This paper presents a nonnegative polynomial that cannot be represented with
nonnegative coefficients in the simplicial Bernstein basis by subdividing the
standard simplex. The example shows that Bernstein Theorem cannot be extended
to certificates of nonnegativity for polynomials with zeros at isolated points

Sloth, Christoffer

English

arXiv

University of Southern Denmark Research Output

Syddansk UniversitetNonnegative Polynomial with no Certificate of Nonnegativity in the Simplicial BernsteinBasisSloth, ChristofferPublished in:arxiv.orgPublication date:2017Document versionEarly version, also known as pre-printCitation for pulished version (APA):Sloth, C. (2017). Nonnegative Polynomial with no Certificate of Nonnegativity in the Simplicial Bernstein Basis.arxiv.org.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 09. Sep. 2018Nonnegative Polynomial with no Certificate ofNonnegativity in the Simplicial Bernstein BasisChristoffer SlothUniversity of Southern DenmarkSDU Roboticschsl@mmmi.sdu.dkAbstractThis paper presents a nonnegative polynomial that cannot be repre-sented with nonnegative coefficients in the simplicial Bernstein basis bysubdividing the standard simplex. The example shows that BernsteinTheorem cannot be extended to certificates of nonnegativity for polyno-mials with zeros at isolated points.1 IntroductionThe positivity and nonnegativity of polynomials play an important role in manyproblems such as Lyapunov stability analysis of dynamical systems [5, 7]. There-fore, much research has been conducted to exploit different certificates of positiv-ity [9, 1]. The certificates of positivity include sum of squares that is extensivelyused in different applications [5, 10].The certification of nonnegativity for polynomials with zeros was addressedin [3, 11], where the paper [3] characterizes polynomials that can be certified tobe nonnegative using Po´lya’s Theorem.This paper exploits a certificate of positivity in the Bernstein basis [4] thathas previously been applied within optimization and control [8, 2, 12]. In par-ticular, a nonnegative polynomial is provided that cannot be represented withnonnegative coefficients in the simplicial Bernstein basis by subdividing thestandard simplex.The paper is organized as follows. Section 2 provides preliminaries on poly-nomials in Bernstein form, which are used in Section 3 that provides the exam-ple, and relies on results presented in Appendix A.2 PreliminariesThis section provides preliminary results and notation on polynomials in theBernstein basis.Let R[X] be the real polynomial ring in n variables. For α = (α0, . . . , αn) ∈Nn+1 we define |α| := ∑ni=0 αi.We utilize Bernstein polynomials defined on simplices via barycentric coor-dinates.1arXiv:1710.05735v1  [math.NA]  12 Oct 2017Definition 1 (Barycentric Coordinates). Let λ0, . . . , λn ∈ R[X] be affine poly-nomials and let v0, . . . , vn ∈ Rn be affinely independent. Ifn∑i=0λi = 1 andx = λ0(x)v0 + · · ·+ λn(x)vn ∀x ∈ Rnthen λ0, . . . , λn are said to be barycentric coordinates associated to v0, . . . , vn.Let λ0, . . . , λn be barycentric coordinates associated to v0, . . . , vn. Then thesimplex generated by λ0, . . . , λn is4 := {x =n∑i=0λivi| λi(x) ≥ 0, i = 0, . . . , n} ⊂ Rn. (1)This work represents polynomials in the simplicial Bernstein basis, wherethe basis polynomials are defined as follows.Definition 2 (Bernstein Polynomial). Let d ∈ N0, n ∈ N, α = (α0, . . . , αn) ∈Nn+10 , and let λ0, . . . , λn ∈ R[X] be barycentric coordinates associated to asimplex ∆. The Bernstein polynomials of degree d on the simplex ∆ areBdα =(dα)λα (2)for |α| = d where (dα)=d!