1 Project description

The notion of convexity underlies important results in many parts of mathematics such as opti-mization, analysis, combinatorics, probability and number theory. The geometric foundations ofthe theory of convex sets date back to work of Minkowski, Caratheodory, and Fenchel around 1900.Since then, this area has expanded into a large number of directions and now includes topics such ashigh-dimensional spaces, convex analysis, polyhedral geometry, computational convexity, approx-imation methods and others. In the context of optimization, both theory and empirical evidenceshow that problems with convex constraints allow efficient algorithms. Many applications in thesciences and engineering involve optimization, and it is always extremely advantageous when theunderlying feasible regions are convex and have practically useful representations as convex sets.

A situation in which convexity has been well-understood is the study of convex polyhedra, whichare the solution sets of finitely many linear inequalities [27, 86]. A context in algebraic geometryin which convexity arises is the theory of toric varieties. These are algebraic varieties derived frompolyhedra [49, 73]. Both convex polyhedra and toric varieties have satisfactory computationaltechniques associated to them. Linear optimization over polyhedra is linear programming whichadmits interior-point algorithms that run in polynomial time. More generally, polyhedra can bemanipulated using well-known methods such as Fourier-Motzkin elimination. In the case of toricgeometry, a similar role is played by the theory of Grobner bases and its computational offshoots.

While many scenarios can be modeled as optimization problems over polyhedra, there is aplethora of applications where it is more natural to optimize over algebraic and semialgebraicsets (sets described by polynomial equations and inequalities [3]). In the semialgebraic category,the subtle interactions among the geometric, algebraic, and computational aspects of optimizationproblems are not well-understood. The central goals of this collaborative research proposal are:

1. to lay the mathematical foundations for convexity in real algebraic geometry,

2. to develop new algorithms for manipulating convex sets described by polynomials,

3. to take the real-world applicability of algebraic geometry methods to a new level.

Numerous preexisting approaches to our main goals are outlined in the proposal. A coarseclassification is in terms of symbolic versus numerical techniques. The former consists of algorithmsin real algebraic geometry such as cylindrical algebraic decomposition [11] or critical point methods[3]. The latter involves methods of convex optimization such as interior-point techniques [44] as wellas the work of Sommese, Wampler, Verschelde and collaborators on numerical algebraic geometry[68, 69]. Past and current research of the PIs and others suggest promising alternative approaches.A central enabling tool is semidefinite programming (SDP) which is an efficient convex optimizationtechnique that generalizes linear programming to symmetric matrices [77, 79]. This proposal isrooted in the general belief that whenever semialgebraic sets enjoy additional properties such asconvexity, much more efficient techniques than those mentioned above should be possible. Thisdistinction is well-known in optimization but a full understanding of the implications of convexity forcomputations in algebraic geometry is sorely needed. Furthermore, these insights can be partiallyextended, through appropriate relaxation techniques, to the case of nonconvex semialgebraic sets.

The investigators have multiple existing research connections and prior results that are veryrelevant to the aims of this project. This collaboration assembles a team that can efficiently utilizedifferent skills and backgrounds to attack goals as broad and ambitious as the ones outlined above.Graduate students and postdocs will be trained in this research area and directly involved in theproject. The PIs anticipate the results to impact theory, algorithms, software and applications.

1

The proposed research at the interface of optimization and algebraic geometry has the attractiveand natural feature that it calls for tools from many branches of the mathematical sciences. It linksareas which are often thought of as separate, such as algebra, analysis and scientific computing. Asa consequence, this Focused Research Group will offer broad training opportunities for students andpostdocs in both classical and modern tools from mathematics, computer science and engineering.

Scientific context and timeliness: Although a number of the questions in this proposal are de-scribed in relatively elementary language, we have found that they cannot be satisfactorily addressedusing the current knowledge in convex, polyhedral and algebraic geometry. New mathematics, andthe associated computational methods, need to be developed. In the next pages, we describe priorresults and our plans for future work towards the successful completion of the proposed agenda.

There are good reasons to believe that our approach will be successful, and that this is theright time to attack these questions. Firstly, in recent years there have been substantial advancesin the core theory of semidefinite programming, sum of squares, and their associated relaxations.These methods, described in detail in Section 2, have enabled the application of convexity-basedtechniques to large-scale, nonconvex optimization problems, once thought to be completely out ofreach. Secondly, there have been significant connections with other areas of mathematics, that haveilluminated and deepened the interaction between real algebraic questions and convexity. Amongthem, we mention operator theory, model theory, tropical geometry, probability theory, etc. Some ofthese connections, and their classical roots, are described in the recent survey by Helton and Putinar[32]. Finally, on the computational side, there are now much improved hardware capabilities, wideavailability of computational resources, and very efficient numerical linear algebra packages (e.g.,BLAS, LAPACK, and ATLAS). The possibility of performing non-linear algebraic computations,but without having to pay the big fixed costs associated with symbolic manipulation, is one of thebest arguments why the mathematical techniques we propose need to be much better understood.

For all these reasons, the time is right for this project. Our approach will leverage on existingadvances in theory and application, take advantage of the synergy between new ideas, connect withother areas of mathematics, and produce new software. Our team has the expertise in all theseareas, and can develop the necessary theory leading to substantial advances in the state of the art.

2 Semidefinite Programming and Sums of Squares

A key connection between real algebraic geometry and convex optimization is given by sums ofsquares (SOS) decompositions of multivariate polynomials. These are essential objects in thecharacterization of the emptiness of basic semialgebraic sets. In this section, we explain these links,as well as a quick introduction to the essential mathematical ideas behind semidefinite optimization.

Semidefinite programming: One of biggest advances in the theory and practice of convexoptimization in recent years is semidefinite programming (SDP). Semidefinite programming is a far-reaching generalization of linear programming (LP), where the nonnegative orthant is replaced bythe cone of positive semidefinite matrices. SDP is today one of the most exciting and active researchareas in optimization. It has attracted researchers with quite diverse backgrounds, including expertsin convex programming, numerical optimization, combinatorial optimization, control theory, andstatistics. This tremendous research activity was spurred by the development of efficient algorithms,the depth and elegance of the underlying optimization theory, and a great number of applications incombinatorial optimization, systems theory, and nonconvex multivariate polynomial optimization.

2

Figure 1: Spectrahedron defined by a three-dimensional family of symmetric 3 × 3-matrices

Geometrically, linear programming amounts to optimizing a linear function over the intersectionof an affine subspace and the nonnegative orthant. The standard form of an LP problem is

minimize cT x

subject to Ax = b, (1)

x ≥ 0.

Similarly, SDP amounts to optimizing a linear function over the intersection of an affine space andthe convex cone of positive semidefinite matrices. The SDP problem in standard primal form is

minimize C • X

subject to Ai • X = bi, i = 1, . . . , m (2)

X # 0,

where C and Ai are symmetric matrices and X •Y := Trace(XY ). The symmetric matrix X is thevariable over which the minimization is performed. The inequality in the second line means thatthe matrix X must be positive semidefinite (all eigenvalues are nonnegative). The set of feasiblesolutions, i.e., the set of matrices X that satisfy the constraints, is always a closed convex set.

Besides the formal parallels between (1) and (2), semidefinite programming is a strict general-ization of linear programming. To see this, suppose that all Ai are diagonal matrices. Then theconstraints in (2) define a polyhedron in the space of symmetric matrices. However, most SDPproblems cannot be reduced to LP. The feasible set in (2) is known as a spectrahedron. It is con-strained by polynomial inequalities (non-negativity of the principal minors of X) and it is generallynot a polyhedron. Figure 1 depicts an example of a three-dimensional spectrahedron.

The salient feature of semidefinite programming is that it is a convex optimization problem, andthat this problem can be solved very efficiently by numerical methods like interior-point algorithms.In the last decade, there has been a significant amount of work on how to solve SDP problemsefficiently, as well as many applications in science and engineering. Among others, we refer to thereferences [77, 79, 84] for theory, algorithms and applications of semidefinite programming.

Why is SDP so useful? Semidefinite representations provide a convenient computational de-scription of a set, that is guaranteed to be convex by construction. In a sense, an SDP representation

3

of a convex set is endowed with a “built-in” certificate of its convexity (through the hyperbolicityof the determinant of the corresponding matrix pencil, see Section 3). Furthermore, there is a verywell-developed calculus for SDP-representable sets [44], that allows to mix and match differentoperations (such as projections, Minkowski sums, etc) while keeping SDP-representability. If a setcan be represented in terms of SDP conditions, we can understand its geometry significantly better,and we gain access to efficient numerical methods for solving any associated optimization problem.

Sum of squares: We now consider multivariate polynomials with real coefficients. Such a poly-nomial is a sum of squares (SOS) if it can be written as a sum of squares of other polynomials, i.e.,

p(x) =∑

i

(qi(x))2, where qi(x) ∈ R[x] = R[x1, . . . , xn].

If p(x) is SOS then clearly p(x) ≥ 0 for all x ∈ Rn. In general, SOS decompositions are not unique.The sum of squares condition is a natural sufficient test for polynomial nonnegativity. Its rich

mathematical structure has been analyzed in detail in the past, notably by Bruce Reznick andcoauthors [12, 65], but until recently the computational implications had not been explored. In thelast few years there have been some very interesting new developments surrounding sums of squares,where several independent approaches have produced a wide array of results linking foundationalquestions in algebra with computational possibilities arising from convex optimization. Most ofthem employ semidefinite programming (SDP) as the essential computational tool.

Deciding whether a multivariate polynomial can be written as a sum of squares is equivalent toa semidefinite programming feasibility problem. The reason is the following well-known result:

Theorem 2.1 A polynomial p(x) is SOS if and only if p(x) = zT Qz, where z is a vector ofmonomials in the xi variables and Q is a positive semidefinite (PSD) matrix of suitable format.

