?W H A T I S . . .

a Spectrahedron?Cynthia Vinzant

A spectrahedron is a convex set that appears in arange of applications. Introduced in [3], the namejoins “spectra”, evoking the eigenvalues of a ma-trix, with “hedron”, suggesting that spectrahedrageneralize convex polyhedra.

First we need to recall some linear algebra. Allthe eigenvalues of a real symmetric matrix arereal, and if these eigenvalues are all nonnegativethen the matrix is positive semidefinite. The set ofpositive semidefinite matrices is a convex cone inthe vector space of real symmetric matrices.

A spectrahedron is the intersection of an affinelinear space with this convex cone of matrices. Ann-dimensional affine linear space of real symmetricmatrices can be parameterized by

A(x) = A0 + x1A1 + · · · + xnAn

as x = (x1, . . . , xn) ranges overRn, whereA0, . . . , Anare real symmetric matrices. This identifies ourspectrahedron with the set of x in Rn for which thematrixA(x) is positive semidefinite. This condition,denoted A(x) � 0, is commonly known as a linearmatrix inequality.

For example, we can write the cylinder

{(x, y, z) ∈ R3 : x2 + y2 ≤ 1, −1 ≤ z ≤ 1}

as a spectrahedron. To do this, parameterize 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

.

Cynthia Vinzant is a Hildebrandt assistant professor andNSF postdoc in the Department of Mathematics at the Uni-versity of Michigan. Her email address is vinzant@umich.edu.

DOI: http://dx.doi.org/10.1090/noti1116

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 theinterior of the cylinder, rank three at most pointson the boundary, and rank two at 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 is theintersection of the nonnegative orthant with anaffine linear space. Any polyhedron is a spectra-hedron parameterized by diagonal matrices sincea diagonal matrix is positive semidefinite exactlywhen the diagonal entries are nonnegative.

Like polyhedra, spectrahedra have faces cut outby tangent hyperplanes, but they may have infinitelymany. For example, one can imagine rolling acylinder on the floor along the 1-dimensionalfamily of its edges.

492 Notices of the AMS Volume 61, Number 5

This brings us to an important motivation forstudying spectrahedra: optimization. The problemof maximizing a linear function over a polyhedronis a linear program. Generalizing polyhedra tospectrahedra leads to semidefinite programming,the problem of maximizing a linear functionover a spectrahedron. Semidefinite programmingproblems can be solved numerically in polynomialtime using interior-point methods and form a broadand powerful tool in optimization.

Angles, Statistics, and Graphs

Semidefinite programming has been used to relaxmany “hard” optimization problems, allowing oneto find a bound on the true solution. This approachhas been most successful in cases where thegeometry of the underlying spectrahedron revealsthe bounds to be close to the true answer.

For a flavor of these applications, consider thespectrahedron (displayed below) of 3× 3 matriceswith 1’s along the diagonal:(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 in R3.To see this, note that we can factor any positivesemidefinite matrix A as a real matrix times itstranspose, A = VVT . The entries of A are then theinner products of the row vectors of V .

The four rank-one matrices on this spectrahe-dron occur exactly when these row vectors lie on acommon line. They correspond to the four ways ofpartitioning the three vectors into two sets.

This elliptope appears in statistics as the 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 function

often (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 MomentsAnother important application of semidefiniteprogramming is to polynomial optimization [1,Chapter 3]. For example, one can bound (frombelow) the global minimum of a multivariatepolynomial p(x) by the maximum value of λ in Rsuch that the polynomial p(x)− λ can be writtenas a sum of squares of real polynomials. (Sums ofsquares are guaranteed to be globally nonnegative!)The expressions of a polynomial as a sum ofsquares form a spectrahedron, and finding this λis a semidefinite programming problem.

For example, take the univariate polynomialp(t) = t4 + t2 + 1. For any choice of the parametera 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 sum of squares.Indeed, if it has rank r , we can write it as a sumof r rank-one matrices

∑ri=1 viv

Ti . Multiplying both

sides by (1, t, t2), we then write p(t) as the sum ofsquares

∑ri=1((1, t, t2) · vi)2.

Here the spectrahedron is a line segment param-eterized by a ∈ [−1,1/2]. Its two rank-two endpoints correspond to the two representations ofp(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 optimizationof a multivariate polynomial over any set definedby 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.

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A NonexampleTo finish, let us return to the question ofwhat a spectrahedron is, giving a nonexam-ple. Projecting our original cylinder ontothe plane x+ 2z = 0 results in the convexhull of two ellipses. This convex set is nota spectrahedron! A matrix is positive semi-definite exactly when its diagonal minorsare nonnegative. Hence any spectrahedronis cut out by finitely many polynomial in-equalities. However, the projection cannotbe written this way. This shows that, unlike

polyhedra, the class of spectrahedra is not closedunder taking projections.

Spectrahedral ConclusionsThe study of spectrahedra brings together op-timization, convexity, real algebraic geometry,statistics, and combinatorics, among other areas.There are effective computer programs like cvxand YALMIP (both for MATLAB) that work withspectrahedra and solve semidefinite programmingproblems.

Spectrahedra are beautiful convex bodies andfundamental objects in optimization and matrixtheory. By understanding the geometry of spec-trahedra, we can fully explore the potential ofsemidefinite programming and its applications.

References[1] Grigoriy Blekherman, Pablo A. Parrilo, and Rekha R.

Thomas, eds., Semidefinite Optimization and ConvexAlgebraic Geometry , MOS-SIAM Series on Optimization13, 2012.

[2] Monique Laurent, Cuts, matrix completions andgraph rigidity, Math. Programming 79 (1997), 255–283.

[3] Motakuri Ramana and A. J. Goldman, Some geomet-ric results in semidefinite programming, J. Global Optim.7 (1995), no. 1.

494 Notices of the AMS Volume 61, Number 5