α0!α1! · · ·αn! and λα :=n∏i=0λαii .Every polynomial P ∈ R[X] of degree no greater than d can be writtenuniquely asP =∑|α|=dbα(P, d,4)Bdα =∑|α|=dbdαBdα = bdBd, (3)where bα(P, d,4) ∈ R is a Bernstein coefficient of P of degree d with respect to4. We say that (3) is a polynomial in Bernstein form.From the definition of Bernstein polynomials on simplex 4 (Definition 2)and (1) it is seen that for any d ∈ N and |α| = dBdα(x) ≥ 0 ∀x ∈ 4.This means that polynomial P in (3) is positive on 4 if bdα > 0 for all α ∈ Nn+1where |α| = d. We call this a certificate of positivity. Similarly, we say thatbdα ≥ 0 for all α ∈ Nn+1 where |α| = d is a certificate of nonnegativity.If a polynomial P of degree p cannot be certified to be positive in the Bern-stein basis of degree p on simplex 4 then two strategies may be exploited torefine the certificate. These are called degree elevation and subdivision and willbe outlined next.To certify the positivity of a polynomial by degree elevation, one may rep-resent the polynomial using Bernstein polynomials of a higher degree as in thefollowing theorem [6].2Theorem 1 (Multivariate Bernstein Theorem). If a nonzero polynomial P ∈R[X1, . . . , Xn] of degree p ∈ N0 is positive on a simplex 4, then there existsd ≥ p such that all entries of b(P, d,4) are positive.A similar result related to subdivision is presented in [6]; however, for sim-plicity the subdivision approach is explained by the following example. Theexample shows how an initial simplex 4 may be subdivided into several sim-plices to certify the positivity of a polynomial on a given domain.Example 1. Consider the following polynomialP = X2 + Y 2 −XYon the standard simplex 4 with vertices v0 = (0, 0), v1 = (1, 0), v2 = (0, 1).Polynomial P has the following nonzero Bernstein coefficients in the Bernsteinbasis with respect to 4b(0,2,0) = 1, b(0,1,1) = −0.5, b(0,0,2) = 1. (4)The nonnegativity of polynomial P cannot be certified directly by the above Bern-stein coefficients since b(0,1,1) is negative; hence, simplex 4 must be subdividedto obtain a certificate of nonnegativity. Consequently, we define a new vertexv˜1 = (1−θ, θ) and a subdivision of 4 given by two simplices 4˜ and 4ˆ with ver-tices v0, v1, v˜1 and v0, v2, v˜1 respectively. The subdivision is shown in Figure 1and gives a collection of simplices, all with a common vertex at the zero of P .v0v1v2v˜14ˆ 4˜Figure 1: Simplex 4 with vertices v0, v1, v2, simplex 4˜ with vertices v0, v1, v˜1,and simplex 4ˆ with vertices v0, v2, v˜1.The nonzero Bernstein coefficients of polynomial P on the two simplices are(by Lemma 1)b˜(0,2,0) = 1, b˜(0,1,1) = −1.5θ + 1, b˜(0,0,2) = 18(1− θ)θ + 14bˆ(0,2,0) = 1, bˆ(0,1,1) = 1.5θ − 0.5, bˆ(0,0,2) = 18(1− θ)θ + 14It is seen that the nonnegativity of the polynomial can be certified by lettingθ = 0.5.It should be noted that there exists nonnegative (sum of squares) polyno-mials that cannot be represented with nonnegative coefficients in the simplicialBernstein basis, i.e., it is not possible to certify the nonnegativity of any non-negative polynomial by employing subdivisioning. This