Hyperbolic Polynomials, Interlacers, and Sums of Squares Cynthia - PowerPoint PPT Presentation

Hyperbolic Polynomials, Interlacers, and Sums of Squares Cynthia Vinzant University of Michigan joint work with Mario Kummer and Daniel Plaumann - 4, Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Hyperbolic Polynomials A homogeneous polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point e ∈ R n if f ( e ) � = 0 and for every x ∈ R n , all roots of f ( te + x ) ∈ R [ t ] are real. Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Hyperbolic Polynomials A homogeneous polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point e ∈ R n if f ( e ) � = 0 and for every x ∈ R n , all roots of f ( te + x ) ∈ R [ t ] are real. x 2 1 − x 2 2 − x 2 3 hyperbolic with respect to e = (1 , 0 , 0) Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Hyperbolic Polynomials A homogeneous polynomial f ∈ R [ x 1 , . . . , x n ] d is hyperbolic with respect to a point e ∈ R n if f ( e ) � = 0 and for every x ∈ R n , all roots of f ( te + x ) ∈ R [ t ] are real. x 2 1 − x 2 2 − x 2 x 4 1 − x 4 2 − x 4 3 3 hyperbolic with not hyperbolic respect to e = (1 , 0 , 0) Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Hyperbolicity Cones Its hyperbolicity cone , denoted C ( f , e ), is the connected component of e in R n \V R ( f ). - 4, Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Hyperbolicity Cones Its hyperbolicity cone , denoted C ( f , e ), is the connected component of e in R n \V R ( f ). G˚ arding (1959) showed that ◮ C ( f , e ) is convex, and - 4, ◮ f is hyperbolic with respect to any point a ∈ C ( f , e ). Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Hyperbolicity Cones Its hyperbolicity cone , denoted C ( f , e ), is the connected component of e in R n \V R ( f ). G˚ arding (1959) showed that ◮ C ( f , e ) is convex, and - 4, ◮ f is hyperbolic with respect to any point a ∈ C ( f , e ). One can use interior point methods to optimize a linear function over an affine section of a hyperbolicity cone, G¨ uler (1997), Renegar (2006). This solves a hyperbolic program . Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Two Important Examples of Hyperbolic Programming f e C ( f , e ) Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Two Important Examples of Hyperbolic Programming Linear Programming f � i x i e (1 , . . . , 1) ( R + ) n C ( f , e ) Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Two Important Examples of Hyperbolic Programming Linear Programming Semidefinite Programming   x 11 . . . x 1 n . . ... . . f � i x i det   . .   x 1 n . . . x nn e (1 , . . . , 1) Id n ( R + ) n C ( f , e ) positive definite matrices - 4, Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Connections to Multiaffine Polynomials and Matroids A polynomial f is multiaffine if it has degree one in each variable. Example: f = x 1 x 2 + x 1 x 3 + x 2 x 3 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Connections to Multiaffine Polynomials and Matroids A polynomial f is multiaffine if it has degree one in each variable. A polynomial f is real stable if it is hyperbolic and ( R + ) n ⊆ C ( f , e ) Example: f = x 1 x 2 + x 1 x 3 + x 2 x 3 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Connections to Multiaffine Polynomials and Matroids A polynomial f is multiaffine if it has degree one in each variable. A polynomial f is real stable if it is hyperbolic and ( R + ) n ⊆ C ( f , e ) Theorem (Choe, Oxley, Sokal, Wagner (2004)) If f is multiaffine and real stable then the monomials in the support of f form the bases of a matroid. Example: f = x 1 x 2 + x 1 x 3 + x 2 x 3 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Connections to Multiaffine Polynomials and Matroids A polynomial f is multiaffine if it has degree one in each variable. A polynomial f is real stable if it is hyperbolic and ( R + ) n ⊆ C ( f , e ) Theorem (Choe, Oxley, Sokal, Wagner (2004)) If f is multiaffine and real stable then the monomials in the support of f form the bases of a matroid. Example: f = x 1 x 2 + x 1 x 3 + x 2 x 3 − → {{ 1 , 2 } , { 1 , 3 } , { 2 , 3 }} Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Connections to Multiaffine Polynomials and Matroids A polynomial f is multiaffine if it has degree one in each variable. A polynomial f is real stable if it is hyperbolic and ( R + ) n ⊆ C ( f , e ) Theorem (Choe, Oxley, Sokal, Wagner (2004)) If f is multiaffine and real stable then the monomials in the support of f form the bases of a matroid. For any representable matroid there is a multiaffine real stable polynomial whose support is the collection of its bases. Example: f = x 1 x 2 + x 1 x 3 + x 2 x 3 − → {{ 1 , 2 } , { 1 , 3 } , { 2 , 3 }} Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Interlacing Derivatives If all roots of p ( t ) are real, then the roots of p ′ ( t ) are real and interlace the roots of p ( t ). Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Interlacing Derivatives If all roots of p ( t ) are real, then the roots of p ′ ( t ) are real and interlace the roots of p ( t ). For any direction a ∈ C ( f , e ) the polynomial � ∂ � � ∂ f � � D a ( f ) = a i = ∂ t f ( ta + x ) � ∂ x i � t =0 i is hyperbolic and interlaces f . 2 1 0 1 2 2 1 0 1 2 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Interlacing Derivatives If all roots of p ( t ) are real, then the roots of p ′ ( t ) are real and interlace the roots of p ( t ). For any direction a ∈ C ( f , e ) the polynomial � ∂ � � ∂ f � � D a ( f ) = a i = ∂ t f ( ta + x ) � ∂ x i � t =0 i is hyperbolic and interlaces f . (Not true for a / ∈ C ( f , e )). 2 2 1 1 0 0 1 1 2 2 2 1 0 1 2 2 1 0 1 2 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