In other words, every SOS polynomial can be written as a quadratic form in a set of monomials,with the associated matrix being positive semidefinite. The vector of monomials z depends ingeneral on the degree and sparsity pattern of p(x). If p(x) has n variables and total degree 2d, thenz can always be chosen as a vector whose entries are monomials of degree less than or equal to d.

Example 2.2 Consider the polynomial p(x1, x2) = x21−x1x2

2+x42+1. It has the SOS representation

p(x1, x2) =1

6

1x2

x22

x1

T

6 0 −2 00 4 0 0

−2 0 6 −30 0 −3 6

1x2

x22

x1

.

The 4 × 4-matrix in the expression is positive semidefinite. Its Cholesky factorization gives rise toan expression of p(x) as a sum of four squares. In particular, we see that p(x) ≥ 0 for all x ∈ R2.

In the representation p(x) = zT Qz of Theorem 2.1, for the right- and left-hand sides to be equal,all coefficients of the two polynomials must be identical. The unknown symmetric matrix Q isthus simultaneously constrained by linear equations and a PSD condition. Hence the problem offinding Q given p(x) is equivalent to an SDP feasibility problem in the standard primal form (2).The conversion step from p(x) to Q is fully algorithmic and has been implemented in the softwarepackage SOSTOOLS [61], developed by Parrilo and collaborators. With this tool we can in principleapply all the available numerical methods for SDP to solve optimization problems involving SOSpolynomials. The PIs have considerable experience in applying this methodology to a wide rangeof problems; see e.g. [45, 46, 47, 51, 53, 55].

4

3 Convex Semialgebraic Sets

A long-term goal of the proposed project is to achieve a full understanding of the interrelationsbetween the algebraic, geometric and computational descriptions of convex sets. These relationshipsare well understood in the case of polyhedra, but the problem is wide open for general semialgebraicsets. Throughout, we motivate these issues with some very concrete questions. As we explain laterin more detail, several of these questions arise from specific applications in engineering problems.

The convex hull of a real algebraic variety: One important topic is the study of convexsets that are defined as the convex hull of (subsets of) a real algebraic variety. While these setshave a simple definition, obtaining “nice” descriptions is a challenging problem, since in general themembership problem for this convex hull is NP-hard. We are interested in obtaining characteriza-tions that are computationally convenient, in the sense that we can combine these descriptions, orperhaps efficiently optimize linear functions over these sets. In certain cases, as we illustrate belowthrough a simple example, suitable descriptions can be obtained using semidefinite programming.

Example 3.1 The convex hull of the monomial (toric) curve{

(t, t2, . . . , t2d+1) : −1 ≤ t ≤ 1}

inR2d+1 is the set of all points (y1, y2, . . . , y2d+1) that satisfy the two SDP feasibility constraints

1 y1 y2 · · · yd

y1 y2 y3 · · · yd+1

y2 y3 y4 · · · yd+2

......

.... . .

...yd yd+1 yd+2 · · · y2d

±

y1 y2 y3 · · · yd+1

y2 y3 y4 · · · yd+2

y3 y4 y5 · · · yd+3

......

.... . .

...yd+1 yd+2 yd+3 · · · y2d+1

# 0. (3)

In the case d = 1, this set is the intersection of two circular cones, and is illustrated in Figure 2.

The correctness of the SDP representation for this example follows from the classical theory ofmoment spaces [36]. It is related to the study of cyclic polytopes [27, 86]. No such results are knownfor more general algebraic varieties, not even toric varieties. This motivates our first question:

Problem 1 Given a real algebraic variety, produce a semidefinite representation of its convex hullif one exists.

A partial, approximate solution can be obtained through the use of sum of squares (SOS)techniques, as can be derived from work of Lasserre [38] and Parrilo [50, 52]. Unfortunately, simplecounterexamples show that this technique can fail in producing an exact semidefinite representation,even if one exists [53]. Recent progress by Helton and Nie [31] shows that under certain positivityconditions on the Hessian of the equations defining the variety, such a representation is possible.

Problem 1 also has an appealing probabilistic interpretation. If the base algebraic varietycorresponds to the probability distributions of a statistical model, the convex hull corresponds tomixture models. There are some beautiful connections between the semidefinite construction andthe classical theorem of De Finetti on exchangeable random variables. In that setting, Example 3.1corresponds to the moment space of a random variable with support on the interval [−1, 1].

Towards a characterization of spectrahedra: Consider a given convex set S in Rn. When isS the feasible set of a linear programming problem? By the Minkowski-Weyl theorem, this is thecase if and only if S is polyhedral, i.e., it has a finite number of extreme points and extreme rays.

5

!1

!0.5

0

0.5

1

0

0.5

1

!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

y1y

2

y3

Figure 2: Convex hull of the parametric curve t '→ (t, t2, t3), for t ∈ [−1, 1]

We can ask the analogous question for semidefinite programming. What structural propertiesshould a closed subset S of Rn have in order to be a spectrahedron, i.e., representable as

{

x ∈ Rn : A1x1 + · · · + Anxn ) C

}

(4)

for some symmetric matrices C, A1, . . . , An? It is clearly necessary that S be convex and semi-algebraic, but this is far from sufficient. Indeed, there are some subtle geometric and algebraicrestrictions. One necessary geometric restriction is that all faces of the convex set S must beexposed, i.e., they should be describable as the intersection of S and a suitable affine subspace [62].

There are further necessary algebraic conditions for S to be a spectrahedron. First, S must bethe closure of a connected component of an open set {x ∈ Rn : p(x) > 0}, where p ∈ R[x1, . . . , xn].The existence of such polynomial p(x) follows by considering the determinant of the matrix pencilC−

∑ni=1 xiAi. Among all the possible choices of such p(x), the one with lowest degree is called the

minimal defining polynomial. It is unique up to a multiplication by a positive constant [33]. For Sto be a spectrahedron, Helton and Vinnikov [33] furthermore showed that S must be rigidly convex,which means that any generic line through the interior of S must intersect the real hypersurface ofp(x) in exactly d points, where p(x) is the minimal defining polynomial of S and d is its degree.

Example 3.2 The closed convex set S = {x ∈ R2 : x41 + x4

2 ≤ 1} is not a spectrahedron becauseany line through the origin intersects S in only two points, and its minimal polynomial has degree 4.

Rigid convexity is a necessary condition for S to be a spectrahedron. In the case of planarconvex sets, the converse was proved by Helton and Vinnikov. This settles the problem for n = 2.The same characterization is false in higher dimensions, and the problem remains open for n ≥ 3.

Theorem 3.3 (Helton-Vinnikov [33]) A compact basic closed semialgebraic set S in the planeR2 is a spectrahedron if and only if S is rigidly convex.

One worthy pursuit is to find appealing and useful classes of SDP representable sets. An exampleof a recent success lies in k-ellipses, a class of planar curves that generalizes circles and ellipses

6

Figure 3: The 3-ellipse is the rigidly convex curve of degree eight which encircles the three points.

studied by Nie, Parrilo and Sturmfels [47]. These are defined as the boundary of a convex set Ek

in the plane consisting of points whose sum of distances to the k given points is at most d. Theseconvex sets are of interest in computational geometry and in optimization, e.g. for the Fermat-Weber facility location problem. The following theorem shows that k-ellipses are spectrahedra. Forexample, the 3-ellipse is illustrated in Figure 3. Its boundary is a rigidly convex curve of degree 8.

Theorem 3.4 (Nie-Parrilo-Sturmfels [47]) The k-ellipse has degree 2k if k is odd and degree2k−

( kk/2

)

if k is even. Its defining polynomial has a determinantal representation

pk(x, y) = det(

x · Ak + y · Bk + Ck

)

, (5)

where Ak, Bk, Ck are symmetric 2k × 2k-matrices whose entries are read off from the k points.

A related but more general question is whether a given convex set S in Rn is an affine projectionof a higher-dimensional spectrahedron, i.e., whether it has a lifted SDP representation:

S ={

x ∈ Rn : ∃y ∈ R

m A1x1 + · · · + Anxn + An+1y1 + · · · + An+mym ) C}

, (6)

for some symmetric matrices C, A1, . . . , An+m. At present, no non-trivial obstructions are known forthe representation (6). At first sight the huge difference in expressive power between the direct (4)versus lifted (6) representations may be surprising, given that such a phenomenon happens neitherin the polyhedral category (by Fourier-Motzkin elimination) nor in the semi-algebraic category (byTarski-Seidenberg). This surprising difference prompts our next question:

Problem 2 Does every compact convex basic semialgebraic set have a lifted SDP representation?

A partial solution of this question was recently found by Helton and Nie [31]. Their affirmativetheorem states that every basic closed semialgebraic set whose boundary satisfies certain naturalconcavity and positive curvature hypothesis does indeed admit a lifted semidefinite representation.These hypotheses are automatically satisfied for a smooth strictly convex set as in Example 3.2.

7

Hyperbolic polynomials and the Lax conjecture: As a direct consequence of Theorem 3.3,a conjecture of Lax from the 1950’s about determinantal representations of hyperbolic polynomialswas settled in [41]. This conjecture (now a theorem) claims that every homogeneous polynomialp(x, y, z) that is hyperbolic with respect to the direction (0, 0, 1) is representable as the determinantof a monic matrix pencil, i.e., p(x, y, z) = det(Ax + By + Iz), where A, B are symmetric d× d ma-trices, and d is the degree of p. Hyperbolic polynomials are fundamental objects in modern convexoptimization, with strong connections to interior point methods [28, 63]. There are also interestinglinks between polynomial hyperbolicity and sum of squares conditions. Recent results by Parriloshow that hyperbolicity of a trivariate polynomial is equivalent to a certain multihomogeneouspolynomial being a sum of squares [54].

