Hyperbolicity cones are amenable
aa r X i v : . [ m a t h . O C ] F e b Hyperbolicity cones are amenable
Bruno F. Louren¸co ∗ Vera Roshchina † James Saunderson ‡ February 15, 2021
Abstract
Amenability is a notion of facial exposedness for convex cones that is stronger thanbeing facially dual complete (or ‘nice’) which is, in turn, stronger than merely being faciallyexposed. Hyperbolicity cones are a family of algebraically structured closed convex conesthat contain all spectrahedra (linear sections of positive semidefinite cones) as specialcases. It is known that all spectrahedra are amenable. We establish that all hyperbolicitycones are amenable. As part of the argument, we show that any face of a hyperbolicity coneis a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicitycones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.
A homogeneous polynomial p of degree d in n variables is hyperbolic with respect to e ∈ R n if p ( e ) > x ∈ R n the univariate polynomial t p ( te − x ) has only real zeros.Associated with a hyperbolic polynomial is the (closed) hyperbolicity coneΛ + ( p, e ) := { x ∈ R n | all zeros of t p ( te − x ) are nonnegative } . (1.1)Hyperbolicity cones are convex cones that generalize spectrahedral cones . Spectrahedralcones are linearly isomorphic to the intersection of a cone of symmetric positive semidef-inite matrices with a linear space. Indeed, if A , A , . . . , A n are symmetric matrices and e ∈ R n is such that P ni =1 A i e i is positive definite, then p ( x ) = det( P ni =1 A i x i ) is hyper-bolic with respect to e , and the associated hyperbolicity cone is the spectrahedral cone { x ∈ R n : P ni =1 A i x i is positive semidefinite } .The generalized Lax conjecture asserts that every hyperbolicity cone is a spectrahedralcone. This is known to hold (in a stronger form) in dimension three [4, 5], but is unresolvedin general. If the generalized Lax conjecture were true, then any geometric property ofspectrahedral cones should also hold for hyperbolicity cones.In this paper, we consider properties related to the faces (see Section 2.1 for a definition)of hyperbolicity cones that are known to hold for spectrahedral cones. A closed convex cone K is amenable if, for each face F of K , there exists a constant κ > x, F ) ≤ κ dist( x, K ) for all x ∈ span( F ) . (1.2) ∗ Department of Statistical Inference and Mathematics, Institute of Statistical Mathematics, Japan.Email: [email protected] † School of Mathematics and Statistics, UNSW Sydney, Australia. Email: [email protected] ‡ Department of Electrical and Computer Systems Engineering, Monash University, Australia.Email: [email protected]
Theorem 1.1.
Every hyperbolicity cone is amenable.
Prior to this, it was known that hyperbolicity cones are facially exposed [10, Theorem 23],but it was not known whether every hyperbolicity cone is nice, let alone amenable. A simpleconsequence of Theorem 1.1 and [6, Proposition 13] is that every hyperbolicity cone is nice.Another basic geometric fact about the faces of spectrahedral cones is that they are,themselves, spectrahedral cones. This follows from the fact that the faces of the positivesemidefinite cone are positive semidefinite cones of smaller dimension. En route to the proofof Theorem 1.1, we show that faces of hyperbolicity cones are, themselves, hyperbolicity cones(Corollary 3.4). Based on this result, we also show that the intersection of two hyperbolicitycones is, again, a hyperbolicity cone (Corollary 3.5). This is a well-known fact under the addi-tional assumption that the hyperbolicity cones share an interior point (and hence a directionof hyperbolicity). However, we could not find this result in the literature in the more subtlecase where the relative interiors of the cones do not intersect.
Outline
In Section 2.1, we recall the required definitions from convex geometry, and state astandard error bound result that will be needed for the proof of Theorem 1.1. In Section 2.2we briefly summarize certain key facts and definitions about hyperbolic polynomials, hyper-bolicity cones, and their derivative relaxations, that will be used throughout. In Section 3.1,we show that faces of hyperbolicity cones are, themselves, hyperbolicity cones, and that theintersection of two hyperbolicity cones is always a hyperbolicity cone. We establish these factsvia an explicit description of a face of a hyperbolicity cone as the (proper) intersection of asubspace with an appropriate derivative relaxation of the hyperbolicity cone. In Section 3.2we use this representation, together with the error bound stated in Section 2.1, to show thatany hyperbolicity cone is amenable. In Section 4 we discuss some further questions related tothe faces of hyperbolicity cones.
