Spectral sets and derivatives of the psd cone Mario Kummer TU - PowerPoint PPT Presentation

Spectral sets and derivatives of the psd cone Mario Kummer TU Berlin August 28, 2020 Mario Kummer Spectral sets and derivatives of the psd cone 1 / 23

Spectrahedral cones A spectrahedral cone is a set of the form S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } , where A 1 , . . . , A n ∈ Sym 2 ( R d ) are real symmetric d × d matrices. Mario Kummer Spectral sets and derivatives of the psd cone 2 / 23

Spectrahedral cones A spectrahedral cone is a set of the form S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } , where A 1 , . . . , A n ∈ Sym 2 ( R d ) are real symmetric d × d matrices. ◮ Feasible sets of semidefinite programming. Mario Kummer Spectral sets and derivatives of the psd cone 2 / 23

Spectrahedral cones A spectrahedral cone is a set of the form S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } , where A 1 , . . . , A n ∈ Sym 2 ( R d ) are real symmetric d × d matrices. ◮ Feasible sets of semidefinite programming. ◮ Polyhedral cones: Take A ( x ) to be diagonal. Mario Kummer Spectral sets and derivatives of the psd cone 2 / 23

Spectrahedral cones A spectrahedral cone is a set of the form S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } , where A 1 , . . . , A n ∈ Sym 2 ( R d ) are real symmetric d × d matrices. ◮ Feasible sets of semidefinite programming. ◮ Polyhedral cones: Take A ( x ) to be diagonal. Question ◮ Which sets K ⊂ R n are spectrahedral? Mario Kummer Spectral sets and derivatives of the psd cone 2 / 23

Spectrahedral cones S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } . ◮ Fix e ∈ int ( S ). W.l.o.g. A ( e ) = I d . ◮ The polynomial det A ( x ) is hyperbolic in the following sense: Mario Kummer Spectral sets and derivatives of the psd cone 3 / 23

Spectrahedral cones S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } . ◮ Fix e ∈ int ( S ). W.l.o.g. A ( e ) = I d . ◮ The polynomial det A ( x ) is hyperbolic in the following sense: Definition A homogeneous polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e ∈ R n if h ( e ) � = 0 and if h ( te − v ) has only real roots for all v ∈ R n . Mario Kummer Spectral sets and derivatives of the psd cone 3 / 23

Spectrahedral cones S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } . ◮ Fix e ∈ int ( S ). W.l.o.g. A ( e ) = I d . ◮ The polynomial det A ( x ) is hyperbolic in the following sense: Definition A homogeneous polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e ∈ R n if h ( e ) � = 0 and if h ( te − v ) has only real roots for all v ∈ R n . The hyperbolicity cone is C ( h , e ) = { v ∈ R n : h ( te − v ) has only nonnegative roots } . Mario Kummer Spectral sets and derivatives of the psd cone 3 / 23

Spectrahedral cones S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } . ◮ Fix e ∈ int ( S ). W.l.o.g. A ( e ) = I d . ◮ The polynomial det A ( x ) is hyperbolic in the following sense: Definition A homogeneous polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e ∈ R n if h ( e ) � = 0 and if h ( te − v ) has only real roots for all v ∈ R n . The hyperbolicity cone is C ( h , e ) = { v ∈ R n : h ( te − v ) has only nonnegative roots } . ◮ det A ( te − v ) = det( tI d − A ( v )). Mario Kummer Spectral sets and derivatives of the psd cone 3 / 23

Spectrahedral cones S = { x ∈ R n : A ( x ) = x 1 A 1 + . . . + x n A n is positive semidefinite } . ◮ Fix e ∈ int ( S ). W.l.o.g. A ( e ) = I d . ◮ The polynomial det A ( x ) is hyperbolic in the following sense: Definition A homogeneous polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e ∈ R n if h ( e ) � = 0 and if h ( te − v ) has only real roots for all v ∈ R n . The hyperbolicity cone is C ( h , e ) = { v ∈ R n : h ( te − v ) has only nonnegative roots } . ◮ det A ( te − v ) = det( tI d − A ( v )). ◮ S = C (det A ( x ) , e ). Mario Kummer Spectral sets and derivatives of the psd cone 3 / 23

The Generalized Lax Conjecture Conjecture. Let h ∈ R [ x 1 , . . . , x n ] be hyperbolic with respect to e ∈ R n . Then C ( h , e ) is spectrahedral. Mario Kummer Spectral sets and derivatives of the psd cone 4 / 23

The Generalized Lax Conjecture Conjecture. Let h ∈ R [ x 1 , . . . , x n ] be hyperbolic with respect to e ∈ R n . Then C ( h , e ) is spectrahedral. True if: ◮ deg h ≤ 2. ◮ n ≤ 3. (Helton–Vinnikov) ◮ n = 4 and deg h = 3. (Buckley–Koˇ sir) Mario Kummer Spectral sets and derivatives of the psd cone 4 / 23