The Convex Cone of Interlacers Int ( f , e ) = { g ∈ R [ x 1 , . . . , x n ] d − 1 : g ( e ) > 0 and g interlaces f } 3 2 1 0 1 2 3 3 2 1 0 1 2 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

The Convex Cone of Interlacers Int ( f , e ) = { g ∈ R [ x 1 , . . . , x n ] d − 1 : g ( e ) > 0 and g interlaces f } 3 2 1 0 1 2 Theorem If f is square free and hyperbolic w.r.t. e ∈ R n , then 3 3 2 1 0 1 2 Int ( f , e ) = { g : D e f · g − f · D e g ≥ 0 on R n } . Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

The Convex Cone of Interlacers Int ( f , e ) = { g ∈ R [ x 1 , . . . , x n ] d − 1 : g ( e ) > 0 and g interlaces f } 3 2 1 0 1 2 Theorem If f is square free and hyperbolic w.r.t. e ∈ R n , then 3 3 2 1 0 1 2 Int ( f , e ) = { g : D e f · g − f · D e g ≥ 0 on R n } . This is a convex cone in R [ x 1 , . . . , x n ] d − 1 . Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Special Interlacers g = D a f Theorem If f ∈ R [ x 1 , . . . , x n ] d is square-free and hyperbolic w.r.t e ∈ R n , C ( f , e ) = { a ∈ R n : D e f · D a f − f · D e D a f ≥ 0 R n } . on 2 1 0 1 2 2 1 0 1 2 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Special Interlacers g = D a f Theorem If f ∈ R [ x 1 , . . . , x n ] d is square-free and hyperbolic w.r.t e ∈ R n , C ( f , e ) = { a ∈ R n : D e f · D a f − f · D e D a f ≥ 0 R n } . on 2 1 This writes the hyperbolicity cone C ( f , e ) as a slice of the cone of 0 nonnegative polynomials . 1 2 2 1 0 1 2 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Example: the Lorentz cone 1.0 0.5 f ( x ) = x 2 1 − x 2 2 − . . . − x 2 e = (1 , 0 , . . . , 0) n 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Example: the Lorentz cone 1.0 0.5 f ( x ) = x 2 1 − x 2 2 − . . . − x 2 e = (1 , 0 , . . . , 0) n 0.0 0.5 D e f · D a f − f · D e D a f 1.0 j � =1 2 a j x j ) − ( x 2 j � =1 x 2 = (2 x 1 )(2 a 1 x 1 − � 1 − � j )(2 a 1 ) 1.0 0.5 0.0 0.5 1.0 Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares

Example: the Lorentz cone 1.0 0.5 f ( x ) = x 2 1 − x 2 2 − . . . − x 2 e = (1 , 0 , . . . , 0) n 0.0 0.5 D e f · D a f − f · D e D a f 1.0 j � =1 2 a j x j ) − ( x 2 j � =1 x 2 = (2 x 1 )(2 a 1 x 1 − � 1 − � j )(2 a 1 ) 1.0 0.5 0.0 0.5 1.0 � � j x 2 � j − 2 � = 2 a 1 j � =1 a j x 1 x j Cynthia Vinzant Hyperbolic Polynomials, Interlacers, and Sums of Squares