is exemplified in thenext section.33 CounterexampleIn this section, a nonnegative polynomial is presented for which there exists nocertificate of nonnegativity in the simplicial Bernstein basis.Consider the polynomialP = 21X41 +24X31X2−36X31 +18X21X22−24X21X2+18X21 +12X1X32−12X1X22 +30X42 .The graph of P on the standard simplex is shown in Figure 2.00.5100.510102030x1x2Figure 2: Graph of the polynomial P on the standard simplex 4.We aim at proving the nonnegativity of P on the standard simplex 4 withvertices v0 = (0, 0), v1 = (1, 0), and v2 = (0, 1). The polynomial P can bewritten as the following sum of squaresP =[X1 X21 X1X2 X22] 18 −18 −12 −6−18 21 12 0−12 12 18 6−6 0 6 30︸ ︷︷ ︸=MPX1X21X1X2X22 ,where the matrix MP is positive definite; hence, the polynomial is positiveeverywhere except from being zero at (x1, x2) = (0, 0).Polynomial P can be represented in the Bernstein basis asP =∑|α|=dbαBdα,where Bdα is the Bernstein basis defined on the standard simplex4. The nonzerocoefficients of P areb(2,2,0) = 3, b(1,2,1) = 1, b(1,1,2) = −1, b(0,4,0) = 3, b(0,0,4) = 30.It is seen that one coefficient is negative (b(1,1,2) < 0); thus, the nonnegativityof P cannot be directly certified.Next we show that there exists no certificate of nonnegativity for the poly-nomial P in the Bernstein basis, since the coefficient with index (1, 1, 2) willremain negative on one simplex of any triangulation of 4. In particular, thecoefficient remains negative on any nondegenerate simplex with vertex v0, a4vertex on the line between v0 and v2, and any third vertex in 4, i.e., for anyBernstein basis defined on the simplex 4˜ with verticesv˜0 = v0v˜1 = β0v0 + β1v1 + β2v2v˜2 = ρv0 + (1− ρ)v2where β0 + β1 + β2 = 1, β0, β1, β2 ≥ 0, β1 > 0, and ρ ∈ [0, 1). The simplices 4and 4˜ are illustrated in Figure 3.v0v1v2v˜1v˜2Figure 3: Simplex 4 with vertices v0, v1, v2 and simplex 4˜ with verticesv0, v˜1, v˜2.The polynomial P can be represented in the Bernstein basis asP =∑|α|=db˜αB˜dα,where B˜dα is the Bernstein basis defined on the standard simplex 4˜ and whereby Lemma 2 and Lemma 1 the coefficients bα and b˜γ are related asb˜γ =γ2∑k=0(γ2γ2 − k)ργ2−k(1−ρ)k∑|α| = dγ0 + γ2 − k ≤ α0k ≤ α2γ1!βα0−(γ0+γ2−k)0 βα11 βα2−k2(α0 − (γ0 + γ2 − k))!α1!(α2 − k)! bαFor the polynomial P , it is seen thatb˜(1,1,2) = β1(1− ρ)2b(1,1,2),which is negative for any admissible β1 and ρ (β1 > 0 and ρ ∈ [0, 1)), sinceb(1,1,2) = −1. Therefore, there exists no certificate of nonnegativity of P on 4in the simplicial Bernstein basis.References[1] A. Ahmadi and A. Majumdar. DSOS and SDSOS optimization: LP andSOCP-based alternatives to sum of squares optimization. In proceedings ofthe 48th Annual Conference on Information Sciences and Systems, pages1–5, March 2014.5[2] M. Ben Sassi, R. Testylier, T. Dang, and A. Girard. Reachability analysisof polynomial systems using linear programming relaxations. In AutomatedTechnology for Verification and Analysis, pages 137–151. Springer BerlinHeidelberg, 2012.[3] M. Castle, V. Powers, and B. Reznick. Po´lya’s theorem with zeros. Journalof Symbolic Computation, 46(9):1039–1048, 2011.[4] R. T. Farouki. The Bernstein polynomial basis: A centennial retrospective.Computer Aided Geometric Design, 29(6):379 – 419, 2012.[5] D. Henrion and A. Garulli, editors. Positive Polynomials in Control, volume312 of Lecture Notes in Control and Information Science. Springer BerlinHeidelberg, 2005.