HYPERBOLIC POLYNOMIALS, INTERLACERS AND SUMS OF SQUARES

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1 HYPERBOLIC POLYNOMIALS, INTERLACERS AND SUMS OF SQUARES DANIEL PLAUMANN Universität Konstanz joint work with Mario Kummer (U. Konstanz) Cynthia Vinzant (U. of Michigan) Out[121]= POLYNOMIAL OPTIMISATION Isaac Newton Institute Cambridge 19 July 2013 In[137]:= p1 = ContourPlot3D@x ã 1.6, 8x, -2, 2<, 8z, -4, Mesh Ø None, ContourStyle Ø 8Magenta<, Boxe p2 = ContourPlot3D@z ã 1.5, 8x, -2, 2<, 8z, -4, Mesh Ø None, ContourStyle Ø 8Green<, Boxed Ø Show@8f, p1<d Show@8f, p2<d Show@8f, p1, p2<d Universität Konstanz

2 Two flavours of real polynomials Let f R[x] d be homogeneous of degree d in x = (x 1,..., x n ). Positive polynomials. f 0 on R n. Ideal representation: Sum of squares f = f fr 2

3 Two flavours of real polynomials Let f R[x] d be homogeneous of degree d in x = (x 1,..., x n ). Positive polynomials. f 0 on R n. Ideal representation: Sum of squares f = f fr 2 Hyperbolic polynomials. Ideal representation: Determinantal polynomial. f = det(m), M(x) = x 1 M x n M n with M 1,..., M n Sym d (R), M(e) positive definite. f R[x] is hyperbolic with respect to e R n if p(e) > 0 and for all v R n f (v te) R[t] has only real zeros in t.

4 Two flavours of real polynomials Let f R[x] d be homogeneous of degree d in x = (x 1,..., x n ). Positive polynomials. f 0 on R n. Ideal representation: Sum of squares f = f fr 2 Existence only for n 2 or d 2 or (n, d) = (3, 4) (Hilbert 1888) Hyperbolic polynomials. Ideal representation: Determinantal polynomial. f = det(m), M(x) = x 1 M x n M n with M 1,..., M n Sym d (R), M(e) positive definite. Existence only for n 3 (Helton-Vinnikov 2004) f R[x] is hyperbolic with respect to e R n if p(e) > 0 and for all v R n f (v te) R[t] has only real zeros in t.

5 Hyperbolicity cones f hyperbolic with respect to e: f (e) > 0 and f (v te) R[t] has only real roots for all v R n. Hyperbolic polynomials can be thought of as generalised characteristic polynomials.

6 Hyperbolicity cones In[121]:= f = ContourPlot3D@2 x^4 + y^4 + z^4-3 y^2 x^2-3 z^2 x^2 + y 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4<, Mesh Ø None, ContourStyle Ø 8 Blue, Opacity@.5D<, Boxed Ø False, Axes Ø f hyperbolic with respect to e: f (e) > 0 and f (v te) R[t] has only real roots for all v R n. Out[121]= Hyperbolic polynomials can be thought of as generalised characteristic polynomials. n = 3, d = 4 C e ( f ) = {v R n All roots of f (v te) are non-negative}, is a closed convex cone, called the hyperbolicity cone. In[137]:= p1 = ContourPlot3D@x ã 1.6, 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4< Mesh Ø None, ContourStyle Ø 8Magenta<, Boxed Ø False, Axe p2 = ContourPlot3D@z ã 1.5, 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4< Mesh Ø None, ContourStyle Ø 8Green<, Boxed Ø False, Axes Ø Show@8f, p1<d Show@8f, p2<d Show@8f, p1, p2<d

7 Hyperbolicity cones In[121]:= f = ContourPlot3D@2 x^4 + y^4 + z^4-3 y^2 x^2-3 z^2 x^2 + y 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4<, Mesh Ø None, ContourStyle Ø 8 Blue, Opacity@.5D<, Boxed Ø False, Axes Ø f hyperbolic with respect to e: f (e) > 0 and f (v te) R[t] has only real roots for all v R n. Out[121]= Hyperbolic polynomials can be thought of as generalised characteristic polynomials. n = 3, d = 4 C e ( f ) = {v R n All roots of f (v te) are non-negative}, is a closed convex cone, called the hyperbolicity cone. In[137]:= p1 = ContourPlot3D@x ã 1.6, 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4< Mesh Ø None, ContourStyle Ø 8Magenta<, Boxed Ø False, Axe p2 = ContourPlot3D@z ã 1.5, 8x, -2, 2<, 8z, -4, 4<, 8y, -4, 4< Mesh Ø None, ContourStyle Ø 8Green<, Boxed Ø False, Axes Ø Show@8f, p1<d Show@8f, p2<d Show@8f, p1, p2<d Let f = det(m), M(x) = x 1 M x n M n symmetric, M(e) 0. C e ( f ) = {v R n M(v) 0}. is the spectrahedral cone defined by M 1,..., M n. For if M(e) = I d, then f (v te) is the characteristic polynomial of M(v).