Reduced SDP representations: A natural question motivated by our earlier results in [33, 41],is how to systematically produce minimal determinantal representations for a rigidly convex curve,when a non-minimal one is available. This is likely an easier task than finding a representationdirectly from the defining polynomial, since in this case we have a certificate of its rigid convexity.

Concretely, given two real symmetric n × n matrices A, B such that the polynomial

p(x, y) = det(A · x + B · y + In)

has degree r < n, we want to explicitly compute r × r matrices A, B such that

p(x, y) = det(A · x + B · y + Ir).

The existence of such matrices is guaranteed by the results in [33, 41]. In fact, explicit formulasare presented in [33] in terms of transcendental objects (theta functions of a Jacobian variety), butit would be interesting to know if simpler and more explicit constructions exist in this special case.

Problem 3 Given a symmetric matrix representation of a hyperbolic polynomial whose total degreeis less than the size of the matrices, is it always possible to reduce the matrix representation to havesmaller size? If yes, how to design a practical algorithm for finding this smaller representation?

Matrix variables: Many problems in linear control and systems problems lead directly to matrixinequalities (MIs). There are two fundamentally different classes of linear systems problems. First,there are MIs where the unknowns are g-tuples of symmetric matrices and appear in the formulas ina manner that respects matrix multiplication. Typically one has polynomials and rational functionsof matrix variables in these situations. Most of the classic MIs of control theory are like this. OtherMIs are like the ones discussed so far, and have unknowns which are tuples of numbers. These twoclasses behave very differently, as will soon emerge in this discussion.

A toy model for such and for more complicated situations are matrix or noncommutative spec-trahedra, or alternatively sets that are NCLMI representable. These sets can be written as

S =⋃

N

{

X ∈ (SRN×N )g : A1 ⊗ X1 + · · · + Ag ⊗ Xg ) C ⊗ IN×N

}

(7)

for some symmetric matrices C, A1, · · · , An. For such a representation to hold, it is clearly necessarythat S be convex and semialgebraic in a noncommutative sense. Helton and McCullough conjecturethat this is also sufficient. This lies in dramatic contrast to the case of “commutative” spectrahedra.A substantial body of theorems supporting this contention have been found in the last 5 years byHelton and collaborators; these are summarized in [30]. This conjecture and determining whichsets transform to a noncommutative convex and semialgebraic set are open questions.

8

4 Sparsity and graphical structure

Polynomial equations and inequalities that appear in scientific and engineering applications areusually far from being generic. Rather, it is often the case that there is a significant amount ofadditional mathematical structure. This can take many different forms. The most important onesinclude symmetry, sparsity, and graphical properties. We discuss some of these possibilities below.

Symmetries: An important feature of problems arising in both physical and engineering appli-cations is the presence of symmetries, i.e., invariance under a group of transformations. Thesesymmetries can be often exploited for efficient computation. See [80, 85] for the case of solutionsof polynomial systems, and work of Gatermann and Parrilo [22] for semidefinite programming andsums of squares. There are many examples where a careful exploitation of symmetries leads toa surprising reduction in the computational requirements, enabling a wide variety of applications.Among many others, two recent examples are the analysis of the mixing rate of reversible symmetricMarkov chains [8] and the SDP-based bounds for kissing numbers and spherical codes [2].

Graphical structure: A different kind of structure, often decisive in the context of large-scalemodels, concerns the coupling between variables in the underlying system of equations and inequal-ities. For many problems, the polynomials appearing in the objective or constraints often consist ofa sum of terms, each involving a relatively small number of variables. This can happen for physicalreasons, such as locality of reference in a physical system, limited connectivity restrictions, or asa byproduct of a discretization process for an underlying continuous model. These properties canbe formalized in terms of the graphs (or hypergraphs) associated with the constraints. There area number of existing approaches to the exploitation of this graphical structure in different kindsof models. In statistics and learning theory, the theory of graphical models [40, 58] is a well-developed approach to model probabilistic inference in terms of graphs. Belief propagation andother message-passing algorithms for graphical models have important connections to maximumlikelihood decoding and statistical physics. A key unifying concept in all these approaches is thepossibility of factorizing a global probability distribution in a large number of variables in terms oflocal potential functions, that depend only on the maximal cliques of the underlying graph. Severalof these notions can be naturally adapted to handle systems of polynomial equations and inequal-ities, by taking into account the structure of the constraints defining semialgebraic sets. While afully satisfactory general theory is not yet available, many interesting questions can be asked:

Problem 4 How does the (hyper)graphical structure of a semialgebraic set affect its solvability?Are there nice structures (e.g., trees, chordal, etc) for which the problem becomes provably easier?

For polynomial optimization and semidefinite relaxations, some recent results in this directionappear in [39, 46, 82, 83]. In Section 5, we shall discuss the recent work of Nie and Demmel [46]on minimization of polynomials that are sums of terms each involving a small number of variables.

PSD matrix completion and projections of spectrahedra: A well-known question in matrixanalysis is the positive semidefinite completion problem where a partially specified symmetric matrixis to be completed in such a way that the full matrix is PSD. An appealing solution to this problemis given by the following theorem which completely characterizes the existence of a completion,provided the graph specified by the sparsity pattern of the specified entries has a certain property:

9

Theorem 4.1 (Grone-Johnson-Sa-Wolkowicz [26]) All partially specified positive definite ma-trices with sparsity graph G have a positive definite completion if and only if G is chordal.

A natural geometric interpretation of Theorem 4.1 is the following: under the chordal assumption, itis possible to “eliminate” a set of variables, and provide a semidefinite description of the projectionof the feasible set on the remaining variables. This is a quite remarkable result, since in general,as we have seen, the projection of a spectrahedron is not spectrahedral. From this viewpoint, itis desirable to have a purely geometric characterization of when the projection of a spectrahedronremains describable by semidefinite inequalities, thus suggesting the following natural question:

Problem 5 When is a projection of a spectrahedron again a spectrahedron? Can the chordalcompletion theorem be generalized to nonnegative (or SOS) polynomials?

This circle of ideas has concrete computational implications. Theorem 4.1 has been profitably used,for instance, in the sparsity-exploiting techniques for solving SDPs developed by Fukuda et al. [20].

Sparsity and SOS The Newton polytope New(p) of a polynomial p ∈ R[x1, . . . , xn] is the convexhull of the set of exponent vectors that appear in p. This suggests a notion of sparsity that scaleswith the size of its Newton polytope. This polyhedral sparsity drives the complexity of manyalgebraic algorithms for polynomial systems [74, §3]. In the context of polynomial optimization,exploiting the Newton polytope allows a notable reduction in the computational cost of checkingSOS conditions of multivariate polynomials, as follows from the following result due to Reznick:

Proposition 4.2 ([64], Theorem 1) If p(x) =∑

i(qi(x))2, then New(qi) ⊆12New(p).

The theorem makes possible, without loss of generality, to restrict the set of monomials appearingin the representation p(x) = zT Qz of Theorem 2.1 to those with exponents in the Newton polytopeof p, scaled by a factor of 1

2 . This reduces the size of the desired matrix Q, thus simplifying theSDP problem.

Example 4.3 Consider the following polynomial of degree 2d = 16 in n = 4 variables:

p(x, y, z, w) = (w4 + 1)(x4 + 1)(y4 + 1)(z4 + 1) + 2w + 3x + 4y + 5z.

The naive approach would require a matrix Q of size(n+d

d

)

= 495. However, the Newton polytopeNew(p) is the 4-cube, and the polynomials qi in the SOS decomposition of p have at most 34 = 81distinct monomials, and hence the decomposition can be computed by solving a much smaller SDP.

There is an appealing interpretation of Theorem 4.2, in terms of McMullen’s polytope algebra [43].Here the sum of two polytopes is their convex hull, and their product is their Minkowski sum

P1 ⊕ P2 := conv(P1, P2) and P1 . P2 := P1 + P2.

With this notation, Proposition 4.2 implies that the Newton polytope of p is the sum of the squaresof the Newton polytopes of the qi. This motivates the following intriguing question:

Problem 6 When is a polytope a sum of squares in the polytope algebra? How to find the minimumnumber of squares over all SOS decompositions of a fixed element in the polytope algebra?

10

5 Numerical Polynomial Optimization and Applications

Optimization problems involving polynomials arise frequently in scientific and engineering appli-cations. Although there are a number of relatively “good” theoretical methods to solve theseproblems, often there are significant differences between the theoretical guarantees and practicalalgorithmic performance. Furthermore, in many practical applications these problems need to besolved within reasonable time, and in a consistent and reliable fashion. Therefore, it is crucial todevelop efficient numerical algorithms, where both performance and stability are taken into account.

Structured optimization: Various applications of polynomial optimization involve scenarioswhere the objective function and the constraints are sums of small polynomials. Here, by “small”it is meant that each polynomial involves only relatively few of the variables. For these specialproblems, Nie and Demmel [46] recently proposed a sparse SOS relaxation technique that is signif-icantly more efficient than previous methods. An important application is nonlinear least squaresproblems, and in particular, nonlinear differential equations, and sensor network localization.

In the sensor network localization problem, we seek to compute vectors x1, x2, · · · , xn ∈ Rd (thesensors) such that distances between xi’s and distances to some other known vectors a1, · · · , am

(called anchors) are equal to some measured values dij and eik. This problem is also known asgraph realization problem or distance geometry problem. It takes the form of a polynomial system,

‖xi − xj‖22 = d2

ij for all (i, j) ∈ A,

‖xi − ak‖22 = e2

ik for all (i, k) ∈ B,

where A and B are index sets that specify some (but not all) of the distances. The above equationsare difficult to solve directly. One approach is to rewrite them as a nonlinear least squares problem:

minx1,...,xn

∑

(i,j)∈A

(

‖xi − xj‖22 − d2

ij

)2+

∑

(i,k)∈B

(

‖xi − ak‖22 − e2

ik

)2.

