Real solving polynomial equations with semidefinite programming

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1 .5cm. Real solving polynomial equations with semidefinite programming Jean Bernard Lasserre - Monique Laurent - Philipp Rostalski LAAS, Toulouse - CWI, Amsterdam - ETH, Zürich LAW 2008 Real solving polynomial equations with semidefinite programming p.1

2 The problem Given polynomials h 1,...,h m R[x] = R[x 1,...,x n ] Compute all common real roots (assuming finitely many), i.e. compute the real variety V R (I) of the ideal I := (h 1,...,h m ) Find a basis of the real radical ideal I(V R (I)) V R (I) := {v R n f(v) = 0 f I} I(V R (I)) := {f R[x] f(v) = 0 v V R (I)} }{{} = {f R[x] m N s i R[x] f 2m + i s2 i I} Real Nullstellensatz Real solving polynomial equations with semidefinite programming p.2

3 Our contribution 1. A semidefinite characterization of I(V R (I)) [as the kernel of some positive semidefinite moment matrix] 2. Assuming V R (I) <, an algorithm for finding: a generating set (border or Gröbner basis) of I(V R (I)) the real variety V R (I) Remarks about the method: real algebraic in nature: no complex roots computed works if V R (I) is finite (even if V C (I) is not) no preliminary Gröbner basis of I is needed numerical, based on semidefinite programming (SDP) Real solving polynomial equations with semidefinite programming p.3

4 Plan of the talk 1. The moment-matrix method for V R (I) 2. Adapt the moment-matrix method for V C (I) [drop PSD] 3. Relate to the prolongation-projection algorithm of Zhi and Reid for V C (I) 4. Adapt the prolongation-projection algorithm for V R (I) [add PSD] Real solving polynomial equations with semidefinite programming p.4

5 The complex case is well understood Given an ideal I R[x] with V C (I) <, find the (complex) variety V C (I) and the radical ideal I(V C (I)). Linear algebra in the finite dimensional space R[x]/I Need a linear basis of R[x]/I and a normal form algorithm V C (I) can be computed e.g. with: Linear algebra methods: Eigenvalue method [Stetter-Möller, Stickelberger, Rouillier] Homotopy methods [Verschelde]... Seidenberg [1974]: I(V C (I)) = (I {q 1,...,q n }), where q i is the square-free part of p i, the monic generator of I R[x i ]. Real solving polynomial equations with semidefinite programming p.5

6 The eigenvalue method for V C (I) <, i.e. dim R[x]/I < Stickelberger theorem: Let m f be the multiplication by f linear operator in R[x]/I. 1. The eigenvalues of m f are {f(v) v V C (I)}. 2. The eigenvectors of m T f give the points v V C(I). M T f ζ B,v = f(v)ζ B,v v V C (I) where M f is the matrix of m f in a base B of R[x]/I and ζ B,v := (b(v)) b B Moreover, when B is a set of monomials and 1 B, a border basis of I can be read directly from the multiplication matrices M x1,...,m xn. Real solving polynomial equations with semidefinite programming p.6

7 Finding a linear basis B of R[x]/I and a basis G of the ideal I Typically: G is a Gröbner basis and B is the set of standard monomials for a given monomial ordering (e.g. via Buchberger s algorithm) More generally: Assume B = {b 1 = 1,b 2,...,b N } is a set of monomials with border B := (x 1 B... x n B) \ B. Write any border monomial x i b j = N k=1 a (ij) k b k } {{ } Span(B) + g (ij) }{{} I Then: G := {g (ij) x i b j B} is a (border) basis of I and carries the same information as the multiplication matrices M x1,...,m xn Real solving polynomial equations with semidefinite programming p.7

8 Counting real roots with the Hermite quadratic form For f R[x] Hermite bilinear form: H f : R[x]/I R[x]/I R (g,h) Tr(M fgh ) Theorem: For f = 1 rank(h 1 ) = V C (I), Sign(H 1 ) = V R (I), Rad (H 1 ) = I(V C (I)) rank(h f ) = {v V C (I) f(v) 0} Sign(H f ) = {v V R (I) f(v) > 0} {v V R (I) f(v) < 0} Real solving polynomial equations with semidefinite programming p.8

