore Digital Library
Author	Allamigeon, X. Gaubert, S. Magron, V. Werner, B.
Copyright Year	2013
Description	Author affiliation: INRIA & LIX, Ecole Polytech., Palaiseau, France (Magron, V.; Werner, B.) \|\| INRIA & CMAP, Ecole Polytech., Palaiseau, France (Allamigeon, X.; Gaubert, S.)
Abstract	We consider the problem of certifying an inequality of the form f(x) ≥ 0, ∀x ∈ K, where f is a multivariate transcendental function, and K is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and max-plus approximation. We assume that f is given by a syntaxic tree, the constituents of which involve semialgebraic operations as well as some transcendental functions like cos, sin, exp, etc. We bound some of these constituents by suprema or infima of quadratic forms (max-plus approximation method, initially introduced in optimal control), leading to semialgebraic optimization problems which we solve by semidefinite relaxations. The max-plus approximation is iteratively refined and combined with branch and bound techniques to reduce the relaxation gap. Illustrative examples of application of this algorithm are provided, explaining how we solved tight inequalities issued from the Flyspeck project (one of the main purposes of which is to certify numerical inequalities used in the proof of the Kepler conjecture by Thomas Hales).
Starting Page	2244
Ending Page	2250
File Size	403759
Page Count	7
File Format	PDF
ISBN	9783033039629
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2013-07-17
Publisher Place	Switzerland
Access Restriction	Subscribed
Rights Holder	EUCA
Subject Keyword	Non-linear Inequalities Polynomial Optimization Problems Semidefinite Programming Approximation methods Maxplus approximation Optimization Certification Convergence Flyspeck Project Upper bound Transcendental Functions Abstracts Branch and Bound Approximation algorithms Quadratic Cuts Polynomials Semialgebraic Relaxations Sum of Squares
Content Type	Text
Resource Type	Article

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