This is a sparse polynomial optimization problem, and the sparse SOS method of Demmel and Niecan be applied. As shown in [45], this approach turned out to be surprisingly successful in practice.

Numerical issues: The interface between numerical and algebraic computation imposes seri-ous additional considerations. For instance, many of algebraic algorithms (e.g., Grobner bases,resultants, etc) rely on monomial bases and associated term orderings. Yet, it is well-known fromnumerical analysis and the theory of orthogonal polynomials that monomial bases can be extremelyill-conditioned. There are a number of possible remedies, and symbolic-numeric techniques havebeen investigated by a number of researchers (e.g., [18, 21, 69, 72]). In the SOS context, a promisingstarting point is the interpolation approach of Lofberg and Parrilo [42], where the SOS conditionsare expressed using Chebyshev polynomials and the constraints are written as interpolation restric-tions. Besides increased numerical stability, an appealing feature of this approach is the fact thatin this basis, all the matrices defining the corresponding semidefinite program have rank one, aproperty that interior-point solvers such as SDPT3 [78] can exploit for speed and accuracy.

Problem 7 Develop numerically stable methods for Sum of Squares (SOS) programming.

11

Matrix structure: For problems involving rational functions of matrix variables there are bigadvantages in exploiting the underlying matrix structure. Camino, Helton and Skelton [10] foundthat many optimization problems and algorithms lead to subproblems having the structure L(X) =C where the linear operator L has the Sylvester form

L(X) =∑

j

(AjXjBj + BTj XjA

Tj ). (8)

Here Aj , Bj , C are given square matrices and X is an unknown symmetric matrix. This motivatesthe need of an iterative linear solver for such SDP problems. Work in progress by de Oliveira andHelton [14] addresses this issue. For randomly generated problems, conjugate gradient techniqueswith preconditioner behaves impressively well for large problems (e.g., over 20,000 unknowns).These algorithms are hybrid methods, that use computer algebra to create the subproblem (8) priorto invoking a numerical solver. However, there are a significant number of benchmarks problemsin control that are ill-conditioned and cause such methods to fail. Reliable methods exploiting theSylvester structure would allow a big increase in the size of LMIs which could be solved.

There are many potential advantages in using noncommutative symbolic computation for ma-trix problems with many commuting variables. Invoking this algebraic structure yields a problemwith few aggregated variables, from which efficient formulas can be obtained symbolically. Heltonand de Oliveira are the main providers of general noncommutative algebra capability to Mathemat-ica, via their package NCAlgebra [29]. Matrix-based symbolic software for Control is also underdevelopment, e.g. [14]. Helton’s group is particularly suited to exploring these problems because ofNCAlgebra and their extensive experience with it. The expertise of the other members of the FRGwill be a great benefit to the noncommutative computer algebra effort in the NCAlgebra context.

Software development: A crucial factor in the successful dissemination of our research findingswill be the availability of high-quality software that implements the developed techniques. Allthe PIs in this project have very significant computational experience, and in particular, they havebeen responsible or closely involved in the development of mathematical software such as Gfan [35],NCAlgebra [29], or SOSTOOLS [61]. As part of this project we intend to further develop thesetools, and to the extent possible, provide a closer integration between them and other softwarefor algebraic-geometric computations such as Macaulay2 [24] or Singular [25]. In this direction,a “baby” version of SOSTOOLS for Macaulay2 called SOS.m2 has been developed by Peyrl andParrilo [60], that allows the numerical computation of rational SOS decompositions from withinthe symbolic M2 environment; see [59] for the theory behind the algorithm.

We have good theoretical reasons, as well as abundant preliminary evidence, to believe that thetechniques will prove to be useful in the practical solution of structured polynomial inequalities.More importantly, the availability of high-quality software to reliably provide solutions or infeasi-bility certificates for reasonably-sized problems, would have an immediate appeal and visibility.

Problem 8 Combine polynomial optimization with numerical algebraic geometry to solve a widerrange of non-linear systems arising in applications, including statistics and biology.

Benchmark libraries: An important component in the practical evaluation and comparisonof different approaches to our problems is the availability of well-documented, standard suitesof test problems. For instance, for the case of polynomial system solving, there is the PoSSocollection of test instances, and for SDP there are both the SDPLIB [6] and DIMACS Challengelibraries (dimacs.rutgers.edu/Challenges/Seventh/Instances/). Similarly to these, we shall

12

prepare a comprehensive, well-maintained, web-enabled database of semialgebraic problems andtheir solutions. This resource will be hosted at the website that will coordinate the activities of theFRG. It will then be possible to test different solvers, and to chart in an objective and accurateway the achieved progress in the development of numerical software for convex algebraic geometry.

6 Deformations and variation of parameters

Parameter spaces form a central role in algebraic geometry, and we expect that a similar focus onmoduli and deformation theory will be fruitful also in convex algebraic geometry. For instance, insemidefinite programming it is natural to vary the entries of the given symmetric matrices, and wecan ask how the optimal value depends on the given matrix entries. This function is a piecewisealgebraic function, and the degree of the irreducible pieces of this function is the algebraic degree ofsemidefinite programming. An exact formula for this algebraic degree of SDP was derived, by meansof intersection theory and algebraic combinatorics, in recent work of Bothmer, Nie, Ranestad andSturmfels [23, 48]. The geometric analysis underlying the combinatorial formulas given in thesepapers involves resolution of singularities of determinantal varieties. A first of example of therelevant singularities are the four corners of the convex Cayley cubic surface shown in Figure 1.

This work opens up the possibility of developing of a new class of algorithms for parametric SDP.For instance, one may apply numerical algebraic geometry to spectrahedra and design geometricallymeaningful homotopies for tracking the optimal solution of an SDP through parameter space.Sensitivity analysis for SDP requires the study of the normal cone to a spectrahedron at a boundarypoint. These normal cones are semi-algebraic sets, and a first step in their computation is theidentification of the active constraints at the optimal solution. We propose to address this question:

Problem 9 How to efficiently compute and best represent the active constraints and the tangentcone at the optimal solution of an SDP?

Our approach to this problem builds on known results in the optimization literature, specifically,on the work of Pataki [56, 57]. It may involve a further development of the simplex-like algorithmproposed in [56], and, ideally, it will lead to a rounding algorithm for SDP. The input to ourrounding algorithm would be the floating-point output of an interior-point computation, and itsoutput is an exact representation of algebraic numbers that form the coordinates of the optimalsolution. A related goal is to gain a combinatorial understanding of the face lattices of spectrahedra.

When solving optimization problems such as linear or semidefinite programming, the problemdata and the solution are usually rational or real numbers. In principle, however, it is possible toconsider convex optimization problems over other ordered fields, especially, real closed fields. Wepropose to study SDP over the field of real Puiseux series R{{ε}}. Linear programming problemsover this field can be solved using the simplex algorithm, but it is not clear how to generalizeinterior-point methods to this setting, and so the extension from LP to SDP remains a challenge.

Perturbation analysis using the field over R{{ε}} is fundamental for algorithms in real algebraicgeometry [3]. It also has a long tradition in spectral theory and applied linear algebra, starting withthe seminal work of Visik and Ljusternik [81] and culminating in recent results in max-plus linearalgebra by Gaubert and collaborators [1]. A fundamental result in this theory characterizes theorders of the eigenvalues of a square matrix A with entries in R{{ε}}, and in [1] this is extended toarbitrary one-dimensional matrix pencils. In the emerging context of tropical algebraic geometry[66], these result can be understood as finding the roots of a tropical characteristic polynomial. Inthis project we hope to extend this theory to find a similarly meaningful tropicalization of SDP:

13

Problem 10 Investigate the tropical geometry of semidefinite programming, and use this to developa practical algorithm for solving semidefinite programs over the field of real Puiseux series.

The following two pieces of evidence suggest that tropicalization of SDP will be both mathemat-ically deep and useful for applications. First, the notion of tropical convexity [15] provides such atheory in the case of linear programming. An explanation of how tropical polytopes arise naturallyfrom classical polytopes over R{{ε}} was given by Develin and Yu in [16, §2]. Second, the beautifulpicture painted by Speyer in [70] identifies Vinnikov curves over R{{ε}} with tropical curves knownas honeycombs, and thus points to the correct notion of tropical spectrahedra in dimension two.Our aim is to generalize these two pieces of evidence and to study their algorithmic implications.

7 Further Topics in Convex Algebraic Geometry

Besides semidefinite programming, there are other tractable convex optimization formulations thatare of interest from an algebraic perspective. We hope to study the following three in this project.

Geometric programming and entropy maximization: Geometric Programming (GP) is animportant paradigm for many applications [9, 17]. Its objective and inequality constraints areexpressed by posynomials, which are polynomial-like expressions p(x) =

∑

k ck xa1k

1 xa2k

2 · · ·xankn

with positive coefficients ck > 0 and arbitrary real exponents aij . Geometric programs are notconvex in their natural formulation, but they can be easily convexified by a logarithmic transfor-mation on both the domain and the range. That is, if p : Rn

+ → R+ is a posynomial, then thetransformed function y '→ log p(ey1 , . . . , eyn) is a convex function. Geometric programming is thenatural computational tool in a number of problems ranging from relaxations for combinatorialoptimization problems, to maximum likelihood estimation and coding theory. There are severalalgorithms for minimizing a posynomial, for instance, the method of Iterative Proportional Scalingfrom statistics [74, §8.4]. We propose to study the algebraic properties of these specialized convexoptimization methods, and how they may apply to closely related non-convex problems. Work byParrilo, Sturmfels and their collaborators in systems biology [13, 37] raises the following question:

Problem 11 Can geometric programming or iterative proportional scaling be deformed to find (op-timal) stationary points for arbitrary (non-toric) dynamical systems in chemical reaction kinetics?