9 To find V R (I) and a basis of the real radical ideal I(V R (I)) it suffices to have a linear basis B of R[x]/I(V R (I)) and the multiplication matrices in R[x]/I(V R (I))! New tool: Moment matrices y R Nn 2s Ms (y) := (y α+β ) α,β N n s N n s := {α N n α = i α i s} monomials x α of degree s Motivation: For y = (v α ) α N n 2s =: ζ 2s,v where v R n M s (y) = ζ s,v ζ T s,v 0 and KerM s (y) I(v) Real solving polynomial equations with semidefinite programming p.9

10 Real roots of I = (h 1,...,h m ) and PSD moment matrices Lemma: For v V R (I) and t D := max j deg(h j ) the vector y = ζ t,v = (v α ) α t satisfies: the linear constraints (LC): y T [v V C (I)] (h j x α ) = 0 j = 1...m α s.t. α + deg(h j ) t the PSD constraint: M t/2 (y) 0 [v R n ] Set: K t := {y R Nn t (LC), M t/2 (y) 0} Obviously: K t cone(ζ t,v v V R (I)} Theorem: t s D π s (K t )=cone(ζ s,v v V R (I)} Real solving polynomial equations with semidefinite programming p.10

11 Semidefinite characterization of I(V R (I)) Theorem 1: Let y be a generic element of K t, i.e. y lies in the relative interior of the cone K t. Then (KerM t/2 (y)) I(V R (I)) with equality for t large enough. Geometric property of SDP: y is generic rankm t/2 (y) is maximum KerM t/2 (y) KerM t/2 (z) z K t Thus: for v V R (I), KerM t/2 (y) KerM t/2 (ζ t,v ) I(v). Let {g 1,...,g L } be a basis of I(V R (I)). Real Nullstellensatz: gl 2m + i s2 i = m j=1 u jh j. This implies: g l KerM t/2 (y) for t large enough. Real solving polynomial equations with semidefinite programming p.11

12 Stopping criterion when V R (I) < Theorem 2: Let y be a generic element of K t. Assume one of the following two flatness conditions holds: (F1) rankm s (y) = rankm s 1 (y) for some D s t/2 (Fd) rankm s (y) = rankm s d (y) for some d = D/2 s t/2. Then: I(V R (I)) = (KerM s (y)) Any base B of the column space of M s 1 (y) is a base of R[x]/I(V R (I)) The multiplication matrices can be constructed from M s (y). Real solving polynomial equations with semidefinite programming p.12

13 Sketch of proof: Assume rankm s (y) = rankm s 1 (y) Thm [Curto-Fialkow 1996] π 2s (y) has a flat extension ỹ R Nn, i.e. such that rankm(ỹ) = rankm s (y). Thm [La 2005] As M(ỹ) 0, (KerM s (y))=kerm(ỹ) is a real radical 0-dimensional ideal. I }{{} (LC) (KerM s (y)) I(V }{{} R (I)) y generic Thus: (KerM s (y))=i(v R (I)) B indexes a base of M s 1 (y) = B indexes a base of M(ỹ) = B is a base of R[x]/KerM(ỹ) = R[x]/I(V R (I)) Use linear dependencies in M s (y) to construct the multiplication matrices. Real solving polynomial equations with semidefinite programming p.13

14 The moment-matrix algorithm for V R (I) Input: h 1,...,h m R[x] Output: B base of R[x]/I(V R (I)) The multiplication matrices M xi in R[x]/I(V R (I)) Algorithm: For t D Step 1: Compute a generic element y K t. Step 2: Check if (F1) or (Fd) holds. If yes, return a column basis B of M s 1 (y) and M xi = M 1 B P i, M B := principal submatrix of M s 1 (y) indexed by B P i := submatrix of M s (y) with rows in B and columns in x i B. If no, go to Step 1 with t t + 1. Theorem: The algorithm terminates. Real solving polynomial equations with semidefinite programming p.14