Let C ⊆ R n be a closed convex set. In what follows we denote the interior, relative interiorand the span of C by int C , ri C and span C , respectively. We also recall that a face of C is aclosed convex set F contained in C satisfying the following property: whenever x, y ∈ C are2uch that αx + (1 − α ) y ∈ F for some α ∈ (0 , x, y ∈ F . In this case, we write F E C .In what follows, we assume that R n is equipped with some norm k·k and denote bydist( x, C ) the distance from x to C so that dist( x, C ) := inf {k x − y k | y ∈ C } . Finally, weneed the following error bound result.
Proposition 2.1 (Linear regularity under a constraint qualification) . Let L be a linear sub-space and K a closed convex cone with L ∩ int K 6 = ∅ . Then there exists κ > such that dist( x, L ∩ K ) ≤ κ (dist( x, L ) + dist( x, K )) ∀ x ∈ R n . Proof.
It follows from [1, Corollary 3] that L and K are boundedly linearly regular, i.e. forevery bounded set S there exists a κ S > x, L ∩ K ) ≤ κ S max { dist( x, L ) , dist( x, K ) } ∀ x ∈ S. (2.1)Moreover, because L and K are closed convex cones, by homogeneity one can choose κ S := ¯ κ in (2.1) independent of S (see [1, Theorem 10]). In this section, we recall some basic facts about hyperbolic polynomials, hyperbolicity cones,and their derivative relaxations. Recall that if p is hyperbolic with respect to e then wewrite Λ + ( p, e ) for the associated hyperbolicity cone. It turns out that the dependence of thehyperbolicity cone on the choice of e is quite weak. In fact, if z ∈ int(Λ + ( p, e )) then p ishyperbolic with respect to z , and Λ + ( p, e ) = Λ + ( p, z ), a result of G˚arding [3]. When p and e are clear from the context, we will omit them and write Λ + . Polynomial inequality description of hyperbolicity cones
The definition of the closedhyperbolicity cone Λ + ( p, e ) given in (1.1) is expressed in terms of the nonnegativity of thezeros of p ( te − x ). It is also useful to work with an alternative description of Λ + ( p, e ) interms of polynomial inequalities. Let D ke p ( x ) = d k dt k p ( x + te ) (cid:12)(cid:12)(cid:12) t =0 denote the k th directionalderivative of p in the direction e . Expanding p ( x + te ) in powers of t gives p ( x + te ) = p ( x ) + tD e p ( x ) + t D e p ( x ) + · · · + t d d ! D de p ( x ) . By Descartes’ rule of signs (see, e.g., Proposition 18 and Theorem 20 in [10]), we haveΛ + ( p, e ) = { x ∈ R n : p ( x ) ≥ , D e p ( x ) ≥ , . . . , D d − e p ( x ) ≥ } . (2.2)The interior of Λ + ( p, e ) is thenint(Λ + ) = { x ∈ R n : p ( x ) > , D e p ( x ) > , . . . , D d − e p ( x ) > } . (2.3)3 erivative relaxations If p is hyperbolic with respect to e , then so are the directionalderivatives D me p ( x ) (by Rolle’s theorem, see also [10, Proposition 18]). Denote the associatedhyperbolicity cone by Λ + ( D me p, e ). If p and e are clear from the context we write Λ ( m )+ :=Λ + ( D me p, e ). If m ≤ d − ( m )+ = { x ∈ R n : D me p ( x ) ≥ , D m +1 e p ( x ) ≥ , . . . , D d − e p ( x ) ≥ } . (2.4)For convenience we define Λ ( d )+ = R n . From this description it is clear thatΛ + ⊆ Λ (1)+ ⊆ · · · ⊆ Λ ( d − ⊆ Λ ( d )+ = R n . (2.5)These are known as Renegar derivatives or, in light of (2.5), derivative relaxations of Λ + .The following (standard) technical fact about the coefficients of univariate polynomialswith nonnegative coefficients will be useful later in our arguments. Lemma 2.2.
Let p ( t ) = a d t d + a d − t d − + · · · + a t + a be a univariate polynomial with realzeros, nonnegative coefficients, and a d > . If a k = 0 for some k < d , then a ℓ = 0 for all ℓ < k .Proof. Because all coefficients are nonnegative and a d >
0, we have p ( λ ) > λ . Therefore, all roots of p are nonpositive. Let λ , . . . , λ d ≤ p .We recall that | a k /a d | = | e d − k ( λ , . . . , λ d ) | is the absolute value of the elementary sym-metric polynomial of degree d − k in the roots of p . Since all the roots of p are non-positive, | e d − k ( λ , . . . , λ d ) | = e d − k ( | λ | , . . . , | λ d | ). Therefore, if a k = 0, it follows that ev-ery product of d − k roots of p must vanish. This shows that a ℓ = 0 for ℓ < k , because | a ℓ /a d | = e d − ℓ ( | λ | , . . . , | λ d | ) is the absolute value of a sum of products of more than d − k roots of p . In this section, we prove the two main contributions of this paper. In Section 3.1, we showthat the faces of hyperbolicity cones can be expressed as the intersection of a linear spacewith a suitable derivative relaxation of the hyperbolicity cone, in such a way that the linearspace meets the interior of the derivative relaxation. This new representation of the facesof hyperbolicity cones is the crucial ingredient in our proof that every hyperbolicity cone isamenable, which is the focus of Section 3.2.
Given x ∈ R n , let mult p ( x ) denote the multiplicity of 0 as a zero of t p ( te − x ). It turnsout that this quantity is independent of the choice of e [10, Proposition 22], so we suppress e from the notation.The following result tells us that a point of multiplicity m in a hyperbolicity cone is inthe interior of the m th derivative relaxation. Lemma 3.1.
Let p be hyperbolic with respect to e . If mult p ( x ) = m and x ∈ Λ + ( p, e ) then • D ke p ( x ) = 0 for k = 0 , , . . . , m − and D ke p ( x ) > for k = m, m + 1 , . . . , d ,and so x ∈ int(Λ ( m )+ ) .Proof. Since mult p ( x ) = m , we know that t p ( te − x ) vanishes to exactly order m at t = 0.Since p ( te − x ) = ( − d p ( x + ( − t ) e ), it also holds that t p ( x + te ) vanishes to exactly order m at t = 0. Hence p ( x + te ) = t m m ! D me p ( x ) + t m +1 ( m + 1)! D m +1 e p ( x ) + · · · + t d d ! D de p ( x )and D me p ( x ) = 0 (otherwise mult p ( x ) > m ). We can immediately conclude that D ℓe p ( x ) = 0for ℓ = 0 , , . . . , m −
1. Since x ∈ Λ + , we know that D ke p ( x ) ≥ k and so, since D me p ( x ) = 0, it must be the case that D me p ( x ) >
0. By Lemma 2.2, it follows that D ke p ( x ) > k = m, m + 1 , . . . , d −
1. Moreover D de p ( x ) = d ! p ( e ) >
0, since p is hyperbolic with respectto e . Since D ke p ( x ) > k = m, m + 1 , . . . , d −
1, it follows from (2.4) and (2.3) that x ∈ int(Λ ( m )+ ).We also require the following key fact, which is part of Renegar [10, Theorem 26], aboutmultiplicity and faces of hyperbolicity cones. Lemma 3.2.
Suppose that p is hyperbolic with respect to e . If F is a face of Λ + ( p, e ) and x, z ∈ ri( F ) then mult p ( x ) = mult p ( z ) . We now establish the main result of this subsection, which gives a new description of thefaces of hyperbolicity cones.
Proposition 3.3 (Faces and derivative relaxations) . Suppose that p is hyperbolic with respectto e and F is a face of Λ + ( p, e ) . Let z ∈ ri( F ) and let m = mult p ( z ) . Then z ∈ int(Λ ( m )+ ) and F = span( F ) ∩ Λ ( m )+ . Proof.
Since mult p ( z ) = m and z ∈ ri( F ) ⊆ Λ + , it follows from Lemma 3.1 that p ( z ) = D e p ( z ) = · · · = D m − e p ( z ) = 0 and that z ∈ int(Λ ( m )+ ).Since F is a face of Λ + ( p, e ), we have that F = span F ∩ Λ + ( p, e ). Equivalently, F = { x ∈ span( F ) : p ( x ) ≥ , D e p ( x ) ≥ , . . . , D me p ( x ) ≥ , . . . , D d − e p ( x ) ≥ } . Since z ∈ ri( F ) has multiplicity m , it follows that every point in the relative interior of F hasmultiplicity m . As such p ( x ) = D e p ( x ) = · · · = D m − e p ( x ) = 0 for all x ∈ ri( F ). Since ri( F )is full-dimensional in span( F ), it follows that p ( x ) = D e p ( x ) = · · · = D m − e p ( x ) = 0 for all x ∈ span( F ). As such, F = { x ∈ span( F ) : D me p ( x ) ≥ , . . . , D d − e p ( x ) ≥ } = span( F ) ∩ Λ ( m )+ . This result can be rephrased as saying that faces of hyperbolicity cones are, themselves,hyperbolicity cones. To see this, first we clarify the notation we will use. If L ⊆ R n is asubspace and p is hyperbolic with respect to e ∈ L , then the restriction of p to L , denoted p | L ,is also hyperbolic with respect to e when thought of as a polynomial on L . The correspondinghyperbolicity cone, Λ + ( p | L , e ), is linearly isomorphic to Λ + ( p, e ) ∩ L .5 orollary 3.4. Suppose that p is hyperbolic with respect to e and F is a face of Λ + ( p, e ) .If z ∈ ri( F ) and m = mult p ( z ) then q = D me p | span( F ) is hyperbolic with respect to z and F = Λ + ( q, z ) . To conclude this subsection, let Λ + ( p , e ) and Λ + ( p , e ) be two hyperbolicity cones.If their relative interiors intersect, then Λ + ( p , e ) ∩ Λ + ( p , e ) is also a hyperbolicity coneand the corresponding hyperbolic polynomial is p p restricted to the span of Λ + ( p , e ) ∩ Λ + ( p , e ). However, the situation is less obvious when (ri Λ + ( p , e )) ∩ (ri Λ + ( p , e )) = ∅ .As a consequence of Corollary 3.4, we have a proof of the fact that the intersection of twohyperbolicity cones is also a hyperbolicity cone. Corollary 3.5.
Suppose that p and p are hyperbolic polynomials with respect to e and e ,respectively. Then, Λ + ( p , e ) ∩ Λ + ( p , e ) is also a hyperbolicity cone.Proof. Let K = Λ + ( p , e ) ∩ Λ + ( p , e ). It is a well-known fact from convex analysis that if F i is the smallest face of Λ + ( p i , e i ) containing K , then we haveri K ⊆ ri F i for i = 1 , K = F ∩ F . (See, for example, [7, Proposition 2.2].) In particular, there exists z ∈ ri K such that z ∈ (ri F ) ∩ (ri F ).For i = 1 ,
2, let m i := mult p i ( z ) and q i := D m i e i p i (cid:12)(cid:12) span( F i ) . By Corollary 3.4, q i is hyperbolicwith respect to z and F i = Λ + ( q i , z ). If L = span K , then ( q q ) | L is hyperbolic with respectto z and the corresponding hyperbolicity cone is K = F ∩ F . Using the standard error bound from Proposition 2.1 and our new representation of facesof hyperbolicity cones from Proposition 3.3, we can now prove our main result, that everyhyperbolicity cone is amenable.
Proof of Theorem 1.1.
Let p be hyperbolic with respect to e ∈ R n , and let F E Λ + ( p, e ) bea face of the associated hyperbolicity cone. By Proposition 3.3, there exists some derivativerelaxation Λ ( m )+ satisfying the following two properties: F ∩ (int Λ ( m )+ ) = ∅ and F = span( F ) ∩ Λ ( m )+ . In particular, span( F ) ∩ (int Λ ( m )+ ) = ∅ , and so by Proposition 2.1, there exists κ > x ∈ R n we havedist( x, F ) ≤ κ max(dist( x, span F ) , dist( x, Λ ( m )+ )) ≤ κ max(dist( x, span F ) , dist( x, Λ + ( p, e )) , where the last inequality follows from the fact that Λ + ( p, e ) ⊆ Λ ( m )+ . This implies thatdist( x, F ) ≤ κ dist( x, Λ + ( p, e )) for all x ∈ span F .Therefore Λ + ( p, e ) is amenable. 6 Discussion
It could be fruitful to examine in more depth the geometry and facial structure of hyperbolicitycones and check how it compares with spectrahedral cones. One recent result in this style isthat the tangent cone to a hyperbolicity cone at a point is, again, a hyperbolicity cone [12,Theorem 5.9], which parallels the analogous result for spectrahedral cones. For a directionwhere many questions remain open, it might be interesting to see how spectrahedral conesand hyperbolicity cones fare with regards to projectional exposedness .We say that a closed convex cone K is projectionally exposed if for every face F E K thereexists a projection P (not necessarily orthogonal) such that P ( K ) = F , see [2, 13]. This notionappears in connection to facial reduction algorithms and approaches for regularizing convexconic optimization problems. It turns out that projectional exposedness implies amenability[6, Proposition 9] and in dimension four or less, amenability implies projectional exposedness[7, Corollary 6.4].So far, the largest classes of hyperbolicity cones which are known to be projectionallyexposed are the symmetric cones ([6, Proposition 33]) and the polyhedral cones ([13, Corollary3.4]). However, [7, Corollary 6.4] and Theorem 1.1 imply that all hyperbolicity cones indimension four or less are projectionally exposed, which is a curious fact that, at this moment,does not seem to have an obvious algebraic explanation. Acknowledgements
Bruno F. Louren¸co is supported partly by the JSPS Grant-in-Aid for Young Scientists 19K20217and the Grant-in-Aid for Scientific Research (B)18H03206.Vera Roshchina is grateful to the Australian Research Council for continuing support.Specifically, her ARC DECRA grant DE150100240 helped sustain the initial collaboration onthis project.James Saunderson is the recipient of an Australian Research Council Discovery Early Ca-reer Researcher Award (project number DE210101056) funded by the Australian Government.
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