WHAT IS ... a spectrahedron?

Cynthia Vinzant

A spectrahedron is a convex set that appearsin a range of applications. Introduced in [3], thename “spectra” is used because its definition in-volves the eigenvalues of a matrix and “-hedron”because these sets generalize polyhedra.

First we need to recall some linear algebra. Allthe eigenvalues of a real symmetric matrix arereal and if these eigenvalues are all non-negativethen the matrix is positive semidefinite. The setof positive semidefinite matrices is a convex conein the vector space of real symmetric matrices.

A spectrahedron is the intersection of an affine-linear space with this convex cone of matrices. Ann-dimensional affine-linear space of real symmet-ric matrices can be parametrized by

A(x) = A0 + x1A1 + . . .+ xnAn

as x = (x1, . . . , xn) ranges over Rn, whereA0, . . . , An are real symmetric matrices. Thiswrites our spectrahedron as the set of x in Rn forwhich the matrix A(x) is positive semidefinite.This condition, denoted A(x) � 0, is commonlyknown as a linear matrix inequality.

For example, we can write the cylinder,

{(x, y, z) ∈ R3 : x2 + y2 ≤ 1, −1 ≤ z ≤ 1},as a spectrahedron. To do this, parametrize a3-dimensional affine space of 4× 4 matrices by

1 + x y 0 0y 1− x 0 00 0 1 + z 00 0 0 1− z

.This matrix is clearly positive definite at thepoint (x, y, z) = (0, 0, 0). In fact, it is positivesemidefinite exactly for points in the cylinder.

This matrix has rank four at points in the in-terior of the cylinder, rank three at most pointson the boundary, and rank two for points on thetwo circles on the top and bottom. Here we startto see the connection between the geometry ofspectrahedra and rank. The boundary is “morepointy” at matrices of lower rank.

Another example is a polyhedron, which isthe intersection of the non-negative orthant withan affine-linear space. This is a spectrahedronparametrized by diagonal matrices, since a diag-onal matrix is positive semidefinite exactly whenthe diagonal entries are non-negative.

Like polyhedra, spectrahedra have faces cutout by tangent hyperplanes, but they may haveinfinitely many. For example, one can imaginerolling a cylinder on the floor along the one-dimensional family of its edges.

This brings us to one of the main motivationsfor studying spectrahedra: optimization. Thewell-studied problem of maximizing a linear func-tion over a polyhedron is known as a linear pro-gram. Generalizing polyhedra to spectrahedraleads to semidefinite programming, the problemof maximizing a linear function over a spectra-hedron. Semidefinite programs can be solved inpolynomial time using interior-point methods andform a broad and powerful tool in optimization.

Angles, statistics, and graphs.Semidefinite programs have been used to relax

many “hard” problems in optimization, meaningthat they give a bound on the true solution. Thishas been most successful when the geometry ofthe underlying spectrahedron reveals that thesebounds are close to the true answer.

For a flavor of these applications, consider thespectrahedron of 3×3 matrices with 1’s along thediagonal (shown in yellow below):(x, y, z) ∈ R3 :

1 x yx 1 zy z 1

is positivesemidefinite

.This spectrahedron consists of points (x, y, z) =(cos(α), cos(β), cos(γ)) where α, β, γ are the pair-wise angles between three length-one vectors inR3. To see this, note that we can factor any pos-itive semidefinite matrix A as a real matrix timesits transpose, A = V V T . The entries of A arethen the inner products of the row vectors of V .

1

2

The four rank-one matrices on this spectrahe-dron occur exactly when these row vectors lie ona common line. They correspond to the four waysof partitioning the three vectors into two sets.

This elliptope appears in statistics as a setof correlation matrices and in the remarkableGoemans-Williamson semidefinite relaxation forfinding the maximal cut of a graph (see [2]).

This spectrahedron sticks out at its rank-onematrices, meaning that a random linear functionoften (but not always) achieves its maximum atone of these points. This is good news for themany applications that favor low-rank matrices.

Sums of squares and moments.Another important application of semidefinite

programming is to polynomial optimization [1,Chapter 3]. For example, given a multivariatepolynomial p(x), one can bound (from below)its global minimum by the maximum value ofλ in R so that the polynomial p(x) − λ can bewritten as a sum of squares of real polynomials.(Sums of squares are guaranteed to be globallynon-negative!) Finding this λ is a semidefiniteprogram and the expressions of a polynomial asa sum of squares form a spectrahedron.

For example, take the univariate polynomialp(t) = t4 + t2 + 1. For any choice of the parame-ter a in R we can write our polynomial as

p(t) =(1 t t2

)1 0 a0 1− 2a 0a 0 1

1tt2

.When this 3×3 matrix is positive semidefinite, itgives a representation of p(t) as a sums of squares.Indeed, if it has rank r we can write it as a sum ofr rank-one matrices

∑ri=1 viv

Ti . Multiplying both

sides by the vector of monomials (1, t, t2) writesp(t) as the sum of squares

∑ri=1((1, t, t

2) · vi)2.Here the spectrahedron is a line segment

parametrized by a ∈ [−1, 1/2]. Its two rank-twoend points correspond to the two representationsof p(t) as a sum of two squares:

(t2 − 1)2 + (√

3t)2 and (t2 + 1/2)2 + (√

3/2)2.

This idea extends to relaxations for optimiza-tion of a multivariate polynomial over any set de-fined by polynomial equalities and inequalities.

Dual to this theory is the study of moments,which come with their own spectrahedra. Theconvex hull of the curve {(t, t2, t3) : t ∈ [−1, 1]}(a spectrahedron) is an example shown above.

A non-example.To finish, let us return to the question of

what a spectrahedron is with a non-example.Projecting our original cylinderonto the plane x+ 2z = 0 resultsin the convex hull of two ellipses.This convex set is not a spectra-hedron! Any spectrahedron is cutout by finitely many polynomialinequalities, namely all of the di-agonal minors of the matrix beingnon-negative. However the pro-

jection cannot be written this way. This showsthat, unlike polyhedra, the class of spectrahedrais not closed under taking projections.

Spectrahedral conclusions.The study of spectrahedra brings together

optimization, convexity, real algebraic geome-try, statistics, and combinatorics, among others.There are effective computer programs like cvxand YALMIP (both for MATLAB) that work withspectrahedra and solve semidefinite programs.

Spectrahedra are beautiful convex bodies andfundamental objects in optimization and matrixtheory. By understanding the geometry of spec-trahedra (and their projections) we can fully ex-plore the potential of semidefinite programmingand its many applications.

References

[1] Semidefinite Optimization and Convex Algebraic Ge-ometry, Editors: Grigoriy Blekherman, Pablo A. Par-

rilo and Rekha R. Thomas, MOS-SIAM Series on Op-timization 13 , 2012.

[2] Monique Laurent. Cuts, matrix completions and graph

rigidity. Math. Programming, 79, 255–283, 1997.[3] Motakuri Ramana and A. J. Goldman. Some geomet-

ric results in semidefinite programming. J. Global Op-

tim. 7(1), 33–50, 1995.