15 The algorithm terminates: (F1) holds for t large enough. For t t 0, KerM t/2 (y) contains a Gröbner base {g 1,...,g L } of I(V R (I)) for a total degree ordering. B := {b 1,...,b N }: set of standard monomials base of R[x]/I(V R (I)). Set: s := 1 + max b B deg(b) and assume t t 0, t/2 > s. N L For α s, write x α = λ i b i + u l g l i=1 }{{} deg s 1 Thus: x α N i=1 λ ib i KerM t/2 (y). That is: rankm s (y) = rankm s 1 (y). l=1 }{{} deg α s< t/2 Real solving polynomial equations with semidefinite programming p.15

16 Two small examples Ex. 1: I = (h := x x2 2 ) V R (I) = {0}, V C (I) =. M 1 (y) 0, 0 = y T h = y20 + y 02 = y α = 0 α 0. Any generic y K 2 is y = (y 0, 0,..., 0) with y 0 > 0. Thus: (KerM 1 (y)) = (x 1,x 2 ) = I(V R (I)). Ex. 2: I = (h i := x i (x 2 i + 1) i = 1,...,n) V R (I) = {0}, V C (I) = 3 n. M 2 (y) 0, 0 = y T (x i h i ) = y 4ei +y 2ei i = y α = 0 α 0. Any generic y K 4 is y = (y 0, 0,..., 0) with y 0 > 0. Thus: (KerM 1 (y)) = (x 1,...,x n ) = I(V R (I)). Real solving polynomial equations with semidefinite programming p.16

17 Some algorithmic issues How to find a generic y K t, i.e. with rankm t (y) max.? Solve the SDP program: min y Kt 1 with a SDP solver using the extended self-dual embedding property. Then the central path converges to a solution in the relative interior of the optimum face, i.e., to a generic point y K t. How to compute ranks of matrices? We use SVD decomposition, but this is a sensitive numerical issue... The method may work without (F1) or (Fd): If rankm B (y) = rankm B B (y) and the formal multiplication matrices commute. Real solving polynomial equations with semidefinite programming p.17

18 Extension of the moment-matrix algorithm to V C (I) Omit the PSD condition and work with the linear space: K t = {y R Nn t y T (h j x α ) = 0 j,α with α + deg(h j ) t} The same algorithm applies: For t D Pick generic y K t, i.e. rankm s (y) maximum s t/2 [choose y K t randomly] Check if the flatness condition (F1) or (Fd) holds. If yes, find a basis of R[x]/J where J := (KerM s (y)) satisfies I J I(V C (I)) and thus V C (J) = V C (I). If not, iterate with t + 1. Real solving polynomial equations with semidefinite programming p.18

19 Find the ideal (KerM s (y)) = I in the Gorenstein case The inclusion I (KerM s (y)) I(V C (I)) may be strict for any generic y. Example: For I = (x 2 1,x2 2,x 1x 2 ), V C (I) = {0}, I(V C (I)) = (x 1,x 2 ), dim R[x]/I = 3, dim R[x]/I(V C (I)) = 1, while dim R[x]/(KerM s (y)) = 2 for any generic y! Recall: The algebra A := R[x]/I is Gorenstein if there exists a non-degenerate bilinear form on A satisfying (f,gh) = (fg,h) f,g,h A, i.e. if there exists y K with I = KerM(y) Hence: y K t s.t. rankm s (y) = rankm s 1 (y) and I = (KerM s (y)) IFF A is Gorenstein. Real solving polynomial equations with semidefinite programming p.19

20 Example: the moment-matrix algorithm for real/complex roots I = (x 2 1 2x 1x 3 + 5, x 1 x x 2x 3 + 1, 3x 2 2 8x 1x 3 ), D = 3, d = 2 Ranks of M s (y) for generic y K t, K t : t = s = s = s = s = s = no PSD 8 complex roots t = s = s = s = s = 3 10 with PSD extract 2 real roots Real solving polynomial equations with semidefinite programming p.20