8 Optimisation Semidefinite programming relaxations f 0 on R n? f is a sum of squares in R[x]? SDP

9 Optimisation Semidefinite programming relaxations f 0 on R n? f is a sum of squares in R[x]? SDP Let f be a hyperbolic polynomial. Hyperbolic programme? SDP Optimise over C e ( f )? SDP formulation/relaxation of C e ( f )

10 Optimisation Semidefinite programming relaxations f 0 on R n? f is a sum of squares in R[x]? SDP Let f be a hyperbolic polynomial. Hyperbolic programme? SDP Optimise over C e ( f )? SDP formulation/relaxation of C e ( f ) Hyperbolic programming. Studied by Güler, Lewis, Renegar, Tunçel and others. Can use log f as a barrier function in interior point methods.

11 Interlacers Let f be hyperbolic w.r.t. e, deg( f ) = d, irreducible g be hyperbolic w.r.t. e, deg(g) = d 1

12 Interlacers Let f be hyperbolic w.r.t. e, deg( f ) = d, irreducible g be hyperbolic w.r.t. e, deg(g) = d 1 Call g an interlacer of f if the roots of g(v te) lie between those of f (v te) for all v R n Quartic with an interlacing cubic α 1 α d roots of f (v te) β 1 β d 1 roots of g(v te) α 1 β 1 α 2 β 2 β d 1 α d

13 Interlacers Let f be hyperbolic w.r.t. e, deg( f ) = d, irreducible g be hyperbolic w.r.t. e, deg(g) = d 1 Call g an interlacer of f if the roots of g(v te) lie between those of f (v te) for all v R n Quartic with an interlacing cubic Directional derivatives g = D v f with v C e ( f ) are interlacers. If f = det(m) with M(e) 0, then any symmetric minor of size d 1 of M is an interlacer.

14 Interlacers Plane quartic curve with two interlacing cubics The product of two interlacers of f is non-negative on V R ( f ) = {v R n f (v) = 0}.

15 Interlacers Plane quartic curve with two interlacing cubics The product of two interlacers of f is non-negative on V R ( f ) = {v R n f (v) = 0}. Proposition The set Int e ( f ) of interlacers is a convex cone, namely Int e ( f ) = {g R[x] d 1 D e f g f D e g 0 on R n } Reason. If f, g R[t] have real roots with deg(g) = deg( f ) 1, then g interlaces f if and only if the Wronskian f g f g is everywhere positive or negative (Bézout).

16 Representing the hyperbolicity cone The convex cone of interlacers is Int e ( f ) = {g R[x] d 1 D e f g f D e g 0 on R n }

17 Representing the hyperbolicity cone The convex cone of interlacers is Int e ( f ) = {g R[x] d 1 D e f g f D e g 0 on R n } A directional derivative D v f is an interlacer if and only if the point v is in the hyperbolicity cone C e ( f ).

18 Representing the hyperbolicity cone The convex cone of interlacers is Int e ( f ) = {g R[x] d 1 D e f g f D e g 0 on R n } A directional derivative D v f is an interlacer if and only if the point v is in the hyperbolicity cone C e ( f ). Corollary C e ( f ) = {v R n D e f D v f f D e D v f 0 on R n }

19 Representing the hyperbolicity cone The convex cone of interlacers is Int e ( f ) = {g R[x] d 1 D e f g f D e g 0 on R n } A directional derivative D v f is an interlacer if and only if the point v is in the hyperbolicity cone C e ( f ). Corollary Write C e ( f ) = {v R n D e f D v f f D e D v f 0 on R n } W e,v ( f ) = D e f D v f f D e D v f for the family of Wronkians associated with f. This realises the hyperbolicity cone as a linear slice of the cone of non-negative polynomials.

20 A sums of squares relaxation We obtain an increasing sequence of inner approximations C e (N) ( f ) ={v R n (x x2 n) N W e,v ( f ) is a sum of squares C e ( f ) for N = 0, 1, 2,... If f is strictly hyperbolic (V R ( f ) smooth), then C e (N) ( f ) = C e ( f ) N 0 by a result of Reznick on sums of squares with denominators. }

21 A sums of squares relaxation We obtain an increasing sequence of inner approximations C e (N) ( f ) ={v R n (x x2 n) N W e,v ( f ) is a sum of squares C e ( f ) for N = 0, 1, 2,... If f is strictly hyperbolic (V R ( f ) smooth), then C e (N) ( f ) = C e ( f ) N 0 by a result of Reznick on sums of squares with denominators. Theorem (Netzer-Sanyal 2012) If f is strictly hyperbolic, then C e ( f ) is a projected spectrahedral cone. Theorem (Parrilo-Saunderson 2012) Derivative cones for determinantal polynomials are projected spectrahedral cones. }