Principal minors of positive definite matrices: Consider an n × n symmetric matrix A,and index its principal minors by the corresponding subset of rows and columns. The followingHadamard-Fischer-Koteljanskii inequalities state that the function S '→ det AS is log-submodular:

A[α ∩ β] · A[α ∪ β] ≤ A[α] · A[β] for all α, β ⊂ {1, . . . , n}. (9)

Equivalently, the logarithmic image of the cone of positive semidefinite matrices under the mapdefined by all the principal minors is contained in the cone of submodular functions on {1, 2, . . . , n}.This logarithmic image is the amoeba of the real algebraic subvariety of (R≥0)2

n

that is parametrizedby the principal minors of a symmetric matrix. Holtz and Sturmfels [34] recently described theequations of this variety in terms of hyperdeterminants. The study of amoebas is closely related totropical algebraic geometry, and we are naturally led to following problem formulation:

Problem 12 Characterize the amoeba parametrized by the principal minors of positive definite ma-trices. Determine the tropical variety given by the hyperdeterminantal ideal of Holtz and Sturmfels.

This problem is also fundamental for the study of Gaussian conditional independence modelsin algebraic statistics, as explained in [75, §4] and it is currently open even for 5 × 5-matrices.

14

Positive semidefinite functions on semigroups: Many problems in analysis boil down tounderstanding or characterizing positive definite functions over specific domains. A generalization ofmany instances of this problem is the study of positive definite functions over abelian ∗-semigroups.These are abelian semigroups S with a conjugation ∗ : S → S, s '→ s∗ such that (s∗)∗ = s and(st)∗ = t∗s∗ for all s, t ∈ S. A function f : S → S is positive definite if the matrix (f(s∗t)) with rowsand columns indexed by the elements of S is positive semidefinite. Every abelian group is an abelian∗-semigroup with g∗ := g−1, and, in this case, we have the following classical characterization:

Theorem 7.1 (Bochner’s Theorem, Theorem 2.9 in [4]) Let G be a discrete abelian groupwith g∗ = g−1. A function φ : G → C is positive definite if and only if it is the Fourier transformof a nonnegative Radon measure on the compact dual group G, which is the group characters of G.

Under a certain exponential boundedness assumption there exists a general characterization ofpositive definite functions over ∗-semigroups. Let S∗ be the set of semicharacters of S (functionsρ : S → C such that (i) ρ(0) = 1, (ii) ρ(s + t) = ρ(s)ρ(t) ∀ s, t ∈ S and (iii) ρ(s∗) = ρ(s) ∀ s ∈ S.)A function φ : S → C is called a moment function if there exists a positive Radon measure µ onS∗ with

∫

S∗ |ρ(s)|dµ(ρ) < ∞ for all s ∈ S such that φ(s) =∫

S∗ ρ(s)dµ(ρ) for all s ∈ S.

Theorem 7.2 (Theorem 2.5 in [4]) A function φ : S → C is a moment function, i.e. it admitsthe above integral representation, if and only if φ is positive definite and exponentially bounded.

An intriguing incarnation of this theorem appears in work of Freedman, Lovasz and Schrijver[7, 19] on graph parameters that can be represented as functions counting graph homomorphisms.

The final problem to be stated in this proposal concerns the intimate link between these resultsabout positive semidefinite functions on semigroups and the sum of squares techniques describedearlier. All moment functions are positive definite and the cone of exponentially bounded positivedefinite functions on S is contained in the cone of moment functions on S. This raises the naturalquestion of characterizing the semigroups for which the set of moment functions equals the set ofall positive definite functions and when equality fails, what are necessary and sufficient conditionsfor a positive definite function to be a moment function? Theorem 1.11 in [4, §6] identifies a classof fairly general semigroups where moment functions are precisely the positive definite functions. Asimple example of such a semigroup is (N, +) with trivial conjugation. This is precisely Hamburger’sTheorem [4, Theorem 2.2] which says that a sequence s = (sn : n ∈ N) of real numbers is of theform sn =

∫

xndµ(x) if and only if s is positive definite over N. However, for k ≥ 2 there existpositive definite functions on (Nk, +) which are not moment functions. These arise from positivepolynomials that fail to be sums of squares. Blekherman [5] showed that such examples abound.

Problem 13 Identify natural classes of semigroups for which the moment functions are preciselythe positive definite functions. In such cases, find representations for positive definite functions asmoment functions. When equality fails, provide useful conditions that separate the two classes offunctions. Apply these results to sums of squares and positive polynomials on semi-algebraic sets.

A case of special interest is the study of sums of squares in R[x1, . . . , xn]/I where I is a poly-nomial ideal. Parrilo [51] showed that, for zero-dimensional radical ideals I, every non-negativepolynomial modulo I is SOS modulo I. When the variety of I is a curve or surface, considerablydeeper results due to Scheiderer [67] are available, but little is known in general dimensions. More-over, hardly any algorithmic results exist. These issues arise naturally in polynomial optimizationwhere it is crucial to understand when non-negativity equals the sum of squares condition.

15

8 Research Team

The projects described in this proposal require expertise and interest in, and the ability to combine,diverse areas of mathematics such as convex analysis, operator theory, algebraic geometry, combi-natorics, noncommutative algebra and optimization. They also require substantial experience incomputation and algorithms, as well as their role in science and engineering. This research team isuniquely positioned to undertake the complex, diverse, and interdisciplinary tasks proposed here.

Helton is a well known operator theorist who has played a leading role in many of the resultsthat are central to this proposal. He devotes much effort to promoting interactions between func-tional analysts and engineers; he co-organized a workshop at AIM in 2005 and at the IMA in 2007that mixed mathematicians and engineers. Helton also plays a substantial role in steering inter-disciplinary conferences such as Mathematical Theory of Networks and Systems and InternationalWorkshop on Operator Theory and Applications, which he started in 1981. He has given plenarytalks at various conferences such as GPOTS, IWOTA and SEAM and is on the editorial board ofseven journals and book series. Helton has a long history of advising graduate and undergraduatestudents and mentoring post-docs (see CV). His group develops NCAlgebra, a software package forcomputing noncommutative Grobner bases for applications in engineering. Nie has recently joinedUCSD as an assistant professor and Helton is his natural mentor. They already have a joint paper[31] and plan to initiate several further activities at UCSD.

Sturmfels is widely recognized as one of the worldwide experts in discrete mathematics andcomputational algebraic geometry. During the academic year 2007-08 Sturmfels is a HumboldtSenior Scholar at TU Berlin. He serves on numerous editorial boards, including Annals of Math-ematics, and on the governing bodies of research institutes such as the IMA in Minneapolis andIMDEA in Madrid. He has a tremendously successful mentoring record having advised 25 Ph.D.students and 32 postdocs since 1989 from many different departments besides Mathematics suchas Applied Math, Computer Science and Operations Research. He is currently involved in severalinterdisciplinary projects with engineers in the Bay Area (see the next section) and has had pre-vious successful collaborations with statisticians and biologists. Sturmfels is an expert expositorcapable of reaching diverse audiences. This quality is reflected in the many plenary talks that hehas been invited to as well as his eight books. This Fall he will give the Math Matters public lecturesponsored by the IMA in Minneapolis and the Lindhard lectures at Aarhus University, Denmark.In 2005-2007, Sturmfels served as the Polya Lecturer of the Mathematical Association of America.

Parrilo is the engineer on the team and introduced many of the key ideas and links betweenconvex optimization and computational real algebra in his Ph.D. dissertation in Control and Dy-namical Systems from Caltech. His connection to algebra was further fostered by a postdoctoralposition at UC Berkeley with Sturmfels. Since then, he has remained one of the leading expertsand developers of the core theory, and has initiated a number of original scientific and engineeringapplications. His contributions have been recognized through a number of awards, such as the 2005Donald P. Eckman Award of the American Automatic Control Council, as well as the triennialSIAM Activity Group on Control and Systems Theory (SIAG/CST) Prize. He was also a finalistfor the 2000-2003 Tucker Prize of the Mathematical Programming Society. Parrilo is a model linkbetween pure mathematics and engineering and his involvement in this project promises to furtherlinks between the two communities. In recent years, Helton and Parrilo have co-organized a numberof workshops and sessions in different mathematics and control theory conferences.

Nie will join UCSD in the Fall as a tenure-track assistant professor after having been a postdoc-toral researcher at the IMA in Minneapolis and in Caltech’s Mathematics of Information program.He is one of the strongest young people working at the interface of optimization, real algebraicgeometry and their applications. He is trained broadly as a mathematician having started in Nu-

16

merical Analysis under the supervision of Jim Demmel at UC Berkeley and subsequently writing hisdissertation under both Demmel and Sturmfels. He was one of the top candidates in the post-docpool at the IMA in the special year on Applications of Algebraic Geometry. He has demonstratedthe ability and interest to work in a wide range of mathematical areas and is tremendously energetic.

Thomas has had a long standing connection to optimization but mostly on the discrete side. Shewrote her thesis on algebraic methods in integer programming under the supervision of Sturmfelsat Cornell University, graduating from the Operations Research department there. She has madeseveral fundamental contributions to structural results in integer programming using methods fromalgebra and has also been involved in developing software packages such as Gfan. Having workedmore in algebraic combinatorics in recent years, she is now interested in shifting direction back intooptimization via semidefinite programming and its connections to combinatorics and geometry.She is currently on sabbatical leave at MIT working with Parrilo. In 1999-2000 she was hosted byHelton as a visitor at UCSD where they taught a topics class together on computer algebra.

The research team has a history of strong synergy and collaborations. There are a large num-ber of joint publications such as [31, 47, 48, 55, 76]. Everyone on the team has experience withinterdisciplinary work at the interface with the applied sciences. The collective expertise representsa wide, but coherent, cross-section of different areas of pure and applied mathematics. Our teamis also integrated vertically as it consists of researchers at different stages of their academic career,from well-established senior professors, to successful mid-career researchers, and up-and-comingassistant professors. This composition offers substantial mentoring opportunities within the groupof PIs which extends naturally to the postdocs and graduate students involved with these projects.

9 Broader Impacts

The big picture within which this proposal sits aims at developing new directions for applied math-ematics both in terms of science and curricula by bringing in tools from discrete mathematics,algebra, geometry, optimization, analysis and symbolic computation. Polynomial systems are fastbecoming feasible models for applied problems as a result of their naturality in these problems andcomputational advances. The results of this proposal will directly impact these applications andprovide efficient, reliable computational techniques for them. Most likely, the results will, evensooner, filter down into graduate and undergraduate curricula in mathematics and engineering.Engineers are already on board with these ideas as witnessed by the increasing inclusion of op-timization methods in standard courses within engineering and computer science. This proposalseeks to further synergies among different branches of applied mathematics, and their engineeringand scientific applications. Computational biology and statistics are two unusual new comers to thismelting pot. The current interest level among researchers in pure and applied mathematics to mineideas from the various math areas around which this project revolves was witnessed by the highlysuccessful 2006-7 special year in Applications of Algebraic Geometry at the IMA in Minneapolis.This interdisciplinary program, organized by Parrilo, Sturmfels and Thomas along with DimitrisBertsimas (Sloan School, MIT), Madhu Sudan (Computer Science, MIT) and Mike Stillman (Math,Cornell) attracted scholars and students from many areas beyond mathematics such as statistics,communication, complexity theory and biology. See www.ima.umn.edu/2006-2007/.

Further, specific broader impacts envisioned are summarized below.

Graduate Student Training:. This project will involve seven graduate students and one post-doctoral scholar at the four institutions involved. These students have the unique opportunity tobe introduced to a non-standard interdisciplinary area of mathematics that requires considerable

17

breadth of knowledge along with computational skills. Semidefinite optimization has known andexciting links to many parts of mathematics beyond what is outlined here which will allow studentsto utilize their training diversely both in academia and industry.

Sturmfels is very active in advising graduate students. For instance, he supervised eight doctoralstudents who graduated from Berkeley in 2005, 2006 or 2007. They are: Seth Sullivant, Christo-pher Hillar, David Speyer, Hannah Markwig, Nicholas Eriksson, Jiawang Nie, Debbie Yuster, andJosephine Yu. Graduate students and postdocs supervised by Sturmfels are strongly encouraged todevelop their own research program. Students supported under this project may therefore eitherparticipate in investigating specific questions in optimization and convex algebraic geometry asoutlined above, or they may choose to pursue different directions.

Helton also has a strong record of mentoring students, and he is currently directing threePh.D. students. In addition to working with his own graduate students and postdocs, Helton runsa summer workshop with four to five graduate students and several undergraduates every year,devoted to applied and computational projects. Many of these students get Ph.D.’s in pure mathand the lab experience with computation and applied projects broadens them considerably.

Parrilo is currently supervising six graduate students at MIT, from Electrical Engineering &Computer Science and the Operations Research Center. He has also sponsored three postdoctoralscholars (Johan Lofberg, Danielle Tarraf, and Christian Ebenbauer), of which the first two havealready moved on to academic positions. With the aim of achieving academic openness and cross-fertilization of ideas, MIT has considerable flexibility in allowing faculty to advise students fromdifferent departments. This program will provide Parrilo even greater opportunities to supervisestudents from Mathematics, Applied Mathematics, or the Operations Research Center.

Thomas has supervised three Ph.D. dissertations in the past four years: Edwin O’Shea (Ph.D.2006), Tristram Bogart (Ph.D. 2007), and Anders Jensen (Ph.D. 2007). All three students currentlyhold postdoctoral appointments. She hopes to attract new students to this project next year.

Several prospective students have expressed interest in coming to UC San Diego to work withNie, and he hopes to include some of them in this project.

Strong research and communication skills are key elements in the successful transition of stu-dents from graduate school to academic and industrial careers. It is also important for them tointeract early, and often, with the academic community at large. To facilitate this, students andpostdocs will participate in and give presentations at professional meetings, and interact with theiracademic peers at other institutions. They will also be included in the educational activities below.

Education. Several courses are planned around the topics of this proposal. More ideas will evolveas the project proceeds. Specific plans for the immediate future are as follows.

Nie will develop a graduate course on Global optimization with polynomials to be taught atUCSD. This will be an overview of recent advances in global polynomial optimization via toolssuch as semidefinite programming and aims to provide access to front-line research in this area.Nie has been considerably influenced by his own experiences as a TA in engineering at UC Berkeleywhich both broadened his horizons and provided access to research projects with engineers there.

Since 2006 Parrilo has developed and taught a new graduate course at MIT on Algebraic Tech-niques and Semidefinite Optimization (6.972) patterned after very successful workshops offered inearlier years (in collaboration with Sanjay Lall at Stanford University) at several systems and con-trol conferences; see control.ee.ethz.ch/~parrilo/cdc03_workshop. The lecture notes for thecourse are freely available from MIT’s OpenCourseWare: www.mit.edu/~parrilo/6972.

Thomas plans three new courses at the University of Washington in 2008 related to this proposal.In Spring 2008, she will teach a graduate topics class on Linear and Integer Polyhedra from thepoint of view of linear and semidefinite optimization. She will also design and teach a research

18

class for advanced undergraduates on Solving Polynomial Equations. Both classes will aim towardintroducing students to research, and hope to attract students from various departments besidesMathematics. In Fall 2008 she hopes to teach a class on Discriminants at the graduate level.Thomas is currently supervising an undergraduate research project at MIT with Andreas Schulz.

Sturmfels serves on a committee charged with developing a new curriculum in applied math-ematics at UC Berkeley, and he plans to integrate the topics discussed here into one or two newcourses.

Helton often teaches graduate topics courses that include many topics from this proposal. Lastyear’s efforts contributed heavily to the expository parts of the article with Putinar [32], that inthe future will become an expository book suitable for graduate classes. The graduate course atUCSD is MAE 284 and the NC systems software package is used for many of the core calculationsin the course. One main goal is to eventually have a computer algebra version of this course. Thisof course is highly experimental, since noncommutative (free) algebra is still in its infancy.

Results from this project will feed into both existing and future courses. All such materials willbe available on the project webpage.

Collaborations with other projects and industry: Several of the PIs have previous andongoing collaborations with engineering and industry. Sturmfels is currently involved in interac-tions with several Bay Area engineers who use polynomial optimization, namely, Stephen Boyd(EE Stanford), Laurent El Ghaoui (EECS UC Berkeley), Sanjay Lall (Aero/Astro Stanford), andMartin Wainwright (EECS and Statistics, UC Berkeley). He has also strong connections to biol-ogy and statistics. Helton has a long history of collaboration with the Mechanical and AerospaceEngineering department at UCSD, through colleagues such as Robert Skelton. Besides his naturalconnections to engineering colleagues at MIT, Parrilo has research links with the Aerospace Ad-vanced Technology lab at Honeywell, that is currently interested in sum of squares based techniquesfor the analysis of hybrid dynamical systems. Nie is currently working with Saira Mian in the LifeSciences Division at the Lawrence Berkeley National Laboratory on chromosome packing problems.

Dissemination of results: Here are concrete plans for disseminating the research and educationalresults of this project. See the Management Plan as well.

• A project website containing publications, software packages and other products of this re-search will be hosted and maintained at the University of Washington. It will also link to allcourses taught and talks given under this umbrella. In addition, it will host a database ofproblems and counterexamples that will be fine-tuned over the course of this research.

• The PIs plan to organize three meetings, one in each year, on topics related to this proposal.In 2009 and 2010, meetings are targeted at BIRS and AIM respectively. A final meeting in2011 is planned at the IMA with representation from the various applied mathematiciansinterested in these ideas. Several separate special sessions are already planned: MasakazuKojima and Nie will organize two sessions at the SIAM optimization meeting in May 2008and Thomas and Elizabeth Allman will organize an AMS special session in October 2008 asa follow-up to the IMA year.

• The students and postdocs supported by this project will travel among the four institutionsto give talks and collaborate on projects. This will allow them to benefit from the diversityof our senior personnel and will provide them with valuable professional contacts. Juniorparticipants will also be encouraged to speak at conferences and attend relevant short-courses.

19

• Ioana Dumitriu and Thomas plan to initiate a campus-wide colloquium at the Universityof Washington to showcase applied mathematics in Seattle with a special focus on discretemethods. This activity is aimed at building a community for users of mathematics in thePuget Sound area and to help students find directions and jobs in math related applied work.

• Since September 2003, MIT has engaged in an unprecedented initiative to make all its coursespublicly available on the Internet via the MIT OpenCourseWare (OCW). See ocw.mit.edu.There are currently more than 1,550 courses online, with more being added each term. Thisprogram has been enormously successful, with more than 37,000 visits per day from all overthe world. Several outcomes of this proposal are expected to eventually migrate into graduatecourses, and thus to be promptly available on OCW. An example of this, mentioned earlier, arethe lecture notes of Parrilo’s 6.972 course, that can be found at www.mit.edu/~parrilo/6972.