Constructing hyperbolic polynomials The following polynomials are hyperbolic with respect to e : ◮ det A ( x ) for A ( x ) real symmetric matrix with linear entries and A ( e ) positive definite. Includes spanning tree polynomials of graphs, bases generating polynomials of regular matroids and ternary hyperbolic polynomials. Mario Kummer Spectral sets and derivatives of the psd cone 5 / 23

Constructing hyperbolic polynomials The following polynomials are hyperbolic with respect to e : ◮ det A ( x ) for A ( x ) real symmetric matrix with linear entries and A ( e ) positive definite. Includes spanning tree polynomials of graphs, bases generating polynomials of regular matroids and ternary hyperbolic polynomials. Their hyperbolicity cones are clearly spectrahedral. Mario Kummer Spectral sets and derivatives of the psd cone 5 / 23

Constructing hyperbolic polynomials The following polynomials are hyperbolic with respect to e : ◮ The homogeneous multivariate matching polynomial of an undirected graph G = ( V , E ): ( − 1) | M | · � � � w 2 µ G ( x , w ) = x v · e e ∈ M v �∈ V ( M ) where the sum is over all matchings M of G . (Heilmann–Lieb) ◮ e = (1 V , 0 E ). Mario Kummer Spectral sets and derivatives of the psd cone 6 / 23

Constructing hyperbolic polynomials The following polynomials are hyperbolic with respect to e : ◮ The homogeneous multivariate matching polynomial of an undirected graph G = ( V , E ): ( − 1) | M | · � � � w 2 µ G ( x , w ) = x v · e e ∈ M v �∈ V ( M ) where the sum is over all matchings M of G . (Heilmann–Lieb) ◮ e = (1 V , 0 E ). Their hyperbolicity cones are spectrahedral (Amini). Mario Kummer Spectral sets and derivatives of the psd cone 6 / 23

Constructing hyperbolic polynomials The following polynomials are hyperbolic with respect to e : ◮ The defining polynomial of the k th secant variety of a projectively normal M -curve with “many” pseudolines in P 2 k +2 . (K.–Sinn) Their hyperbolicity cones are spectrahedral for rational and elliptic curves. Mario Kummer Spectral sets and derivatives of the psd cone 7 / 23

Operations preserving hyperbolicity These operations preserve being hyperbolic with respect to e : ◮ Taking products. ◮ Restricting to a linear subspace containing e . ◮ Applying linear changes of coordinates ( e might change). Mario Kummer Spectral sets and derivatives of the psd cone 8 / 23

Operations preserving hyperbolicity These operations preserve being hyperbolic with respect to e : ◮ Taking products. ◮ Restricting to a linear subspace containing e . ◮ Applying linear changes of coordinates ( e might change). These operations also preserve spectrahedrality of the corresponding hyperbolicity cones. Mario Kummer Spectral sets and derivatives of the psd cone 8 / 23

Renegar derivatives Consequence of Rolle’s Theorem: ◮ If a polynomial p ∈ R [ t ] has only real zeros, then its derivative p ′ has only real zeros. Mario Kummer Spectral sets and derivatives of the psd cone 9 / 23

Renegar derivatives Consequence of Rolle’s Theorem: ◮ If a polynomial p ∈ R [ t ] has only real zeros, then its derivative p ′ has only real zeros. This implies: ◮ If a polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e , then its directional derivative n e i · ∂ h � D e h = ∂ x i i =1 is hyperbolic with respect to e as well. Mario Kummer Spectral sets and derivatives of the psd cone 9 / 23

Renegar derivatives Consequence of Rolle’s Theorem: ◮ If a polynomial p ∈ R [ t ] has only real zeros, then its derivative p ′ has only real zeros. This implies: ◮ If a polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e , then its directional derivative n e i · ∂ h � D e h = ∂ x i i =1 is hyperbolic with respect to e as well. Question Is the hyperbolicity cone of D e (det A ( x )) spectrahedral? Mario Kummer Spectral sets and derivatives of the psd cone 9 / 23

Renegar derivatives Consequence of Rolle’s Theorem: ◮ If a polynomial p ∈ R [ t ] has only real zeros, then its derivative p ′ has only real zeros. This implies: ◮ If a polynomial h ∈ R [ x 1 , . . . , x n ] is hyperbolic with respect to e , then its directional derivative n e i · ∂ h � D k e h = ∂ x i i =1 is hyperbolic with respect to e as well. Question Is the hyperbolicity cone of D k e (det A ( x )) spectrahedral? Mario Kummer Spectral sets and derivatives of the psd cone 10 / 23

Example The polynomial h = x 1 · · · x d is hyperbolic with respect to e = (1 , . . . , 1). Mario Kummer Spectral sets and derivatives of the psd cone 11 / 23