22 Results on determinantal representations Effective version of the Helton-Vinnikov theorem for Hermitian determinantal representations. Theorem If f R[x, y, z] is hyperbolic with respect to e, we can construct a representation f = det(m) with M(e) 0 and M Hermitian, starting from any interlacer of f

23 Results on determinantal representations Theorem If f k = det(m) is a determinantal representation with M(e) 0, k 1, then all the Wronskians W e,v ( f ) with v C e ( f ) are sums of squares.

24 Results on determinantal representations Theorem If f k = det(m) is a determinantal representation with M(e) 0, k 1, then all the Wronskians W e,v ( f ) with v C e ( f ) are sums of squares. Example The Vámos polynomial h(x 1,, x 8 ) = x i, I ( [8] 4 )/C i I where C = {{1, 2, 3, 4}, {1, 2, 5, 6}, {1, 2, 7, 8}, {3, 4, 5, 6}, {3, 4, 7, 8}} is the bases-generating polynomial of the matroid with circuits in C. The polynomial h is hyperbolic (Wagner-Wei 2009) but no power of h has a definite determinantal representation (Brändén 2011). We show that W e7,e 8 (h) is not a sum of squares, yielding a new proof of Brändén s result.

25 Results on determinantal representations Theorem If f k = det(m) is a determinantal representation with M(e) 0, k 1, then all the Wronskians W e,v ( f ) with v C e ( f ) are sums of squares. Theorem If f is multi-affine and hyperbolic with respect to e, then f has a determinantal representation f = det(m) with M(e) 0 if and only if the Wronskians W ei,e j are squares in R[x] for all i, j.

26 Results on determinantal representations Theorem If f k = det(m) is a determinantal representation with M(e) 0, k 1, then all the Wronskians W e,v ( f ) with v C e ( f ) are sums of squares. Theorem If f is multi-affine and hyperbolic with respect to e, then f has a determinantal representation f = det(m) with M(e) 0 if and only if the Wronskians W ei,e j are squares in R[x] for all i, j. Example We can use this to reprove a classical result saying that the d-th elementary symmetric polynomial in n variables, which is multi-affine and hyperbolic with respect to (1,..., 1), has a definite determinantal representation if and only if d {1, n 1, n}.

27 The interlacer cone Example Let f = det(x) where X is a d d symmetric matrix of variables. Int Id (det(x)) = { tr(m X adj ) M Sym + d (R) } Sym+ d (R).

28 The interlacer cone Example Let f = det(x) where X is a d d symmetric matrix of variables. Int Id (det(x)) = { tr(m X adj ) M Sym + d (R) } Sym+ d (R). Theorem Let f R[x] d be hyperbolic with respect to e R n and assume that the projective variety V C ( f ) is smooth. The algebraic boundary of the cone Int e ( f ) is the irreducible hypersurface in C[x] d 1 given by g C[x] d 1 p P n 1 such that f (p) = g(p) = 0 and rank ( f (p) g(p) ) 1.

29 References M. Kummer, D. Plaumann, C. Vinzant. Hyperbolic polynomials, interlacers and sums of squares. Preprint (2013). arxiv: P. Brändén. Obstructions to determinantal representability. Adv. Math., 226(2), (2011). arxiv: M. Kummer. A Note on the Hyperbolicity Cone of the Specialized Vámos Polynomial. Preprint (2013). arxiv: T. Netzer, R. Sanyal. Smooth hyperbolicity cones are spectrahedral shadows. Preprint (2012). arxiv: P. Parrilo, J. Saunderson. Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones. Preprint (2012). arxiv: D. Plaumann, B. Sturmfels, C. Vinzant. Computing determinantal representations of Helton-Vinnikov curves. In: Operator Theory: Advances and Applications, Vol. 222 (2010). arxiv: D. Plaumann, C. Vinzant. Determinantal representations of hyperbolic plane curves: An elementary approach. To appear in J. Symbolic Computation (2010). arxiv: Surveys G. Blekherman, P. Parrilo, and R. Thomas (editors). Semidefinite Optimization and Convex Algebraic Geometry, MOS-SIAM Series on Optimization. To appear (2012). D. Plaumann. Geometry of Linear Matrix Inequalities. Lecture Notes, University of Konstanz (2013). (link). V. Vinnikov, LMI Representations of Convex Semialgebraic Sets: Past, Present, and Future. In: Operator Theory: Advances and Applications, Vol. 222 (2011). arxiv: