Semidefinite Programming and its Feasible Sets
Part II: Spectrahedra
Tim Netzer
University of Leipzig, Germany
Polynomial Optimisation Summer School
Isaac Newton Institute for Mathematical Sciences
15 July - 9 August 2013
Spectrahedra
Definition:
The feasible set of a semidefinite program is called a spectrahedron.
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Primal point of view: a spectrahedron is an affine-linear section of
the cone P ⊆ Symd (R) of positive semidefinite matrices.
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Dual point of view: a spectrahedron is a subset of Rn defined by a
linear matrix inequality:
{a ∈ Rn | M0 + a1 M1 + · · · + an Mn 0}
for some M0 , . . . , Mn ∈ Symd (R).
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Most of the time we take the dual point of view.
Spectrahedra
Which sets are spectrahedra?
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None of them is a polyhedron. Deciding spectrahedrality is more
complicated.
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Since spectrahedra are so important in semidefinite programming,
a good characterization is highly desirable!
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Finding defining matrices M0 , . . . , Mn is also important!
Spectrahedra
Some first examples:
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Choosing diagonal matrices M0 , . . . , Mn results in a polyhedron, as
already seen. The name spectrahedron highlights the connection
to polyhedra and indicates that the spectrum of a matrix is
involved in the definition (Ramana & Goldman, 1995).
1 0
1 0
0 1
+ a1
+ a2
0 ⇔ a12 + a22 ≤ 1.
0 1
0 −1
1 0
More generally, every ellipsoid is a spectrahedron (Nie, Parillo &
Sturmfels, 2008)
Spectrahedra
M0 + a1 M1 + · · · + an Mn 0
Some first properties:
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Spectrahedra are closed and convex.
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Spectrahedra are basic closed semialgebraic, i.e. definable by
simultaneous polynomial inequalities
{a ∈ Rn | p1 (a) ≥ 0, . . . , pr (a) ≥ 0}
for some p1 , . . . , pr ∈ R[x1 , . . . , xn ]. E.g. use the principle minors
of M0 + x1 M1 + · · · + xn Mn .
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Spectrahedra only have exposed faces, since this is true for the
cone P of positive semidefinite matrices.
Spectrahedra
Which sets are spectrahedra?
Spectrahedra
Which sets are spectrahedra?
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Now it’s time for some algebraic geometry!
From now on we only consider spectrahedral cones, i.e. sets of the
form
{a ∈ Rn | a1 M1 + · · · + an Mn 0}.
This simplifies many arguments, and still all of the results can
easily be carried over to the affine case.
We can also assume that spectrahedral cones have non-empty
interior, as usual when dealing with convex sets.
If e ∈ Rn is an interior point of the cone
{a ∈ Rn | a1 M1 + · · · + an Mn 0},
we can even assume
e1 M1 + · · · + en Mn 0.
Exercise! So we can even assume e1 M1 + · · · + en Mn = I .
Spectrahedra
Let M1 , . . . , Mn ∈ Symd (R) and e ∈ Rn with e1 M1 + · · · + en Mn = I .
Consider
h := det(x1 M1 + · · · + xn Mn ) ∈ R[x1 , . . . , xn ].
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h is a homogeneous polynomial of degree d.
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For a ∈ Rn we have
ha (t) := h(a − te) = det(a1 M1 + · · · + an Mn − tI ) ∈ R[t],
the characteristic polynomial of the symmetric matrix
a1 M1 + · · · + an Mn .
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So all zeros of ha (t) are real, and the spectrahedal cone
{a ∈ Rn | a1 M1 + · · · + an Mn 0} has the following description:
{a ∈ Rn | all zeros of ha (t) are ≥ 0.}
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The determinant polynomial h contains all information about the
spectrahedral cone!
Spectrahedra
Example:
M1 =
1 0
0 1
M2 =
1 0
0 −1
M3 =
0 1
1 0
e = (1, 0, 0)
h = det(x1 M1 + x2 M2 + x3 M3 ) = x12 − x22 − x32
The spectrahedral cone is the upper half of this double ice-cream-cone
(filled, of course).
Spectrahedra
Hyperbolic Polynomials:
A homogeneous polynomial h ∈ R[x1 , . . . , xn ] is called hyperbolic in
direction e ∈ Rn , if h(e) 6= 0 and
ha (t) := h(a − te)
has only real roots, for all a ∈ Rn . So all lines parallel to e meet the
variety of h in only real points.
Hyperbolic in vertical direction:
Spectrahedra
Not hyperbolic in vertical direction (since of degree 4) :
Spectrahedra
For given hyperbolic h the set
Λe (h) := {a ∈ Rn | all roots of ha (t) = h(a − te) are ≥ 0}
is called the hyperbolicity cone of h (in direction e).
Some observations:
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Hyperbolicity cones are closed convex cones.
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Hyperbolicity cones are basic closed semialgebraic.
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Hyperbolicity cones have only exposed faces.
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Each spectrahedral cone is a hyperbolicity cone!
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Hyperbolicity of a cone is usually easier to check than
spectrahedrality! One has to compute the Zariski closure of the
boundary of the cone, and check the real zero property.
Spectrahedra
Not a spectrahedron:
Spectrahedra
Still unclear:
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We know the cone is hyperbolic.
Spectrahedra
Generalized Lax conjecture:
Every hyperbolicity cone is spectrahedral!
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Since hyperbolicity is easier to check than the spectrahedron
property, this would be a nice characteriziation of spectrahedra.
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Especially good would be a constructive proof of the conjecture,
providing a method to obtain defining matrices M1 , . . . , Mn for a
given hyperbolicity cone.
Spectrahedra
How to attack the generalized Lax conjecture?
First idea: Given h ∈ R[x1 , . . . , xn ] hyperbolic in direction e ∈ Rn , is
h = det(x1 M1 + · · · + xn Mn )
for some M1 , . . . , Mn ∈ Symd (R) with e1 M1 + · · · + en Mn 0?
This is called a definite determinantal representation of h.
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This would show that Λe (h) = {a ∈ Rn | a1 M1 + · · · + an Mn 0}
is spectrahedral.
Theorem (Helton & Vinnikov, 2007)
For n = 3, the answer is yes! So every 3-dimensional hyperbolicity cone
is spectrahedral.
Spectrahedra
Spectrahedra:
Spectrahedra
Theorem (Helton & Vinnikov, 2007)
Every hyperbolic h ∈ R[x1 , x2 , x3 ] admits a def. det. rep.
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This result was formerly known as the Lax conjecture. The proof
uses algebraic geometry and is hard.
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There are more elementary and also constructive approaches due
to Leykin, Plaumann, Sturmfels, Vinzant (2010–2013).
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There is a different, purely algebraic proof, due to Hanselka (yet
unpublished).
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The result fails badly for n ≥ 4 :the set of n-tuples of symmetric
d × d-matrices is of dimension n · d+1
2 , the set of hyperbolic
polynomials of degree d in n variables is of dimension n+d−1
.
d
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Concrete example: x12 − x22 − x32 − x42 is hyperbolic in direction
(1, 0, 0, 0) but does not have a def. det. rep. (exercise!)
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This does not mean the generalized Lax conjecture fails!
Spectrahedra
How to attack the generalized Lax conjecture?
First idea: Find a def. det. rep. of hyperbolic h (true for n ≤ 3, fails
for n ≥ 4).
Second idea: Find a def. det. rep. of some power hr .
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Since Λe (h) = Λe (hr ) this would still show that this cone is
spectrahedral.
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The answer is yes for d = 2 and arbitrary n. Powers are really
necessary here (N. & Thom, 2012).
Spectrahedra
Theorem (Brändèn, 2011)
There is a hyperbolic polynomial h ∈ R[x1 , x2 , x3 , x4 ] of degree 4, of
which no power admits a def. det. rep.
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Brändèn’s first example used 8 variables and is the bases
generating polynomial of the Vàmos matroid:
6
4
5
1
3
2
8
7
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It’s the sum over all products of 4 of the variables x1 , . . . , x8 ,
except for the 5 monomials corresponding to hypersurfaces in the
picture.
The statement uses that the Vàmos matroid has no representation
by subspaces of a vector space.
Due to symmetry, the number of variables can be reduced to 4.
Spectrahedra
Second idea: Find a def. det. rep. of some power hr .
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There is another approach, due to N., Plaumann & Thom (2013),
using the parametrized Hermite matrix H(h) of h (introduced by
Henrion 2010). If some hr admits a def. det. rep., then H(h) is a
sum of squares of polynomial matrices.
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This condition can be checked with a semidefinite program.
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For the Brändén-Vàmos polynomial, numerical experiments show
that the Hermite matrix is not a sum of squares. So slight
pertubations of this polynomial will also behave badly.
Spectrahedra
How to attack the generalized Lax conjecture?
First idea: Find a def. det. rep. of hyperbolic h (true for n ≤ 3, fails
badly for n ≥ 4).
Second idea: Find a def. det. rep. of some power hr (works for
quadratic polynomials, fails probably quite badly for d ≥ 4).
Third idea: Find a def. det. rep. of some multiple q · h with
Λe (qh) = Λe (h).
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This would still solve the generalized Lax conjecture. The third
idea is in fact equivalent to the generalized Lax conjecture.
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The condition Λe (qh) = Λe (h) is equivalent to q having no zeros
inside Λe (h).
q=0
h=0
Spectrahedra
Third idea: Find a def. det. rep. of some multiple q · h with
Λe (qh) = Λe (h).
What cofactor q should we try?
I q = hr does not work in general.
I Let q be the hyperplane orthogonal to e.
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For almost no hyperbolic polynomial h, some q r h has a def. det.
rep. (N. & Thom, 2012).
For every hyperbolic polynomial there exists a linear rational def.
det. rep. of some q r h (N., Plaumann & Thom, 2013). But there
seems to be no connection to the generalized Lax conjecture.