10 Results from prior NSF Support

Bill Helton Bill Helton has been continually supported by NSF grants since obtaining his Ph.D. Hiscurrent grant is DMS-0700758 Operator Theory Arising from Systems Engineering. Over the pastfive years, this support resulted in over 26 research articles and one patent (US patent 7,058,555);see math.ucsd.edu/~helton/frgbib.html for the complete list. In addition are several preprints.These articles concern some aspect of functional analysis often motivated by problems in systemengineering or optimization. Several of them and potential extensions are discussed in the bodyof this proposal. A common theme that emerged was the beginnings of a noncommutative realalgebraic geometry required for optimization problems with matrix unknowns.

Bernd Sturmfels is currently supported by the National Science Foundation grant DMS-0456960Computational Algebraic Geometry. He has worked on a wide range of topics in computational al-gebraic geometry and its applications, both to other fields within mathematics and to areas outsideof pure mathematics, such as statistics and computational biology. These projects involve aboutthirty collaborators, including graduate students and postdoctoral fellows. Since the beginning ofthat grant, more than 44 research articles have been published or accepted for publication. Seepapers 116–161 posted at math.berkeley.edu/~bernd/articles.html.

Pablo A. Parrilo is a member of the Laboratory of Information and Decision Systems and theOperations Research Center, and is affiliated with the Computation for Design and Optimiza-tion program at MIT. Currently he is co-PI of the NSF award ECS-0621922 Optimization andControl of Stochastic Wireless Networks. Some recent results developed in the context of thisprogram include semidefinite programming techniques for the characterization and computationof Nash and correlated equilibria for games with an infinite number of pure strategies [71]. Seewww.mit.edu/~parrilo/pubs for a complete list of publications.

Rekha Thomas was supported by NSF grant DMS-0401047 which expired in August 2007. Sheworked on a number of projects in pure mathematics that involved the application of methods fromthe theory of Grobner bases. A major contribution under that grant was the development of thesoftware package Gfan [35] by Anders Jensen as part of his dissertation. Thomas was an infor-mal adviser to Jensen. The grant support resulted in ten research articles (since 2004) and threebooks (two written and one edited). See www.math.washington.edu/~thomas for the complete list.

Jiawang Nie has not yet been supported by the NSF, except through the IMA at Minneapolis.

20

Management Plan

This project will be headquartered at MIT under the supervision of Parrilo. All administrativeresponsibilities such as progress reports and final reports will be administered through MIT. EachPI will be responsible for supervising and coordinating research and educational activities withintheir group as is usual. However, the PIs will also ensure that significant interactions between thegroups and institutions take place as planned. Specific mechanisms to achieve this are as follows.

Virtual collaboration At the most basic level, a regularly scheduled phone/video conferenceinvolving all PIs and their research groups will take during the first week of every month (12meetings/year). These meetings shall be used to report progress, discuss ideas and problemsunder investigation, propose new directions, and for general brainstorming. More frequent virtualmeetings will be arranged if/when necessary. Nie will be in charge of this activity.

Electronic collaboration and dissemination A wiki system will be hosted and maintainedat a server at the University of Washington for all finished products of this effort such as preprints,software packages, and slides/videos of talks given by the participants of the project. This site willalso contain a private discussion board for questions, problems and ideas that come up and willbe linked to individual emails so that all participants can be updated daily on issues that needattention. Minutes of the phone/teleconference meetings (arranged to be transcribed by Nie) willalso be posted on this site. The goal will be to simulate open and unpolished discussions as ispossible with physical proximity.

Face-to-face meetings All the research teams supported by the program will physically meetat least twice a year. One of these meetings will be a private meeting of the whole team for aweekend. For efficiency of coordination, all three meetings will be at UC Berkeley and the first onewill take place in June 2008 if the grant is awarded. These meetings will involve a section withformal presentations by group members which presents a hands-on mentoring opportunity to trainjunior members in the presentation of mathematics. It will also enable new collaborations amongstudents and postdocs at the different institutions.

The second annual meeting will be at a public conference or workshop. Specifically the PIs willapply to organize workshops at BIRS and AIM in 2009 and 2010. The PIs will work in pairs tocoordinate these applications. Parrilo and Thomas will apply to BIRS in September 2007 for aslot in 2009. The next workshop will be organized by Helton and Nie. The third and final publicmeeting is targeted to be at the IMA in Minneapolis. This meeting will be a capstone for theproject and will aim to attract researchers from all the different communities that are relevant tothis proposal. This meeting will be coordinated by all PIs. Sturmfels and Helton are in charge ofrelegating duties.

Progress evaluation Each meeting will be an occasion to assess progress thus far toward thegoals of this collaborative project. Based on the results of this self-evaluation, different aspects ofthe collaboration will be prioritized, and a list of action items will be produced to guide futureefforts (e.g., progress on software development, extension of results in particular directions). Therewill be a follow-up of these items during the subsequent teleconferences, and at the next regularmeeting. These reports will be maintained by Thomas and Parrilo.

21

References

[1] M. Akian, R. Bapat, and S. Gaubert. Perturbation of eigenvalues of matrix pencils and theoptimal assignment problem. C. R. Math. Acad. Sci. Paris, 339(2):103–108, 2004.

[2] C. Bachoc and F. Vallentin. New upper bounds for kissing numbers from semidefinite pro-gramming. Preprint, available at arXiv:math.MG/0608426, 2006.

[3] S. Basu, R. Pollack, and M.-F. Roy. Algorithms in real algebraic geometry, volume 10 ofAlgorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2003.

[4] C. Berg, J. Christensen, and P. Ressel. Harmonic Analysis on Semigroups, volume 100 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1984.

[5] G. Blekherman. There are significantly more nonnegative polynomials than sums of squares.Israel Journal of Mathematics, 153:355–380, 2006.

[6] B. Borchers. SDPLIB 1.2, library of semidefinite programming test problems. Optim. MethodsSoftw., 11/12(1-4):683–690, 1999. Online version at www.nmt.edu/~borchers/sdplib.html.

[7] C. Borgs, J. Chayes, L. Lovasz, V. T. Sos, and K. Vesztergombi. Counting graph homomor-phisms. In Topics in discrete mathematics, volume 26 of Algorithms Combin., pages 315–371.Springer, Berlin, 2006.

[8] S. Boyd, P. Diaconis, P. A. Parrilo, and L. Xiao. Symmetry analysis of reversible Markovchains. Internet Math., 2(1):31–71, 2005.

[9] S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. A tutorial on geometric programming.Optimization and Engineering, 8(1):67–127, 2007.

[10] J. F. Camino, J. W. Helton, and R. E. Skelton. Solving matrix inequalities whose unknownsare matrices. SIAM J. Optim., 17(1):1–36, 2006.

[11] B. F. Caviness and J. R. Johnson, editors. Quantifier elimination and cylindrical algebraicdecomposition, Texts and Monographs in Symbolic Computation, Vienna, 1998. Springer.

[12] M. D. Choi, T. Y. Lam, and B. Reznick. Sums of squares of real polynomials. Proceedings ofSymposia in Pure Mathematics, 58(2):103–126, 1995.

[13] G. Craciun, A. Dickenstein, A. Shiu, and B. Sturmfels. Toric dynamical systems. Preprint.Available from arXiv:arXiv:0708.3431.

[14] M. C. de Oliveira and J. W. Helton. Computer algebra tailored to matrix inequalities incontrol. Internat. J. Control, 79(11):1382–1400, 2006.

[15] M. Develin and B. Sturmfels. Tropical convexity. Documenta Mathematica, 9:1–27, 2004.

[16] M. Develin and J. Yu. Tropical polytopes and cellular resolutions. Experimental Mathematics,to appear. Available at arXiv: math.CO/0605494.

[17] R. J. Duffin, E. L. Peterson, and C. Zener. Geometric programming: Theory and application.John Wiley & Sons Inc., New York, 1967.

1

[18] M. Elkadi and B. Mourrain. Symbolic-numeric methods for solving polynomial equations andapplications. In Solving polynomial equations, volume 14 of Algorithms Comput. Math., pages125–168. Springer, Berlin, 2005.

[19] M. Freedman, L. Lovasz, and A. Schrijver. Reflection positivity, rank connectivity, and homo-morphism of graphs. J. Amer. Math. Soc., 20(1):37–51, 2007.

[20] M. Fukuda, M. Kojima, K. Murota, and K. Nakata. Exploiting sparsity in semidefinite pro-gramming via matrix completion. I. General framework. SIAM J. Optim., 11(3):647–674,2000/01.

[21] S. Gao, E. Kaltofen, J. May, Z. Yang, and L. Zhi. Approximate factorization of multivariatepolynomials via differential equations. In ISSAC 2004, pages 167–174. ACM, New York, 2004.

[22] K. Gatermann and P. A. Parrilo. Symmetry groups, semidefinite programs, and sums ofsquares. Journal of Pure and Applied Algebra, 192(1-3):95–128, 2004.

[23] H.-C. Graf von Bothmer and K. Ranestad. A general formula for the algebraic degree insemidefinite programming. Preprint, available from arXiv:math.AG/0701877.

[24] D. Grayson and M. E. Stillman. Macaulay 2, a software system for research in algebraicgeometry. Available at www.math.uiuc.edu/Macaulay2/.

[25] G.-M. Greuel, G. Pfister, and H. Schonemann. Singular 3.0. A Computer Algebra Systemfor Polynomial Computations, University of Kaiserslautern, 2005. www.singular.uni-kl.de.

[26] R. Grone, C. R. Johnson, E. M. de Sa, and H. Wolkowicz. Positive definite completions ofpartial Hermitian matrices. Linear Algebra Appl., 58:109–124, 1984.

[27] B. Grunbaum. Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2003. Prepared and with a preface by Volker Kaibel, VictorKlee and Gunter M. Ziegler.

[28] O. Guler. Hyperbolic polynomials and interior point methods for convex programming. Math.Oper. Res., 22(2):350–377, 1997.

[29] J. W. Helton. NCAlgebra, a Mathematica package for noncommutative algebra. University ofCalifornia, San Diego. Available at www.math.ucsd.edu/~ncalg/.

[30] J. W. Helton, S. McCullough, M. Putinar, and V. Vinnikov. Convex matrix inequalities versuslinear matrix inequalities. Preprint.

[31] J. W. Helton and J. Nie. Semidefinite representation of convex sets. Preprint, available atarXiv:0705.4068, 2007.

[32] J. W. Helton and M. Putinar. Positive polynomials in scalar and matrix variables, the spectraltheorem and optimization. Preprint, available from arXiv:math.FA/0612103.

[33] J. W. Helton and V. Vinnikov. Linear matrix inequality representation of sets. Comm. PureAppl. Math., 60(5):654–674, 2007. Available from arXiv:math.OC/0306180.

[34] O. Holtz and B. Sturmfels. Hyperdeterminantal relations among symmetric principal minors.Journal of Algebra, to appear. Available at arXiv:math/0604374, 2007.

2

[35] A. N. Jensen. Gfan, a software system for Grobner fans. Available athome.imf.au.dk/ajensen/software/gfan/gfan.html.

[36] S. Karlin and L. Shapley. Geometry of moment spaces, volume 12 of Memoirs of the AmericanMathematical Society. AMS, 1953.

[37] L. Kuepfer, U. Sauer, and P. A. Parrilo. Efficient classification of complete parameter regionsbased on semidefinite programming. BMC Bioinformatics, 8(12), 2007.

[38] J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM J.Optim., 11(3):796–817, 2001.

[39] J. B. Lasserre. Convergent SDP-relaxations in polynomial optimization with sparsity. SIAMJ. Optim., 17(3):822–843 (electronic), 2006.

[40] S. L. Lauritzen. Graphical models, volume 17 of Oxford Statistical Science Series. The Claren-don Press Oxford University Press, New York, 1996.

[41] A. S. Lewis, P. A. Parrilo, and M. V. Ramana. The Lax conjecture is true. Proc. Amer. Math.Soc., 133(9):2495–2499, 2005.

[42] J. Lofberg and P. A. Parrilo. From coefficients to samples: a new approach to SOS optimization.In Proceedings of the 43th IEEE Conference on Decision and Control, 2004.

[43] P. McMullen. The polytope algebra. Advances in Mathematics, 78:76–130, 1989.

[44] Y. E. Nesterov and A. Nemirovski. Interior point polynomial methods in convex programming,volume 13 of Studies in Applied Mathematics. SIAM, Philadelphia, PA, 1994.

[45] J. Nie. Sum of squares method for sensor network localization. Computational Optimizationand Applications, 2007. To appear.

[46] J. Nie and J. Demmel. Sparse SOS relaxations for minimizing functions that are summationof small polynomials. Preprint, available at arXiv:0606.5476, 2006.

[47] J. Nie, P. A. Parrilo, and B. Sturmfels. Semidefinite representation of the k-ellipse. IMAVolume on Algorithms in Algebraic Geometric, to appear. Available at arXiv:math/0702005.

[48] J. Nie, K. Ranestad, and B. Sturmfels. The algebraic degree of semidefinite programming.Preprint. available from arXiv:math/0611562.

[49] T. Oda. Convex bodies and algebraic geometry, volume 15 of Ergebnisse der Mathematik undihrer Grenzgebiete. Springer-Verlag, Berlin, 1988.

[50] P. A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robust-ness and optimization. PhD thesis, California Institute of Technology, May 2000.

[51] P. A. Parrilo. An explicit construction of distinguished representations of polynomials non-negative over finite sets. Technical Report IfA Technical Report AUT02-02. Available fromcontrol.ee.ethz.ch/~parrilo, ETH Zurich, 2002.

[52] P. A. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Math. Prog.,96(2, Ser. B):293–320, 2003.

3

[53] P. A. Parrilo. Exact semidefinite representations for genus zero curves. In preparation, 2007.

[54] P. A. Parrilo. Hyperbolic polynomials and SOS matrices: relationships and an explicit algo-rithm. In preparation, 2007.

[55] P. A. Parrilo and B. Sturmfels. Minimizing polynomial functions. In S. Basu and L. Gonzalez-Vega, editors, Algorithmic and Quantitative Real Algebraic Geometry, volume 60 of DIMACSSeries in Discrete Mathematics and Theoretical Computer Science, pages 83–100. AMS, 2003.

[56] G. Pataki. Cone-LP’s and semidefinite programs: geometry and a simplex-type method. InInteger programming and combinatorial optimization (Vancouver, BC, 1996), volume 1084 ofLecture Notes in Comput. Sci., pages 162–174. Springer, Berlin, 1996.

[57] G. Pataki. The geometry of semidefinite programming. In Handbook of semidefinite program-ming, volume 27 of Internat. Ser. Oper. Res. Management Sci., pages 29–65. Kluwer Acad.Publ., Boston, MA, 2000.

[58] J. Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. Seriesin Representation and Reasoning. Morgan Kaufmann, San Mateo, CA, 1988.

[59] H. Peyrl and P. A. Parrilo. A Macaulay 2 package for computing sum of squares decompositionsof polynomials with rational coefficients. In SNC ’07: Proceedings of the 2007 internationalworkshop on Symbolic-numeric computation, pages 207–208, New York, 2007. ACM Press.

[60] H. Peyrl and P. A. Parrilo. SOS.m2, a sum of squares package for Macaulay 2. Available atwww.control.ee.ethz.ch/~hpeyrl, 2007.

[61] S. Prajna, A. Papachristodoulou, P. A. Parrilo, and P. Seiler. SOSTOOLS: Sum of squaresoptimization toolbox for MATLAB, 2002-07. Available from www.cds.caltech.edu/sostools

and www.mit.edu/~parrilo/sostools.

[62] M. Ramana and A. J. Goldman. Some geometric results in semidefinite programming. J.Global Optim., 7(1):33–50, 1995.

[63] J. Renegar. Hyperbolic programs, and their derivative relaxations. Found. Comput. Math.,6(1):59–79, 2006.

[64] B. Reznick. Extremal PSD forms with few terms. Duke Math. J., 45(2):363–374, 1978.

[65] B. Reznick. Some concrete aspects of Hilbert’s 17th problem. In Contemporary Mathematics,volume 253, pages 251–272. American Mathematical Society, 2000.

[66] J. Richter-Gebert, B. Sturmfels, and T. Theobald. First steps in tropical geometry. In Idem-potent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317.Amer. Math. Soc., Providence, RI, 2005.

[67] C. Scheiderer. Sums of squares on real algebraic surfaces. Manuscripta Math., 119:395–410,2006.

[68] A. J. Sommese, J. Verschelde, and C. W. Wampler. Introduction to numerical algebraicgeometry. In Solving polynomial equations, volume 14 of Algorithms Comput. Math., pages301–335. Springer, Berlin, 2005.

4

[69] A. J. Sommese and C. W. Wampler. The numerical solution of systems of polynomials. WorldScientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.

[70] D. E. Speyer. Horn’s problem, Vinnikov curves, and the hive cone. Duke Math. J., 127(3):395–427, 2005.

[71] N. D. Stein, A. E. Ozdaglar, and P. A. Parrilo. Characterization and computation of correlatedequilibria in infinite games. In Proc. IEEE Conf. on Decision and Control, 2007.

[72] H. J. Stetter. Numerical polynomial algebra. Society for Industrial and Applied Mathematics(SIAM), Philadelphia, PA, 2004.

[73] B. Sturmfels. Grobner Bases and Convex Polytopes, volume 8 of University Lectures. AmericanMathematical Society, 1995.

[74] B. Sturmfels. Solving systems of polynomial equations, volume 97 of CBMS Regional Confer-ence Series in Mathematics. Published for the Conference Board of the Mathematical Sciences,Washington, DC, 2002.

[75] B. Sturmfels. Open problems in algebraic statistics. IMA Volume on Algorithms in AlgebraicGeometric, to appear. Available at arXiv:0707.4558, 2007.

[76] B. Sturmfels and R. Thomas. Variation of cost functions in integer programming. Math.Programming, 77(3, Ser. A):357–387, 1997.

[77] M. Todd. Semidefinite optimization. Acta Numerica, 10:515–560, 2001.

[78] K. C. Toh, R. H. Tutuncu, and M. J. Todd. SDPT3 - a MATLAB software package forsemidefinite-quadratic-linear programming. www.math.nus.edu.sg/~mattohkc/sdpt3.html.

[79] L. Vandenberghe and S. Boyd. Semidefinite programming. SIAM Review, 38(1):49–95, 1996.

[80] J. Verschelde and K. Gatermann. Symmetric Newton polytopes for solving sparse polynomialsystems. Adv. in Appl. Math., 16(1):95–127, 1995.

[81] M. I. Visik and L. A. Ljusternik. Solution of some perturbation problems in the case of matricesand self-adjoint or non-selfadjoint differential equations. I. Russian Math. Surveys, 15(3):1–73,1960.

[82] M. J. Wainwright and M. I. Jordan. Treewidth-based conditions for exactness of the Sherali-Adams and Lasserre relaxations. Technical Report 671, Department of Statistics, Universityof California, Berkeley, 2004.

[83] H. Waki, S. Kim, M. Kojima, and M. Muramatsu. Sums of squares and semidefinite programrelaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim.,17(1):218–242 (electronic), 2006.

[84] H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite Program-ming. Kluwer, 2000.

[85] P. A. Worfolk. Zeros of equivariant vector fields: algorithms for an invariant approach. J.Symbolic Comput., 17(6):487–511, 1994.

[86] G. M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

5