Formalization of Bernstein Polynomials and Applications to Global Optimization

Muñoz, César; Narkawicz, Anthony

doi:10.1007/s10817-012-9256-3

Cited by 65 publications

(53 citation statements)

References 24 publications

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“…Nevertheless, our next priority is to complete our validation by providing an automatic tool within COQ to bound a polynomial on an interval. Different standard techniques exist that have already been applied in a formal setting, such as sum of squares [17] or Bernstein polynomials [18]. Combined with our Taylor models which prove |f (x) − TM f (x)| < 1 for x ∈ I, we could also derive that |TM f (x) − P (x)| < 2 .…”

Section: Discussionmentioning

confidence: 88%

Certified, Efficient and Sharp Univariate Taylor Models in COQ

Martin-Dorel¹,

Rideau²,

Théry³

et al. 2013

2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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Section: Discussionmentioning

confidence: 88%

Certified, Efficient and Sharp Univariate Taylor Models in COQ

Martin-Dorel¹,

Rideau²,

Théry³

et al. 2013

2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing

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“…The basic and efficient GRIND strategy can be extended with additional theories for reals. PVS also provides some strategies based on numerical computations: numerical performs interval arithmetic to verify inequalities involving transcendental functions [10]; bernstein performs global optimization based on Bernstein polynomials to verify systems of polynomial inequalities [26].…”

Section: Pvs Pvsmentioning

confidence: 99%

Coquelicot: A User-Friendly Library of Real Analysis for Coq

2014

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Abstract. Real analysis is pervasive to many applications, if only because it is a suitable tool for modeling physical or socio-economical systems. As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems. The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results. For instance, power series are not developed further than their definition. Moreover, the definitions of integrals and derivatives are based on dependent types, which make them especially cumbersome to use in practice. To palliate these inadequacies, we have designed a userfriendly library: Coquelicot. An easier way of writing formulas and theorem statements is achieved by relying on total functions in place of dependent types for limits, derivatives, integrals, power series, and so on. To help with the proof process, the library comes with a comprehensive set of theorems that cover not only these notions, but also some extensions such as parametric integrals, two-dimensional differentiability, asymptotic behaviors. It also offers some automations for performing differentiability proofs. Moreover, Coquelicot is a conservative extension of Coq's standard library and we provide correspondence theorems between the two libraries. We have exercised the library on several use cases: in an exam at university entry level, for the definitions and properties of Bessel functions, and for the solution of the one-dimensional wave equation.

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“…The novelty in this work is not only the adaptation of these relaxations in the context of polynomial Lyapunov function synthesis but also a new tighter relaxation will be introduced by exploiting the induction relation between Bernstein polynomials. More details on Bernstein polynomials are available elsewhere [40].…”

Section: Overview Of Bernstein Polynomialsmentioning

confidence: 99%

“…As a first contribution of this paper, we extend the so-called Handelman representations, considered in our previous work [50], using the idea of Bernstein polynomials from approximation theory [6,16,40]. Bernstein polynomials are a special basis of polynomials that have many rich properties, especially over the unit interval [0,1].…”

Section: Introductionmentioning

confidence: 99%

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Sassi

Sankaranarayanan

Chen

et al. 2015

IMA J Math Control Info

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We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two classes of relaxations for encoding polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations based on Handelman representations and Bernstein polynomials, that produce linear programs. Next, we present a series of increasingly powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. Subsequently, we show how these LP relaxations can be used to search for Lyapunov functions for polynomial ODEs by formulating LP instances. We compare our techniques with approaches based on SOS on a suite of automatically synthesized benchmarks. Posi-

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Formalization of Bernstein Polynomials and Applications to Global Optimization

Cited by 65 publications

References 24 publications

Certified, Efficient and Sharp Univariate Taylor Models in COQ

Certified, Efficient and Sharp Univariate Taylor Models in COQ

Coquelicot: A User-Friendly Library of Real Analysis for Coq

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

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