[6] R. Leroy. Certificates of positivity in the simplicial Bernstein basis. hal-00589945, 2009.[7] M. Marshall. Positive Polynomials and Sums of Squares, volume 146.American Mathematical Society, 2008.[8] C. Muoz and A. Narkawicz. Formalization of bernstein polynomials andapplications to global optimization. Journal of Automated Reasoning,51(2):151–196, 2013.[9] P. A. Parrilo. Semidefinite programming relaxations for semialgebraic prob-lems. Math. Programming, 96(2):293–320, 2003.[10] V. Powers. Positive polynomials and sums of squares: Theory and practice.Real Algebraic Geometry, 2011.[11] M. Schweighofer. Certificates for nonnegativity of polynomials with zeroson compact semialgebraic sets. manuscripta mathematica, 117(4):407–428,Aug 2005.[12] C. Sloth and R. Wisniewski. Control to facet for polynomial systems.In Proceedings of the 17th International Conference on Hybrid Systems:Computation and Control, pages 123–132, 2014.A Useful ResultsLemma 1. Let P ∈ R[X]. Define a simplex 4 with vertices v0, v1, v2 ∈ R2,and define the simplex 4˜ with vertices v0, v1, v˜2, wherev˜2 = ρv0 + (1− ρ)v2and ρ ∈ [0, 1). Then P can be represented in two Bernstein bases asP =∑|α|=dbαBdα =∑|γ|=db˜γB˜dγ , (5)where Bdα is the Bernstein basis defined by 4, B˜dγ is the Bernstein basis definedby 4˜, andb˜γ =γ2∑k=0(γ2γ2 − k)ργ2−k(1− ρ)kb(γ0+γ2−k,γ1,k).6Proof. Let λ0, λ1, λ2 ∈ R[X] be barycentric coordinates associated to 4 andλ˜0, λ˜1, λ˜2 ∈ R[X] be barycentric coordinates associated to 4˜. Thenλ0 = λ˜0 + ρλ˜2λ1 = λ˜1λ2 = (1− ρ)λ˜2andλα = (λ˜0 + ρλ˜2)α0 λ˜α11 (1− ρ)α2 λ˜α22 .By binomial theoremλα =(α0∑k=0(α0k)λ˜α0−k0 ρkλ˜k2)λ˜α11 (1− ρ)α2 λ˜α22=α0∑k=0(α0k)ρk(1− ρ)α2 λ˜α0−k0 λ˜α11 λ˜α2+k2 .Therefore, the Bernstein bases Bdγ and B˜dγ are related asBdα =α0∑k=0(α2 + kk)ρk(1− ρ)α2 B˜d(α0−k,α1,α2+k),which implies that coefficients bα and b˜α are related asb˜γ =γ2∑k=0(γ2γ2 − k)ργ2−k(1− ρ)kb(γ0+γ2−k,γ1,k).Lemma 2. Let P ∈ R[X]. Define a simplex 4 with vertices v0, v1, v2 ∈ R2,and define the simplex 4˜ with vertices v0, v˜1, v2 ∈ R2, wherev˜1 = β0v0 + β1v1 + β2v2and β0 + β1 + β2 = 1, β0, β2 ≥ 0, β1 > 0. Then P can be represented in twoBernstein bases asP =∑|α|=dbαBdα =∑|γ|=db˜γB˜dγ (6)where Bdα is the Bernstein basis defined by 4, B˜dγ is the Bernstein basis definedby 4˜, andb˜γ =∑|α| = dα0 ≥ γ0α2 ≥ γ2γ1!(α0 − γ0)!α1!(α2 − γ2)!βα0−γ00 βα11 βα2−γ22 bα.7Proof. Let λ0, λ1, λ2 ∈ R[X] be barycentric coordinates associated to 4 andλ˜0, λ˜1, λ˜2 ∈ R[X] be barycentric coordinates associated to 4˜. Thenλ0 = λ˜0 + β0λ˜1λ1 = β1λ˜1λ2 = β2λ˜1 + λ˜2andλα = (λ˜0 + β0λ˜1)α0(β1λ˜1)α1(β2λ˜1 + λ˜2)α2 .By binomial theoremλα =(α0∑k=0(α0k)λ˜α0−k0 βk0 λ˜k1)βα11 λ˜α11 α2∑j=0(α2j)βj2λ˜j1λ˜α2−j2=∑|γ| = dγ0 ≤ α0γ2 ≤ α2(α0α0 − γ0)(α2α2 − γ2)βα0−γ00 βα11 βα2−γ22 λ˜γ .Therefore, the Bernstein bases Bdγ and B˜dγ are related asBdα =∑|γ| = dγ0 ≤ α0γ2 ≤ α2γ1!(α0 − γ0)!α1!(α2 − γ2)!βα0−γ00 βα11 βα2−γ22 B˜dγ ,which implies that coefficients bα and b˜α are related asb˜γ =∑|α| = dα0 ≥ γ0α2 ≥ γ2γ1!(α0 − γ0)!α1!(α2 − γ2)!βα0−γ00 βα11 βα2−γ22 bα.8

Nonnegative Polynomial with no Certificate of Nonnegativity in the Simplicial Bernstein Basis

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Nonnegative Polynomial with no Certificate of Nonnegativity in the Simplicial Bernstein Basis

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