Spectrahedra
Positive results, finally:
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The hyperbolicity cone of the Brändén–Vàmos polynomial h is
spectrahedral, since some suitable qh has a def. det. rep.
(Kummer, 2013).
Spectrahedra
Positive results, finally:
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The elementary symmetric polynomials sk,n ∈ R[x1 , . . . , xn ] are
hyperbolic in direction (1, . . . , 1), and suitable multiples have def.
det. rep. Their hyperbolicity cones are thus spectrahedral
(Brändén, 2012).
Start with a finite graph G = (V , E ) and consider its spanning
tree polynomial
X
Y
h=
xe .
T ⊆E
spanning tree
e∈T
By Kirchhoff’s matrix tree theorem, h has a def. det. rep., and the
elementary symmetric polynomials sk,n arise as cofactors of such h.
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This implies that all derivative cones of polyhedral cones are
spectrahedral.
Spectrahedra
Spectrahedra
Finally, a new approach due to N. & Thom (2012)
Let h ∈ R[x1 , . . . , xn ] be hyperbolic in direction e ∈ Rn . Then
ha (t) = h(a − te) ∈ R[t]
is the characteristic polynomial of a1 M1 + · · · + an Mn in case the Mi
yield a def. det. rep. of h. By Cayley–Hamilton:
ha (a1 M1 + · · · + an Mn ) = 0.
Now define an abstract (non-commutative) ∗-algebra Ae (h) with
symmetric generators z1 , . . . , zn and relations
ha (a1 z1 + · · · + an zn ) = 0 ∀a ∈ Rn .
Finitely many relations suffice. We call Ae (h) the generalized Clifford
algebra associated with h.
Spectrahedra
Theorem
If −1 is not a sum of hermitian squares in Ae (h), then the generalized
Lax conjecture is true for h.
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The hypothesis might always be fulfilled.
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The condition −1 ∈
/ ΣAe (h)2 can (in principle) be checked with a
semidefinite program.
Semidefinite programming can be used to check whether a set is
feasible for semidefinite programming!
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Defining matrices for the hyperbolicity cone Λe (h) can (in
principle) be extracted from this semidefinite programming
approach.
Spectrahedra
Incomplete bibliography I:
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P. Brändén: Obstructions to determinantal representability, Adv.
Math. 226 (2), 1202-–1212 (2011).
P. Brändén: Hyperbolicity cones of elementary symmetric
polynomials are spectrahedral, Preprint (2012).
L. Gårding: An inequality for hyperbolic polynomials, J. Math.
Mech. 8, 958-965 (1959).
J.W. Helton & V. Vinnikov: Linear matrix inequality
representations of sets, Comm. Pure Appl. Math. 60 (5), 654–674
(2007).
D. Henrion: Detecting rigid convexity of bivariate polynomials,
Lin. Alg. Appl. 432 (5), 1218–1233 (2010).
M. Kummer: A Note on the Hyperbolicity Cone of the Specialized
Vàmos Polynomial, Preprint (2013).
P.D. Lax: Differential equations, difference equations and matrix
theory, Comm. Pure Appl. Math. 6, 175-–194 (1958).
Spectrahedra
Incomplete bibliography II:
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A.S. Lewis, P.A. Parrilo & M.V. Ramana: The Lax conjecture is
true, Proc. Amer. Math. Soc. 133 (9), 2495–2499 (2005).
T. Netzer, D. Plaumann & A. Thom: Determinantal
representations and the Hermite matrix, Mich. Math. J. (2),
407–420 (2013).
T. Netzer & A. Thom: Polynomials with and without
determinantal representations, Lin. Alg. Appl. 437 (7), 1579–1595
(2012).
T. Netzer & A. Thom Hyperbolic polynomials and generalized
Clifford algebras, Preprint (2012).
J. Nie, P.A. Parrilo & B. Sturmfels: Semidefinite Representation of
the k-Ellipse, in: Algorithms in Algebraic Geometry, IMA Volumes
in Mathematics and its Applications, Vol. 146, Springer (2008).
W. Nuij: A note on hyperbolic polynomials, Math. Scand. 23,
69–72 (1969).
Spectrahedra
Incomplete bibliography III:
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D. Plaumann & A. Leykin: Determinantal representations of
hyperbolic curves via polynomial homotopy continuation, Preprint
(2012).
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D. Plaumann, B. Sturmfels & C. Vinzant: Computing linear matrix
representations of Helton-Vinnikov curves, in: Mathematical
Methods in Systems, Optimization and Control, Operator Theory:
Advances and Applications, Birkhäuser, Basel (2011).
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D. Plaumann & C. Vinzant: Determinantal representations of
hyperbolic plane curves: An elementary approach, Preprint (2012).
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M.V. Ramana & A.J. Goldman: Some geometric results in
semidefinite programming, J. Global Optim. 7 (1), 33-50 (1995).
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J. Renegar: Hyperbolic programs, and their derivative relaxations,
Found. Comput. Math. 6 (1), 59–79 (2006).