21 8 complex / 2 real roots: v 1 = v 2 = v 3 = v 4 = v 5 = v 6 = v 7 = v 8 = h h h h h h h h i 1.101, 2.878, i i, i, i i i, i, i i, i, i i, i, i i i, i, i i i, i, i i 0.966, 2.813, i i Real solving polynomial equations with semidefinite programming p.21

22 Extracting real roots without (F1) or (Fd) I = (5x 9 1 6x5 1 x 2 + x 1 x x 1x 3, 2x 6 1 x 2 + 2x 2 1 x x 2x 3,x x ) D = 9, d = 5, V R (I) = 8, V C (I) = 20 order rank sequence of extract. order s accuracy comm. error t M s (y) (1 s t/2 ) MON/SVD MON/SVD MON/SVD (3)/ ( ) / / (3)/3(3) e-5/ e-5/ Quotient basis: B = {1,x 1,x 2,x 3,x 2 1,x 1x 2,x 1 x 3,x 2 x 3 } border basis G of size 10 8 x 1 = ( 0.515, , ) x 2 = ( 0.502,0.119, ) >< x 3 = (0.502, 0.119,0.0124) x 4 = (0.515, , ) Real solutions: x 5 = (0.262, 0.444, ) x 6 = ( 2.07e-5,0.515, 1.27e-6) >: x 7 = ( 0.262, 0.444, ) x 8 = ( 1.05e-5, 0.515, 7.56e-7) Real solving polynomial equations with semidefinite programming p.22

23 Link with the elimination method of Zhi and Reid Theorem: If (F1) holds, i.e. for some D s t/2 rankm s (y) = rankm s 1 (y) for generic y K t, then dimπ 2s (K t ) = dim π 2s 1 (K t ) = dimπ 2s (K t+1 ) Theorem (based on [Zhi-Reid 2004]): If for some D s t (ZR) dim π s (K t ) = dim π s 1 (K t ) = dimπ s (K t+1 ) then one can construct a base of R[x]/I and the multiplication matrices in R[x]/I [and thus extract V C (I)]. Hence: The Zhi-Reid criterion (ZR) may be satisfied earlier than the flatness criterion (F1). Real solving polynomial equations with semidefinite programming p.23

24 Example: I = (x 2 1 2x 1 x 3 + 5,x 1 x x 2 x 3 + 1, 3x 2 2 8x 1 x 3 ) t = s = s = s = s = s = rankm 3 (y)=rankm 2 (y) for y K 9 t = s = s = s = s = s = s = s = s = dim π 3 (K 6 ) = dim π 2 (K 6 ) = dim π 3 (K 7 ) Real solving polynomial equations with semidefinite programming p.24

25 Extending the Zhi-Reid criterion to the real case In the complex case, K t = H t where H t := {h j x α j,α with deg(h j x α ) t}. In the real case, K t is a cone, contained in the linear space P t, with the same dimensions: dim K t = dim P t, where P t := H t {fx α f KerM t/2 (y), deg(x α ) t/2 } Theorem: If for some D s t (ZR+) dim π s (P t ) = dim π s 1(P t ) = dimπ s((p t P t ) ) then one can construct a base of J with I J I(V R (I)) and thus extract V R (I) = V C (J) R n. Real solving polynomial equations with semidefinite programming p.25

26 Link with the flatness criterion Theorem: In the PSD case, the flatness criterion (F1): rankm s (y) = rankm s 1 (y) for generic y K t is equivalent to the stronger version of the (ZR) criterion: (ZR++) dimπ s 1 (P t ) = dim π 2s(P t ) = dimπ 2s((P t P t ) ) in which case we find the real radical ideal J = I(V R (I)). Hence: the algorithm based on (ZR) may stop earlier than the moment-matrix algorithm, based on (F1). Future work: Adapt other known efficient algorithms for complex roots to real roots by incorporating SDP conditions. Real solving polynomial equations with semidefinite programming p.26