3,702 research outputs found

    RealCertify: a Maple package for certifying non-negativity

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    Let Q\mathbb{Q} (resp. R\mathbb{R}) be the field of rational (resp. real) numbers and X=(X1,,Xn)X = (X_1, \ldots, X_n) be variables. Deciding the non-negativity of polynomials in Q[X]\mathbb{Q}[X] over Rn\mathbb{R}^n or over semi-algebraic domains defined by polynomial constraints in Q[X]\mathbb{Q}[X] is a classical algorithmic problem for symbolic computation. The Maple package \textsc{RealCertify} tackles this decision problem by computing sum of squares certificates of non-negativity for inputs where such certificates hold over the rational numbers. It can be applied to numerous problems coming from engineering sciences, program verification and cyber-physical systems. It is based on hybrid symbolic-numeric algorithms based on semi-definite programming.Comment: 4 pages, 2 table

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

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    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.Comment: 16 pages, 3 figures, 2 table

    Prediction based task scheduling in distributed computing

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

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    We consider the problem of certifying an inequality of the form f(x)0f(x)\geq 0, xK\forall x\in K, where ff is a multivariate transcendental function, and KK is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and max-plus approximation. We assume that ff is given by a syntaxic tree, the constituents of which involve semialgebraic operations as well as some transcendental functions like cos\cos, sin\sin, exp\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).Comment: 7 pages, 3 figures, 3 tables, Appears in the Proceedings of the European Control Conference ECC'13, July 17-19, 2013, Zurich, pp. 2244--2250, copyright EUCA 201

    Formal Proofs for Nonlinear Optimization

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    We present a formally verified global optimization framework. Given a semialgebraic or transcendental function ff and a compact semialgebraic domain KK, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of ff over KK. This method allows to bound in a modular way some of the constituents of ff 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.Comment: 24 pages, 2 figures, 3 table

    Certifying isolated singular points and their multiplicity structure

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    This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construc-tion uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new vari-ables. We show that the roots of this new system include the original singular root but now with multiplicity one, and the new variables uniquely determine the multiplicity structure. Both constructions are "exact", meaning that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system
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