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This paper studies how to compute all real eigenvalues, associated to real eigenvectors, of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle ones cannot. We propose a new approach for computing all real eigenvalues sequentially, from the largest to the smallest. It uses Jacobian semidefinite relaxations in polynomial optimization. We show that each eigenvalue can be computed by solving a finite hierarchy of semidefinite relaxations. Numerical experiments are presented to show how to do this.

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Let A ⊆ N n be a finite set, and K ⊆ R n be a compact semialgebraic set. An A-truncated multisequence (A-tms) is a vector y = (yα) indexed by elements in A. The A-truncated K-moment problem (A-TKMP) concerns whether or not a given A-tms y admits a K-measure µ, i.e., µ is a nonnegative Borel measure supported in K such that yα = K x α dµ for all α ∈ A. This paper proposes a numerical algorithm for solving A-TKMPs. It aims at finding a flat extension of y by solving a hierarchy of semidefinite relaxations {(SDR) k } ∞ k=1 for a moment optimization problem, whose objective R is generated in a certain randomized way. If y admits no K-measures and R[x] A is K-full (there exists p = α∈A pαx α that is positive on K), then (SDR) k is infeasible for all k big enough, which gives a certificate for the nonexistence of representing measures. If y admits a K-measure, then for almost all generated R, this algorithm has the following properties: i) we can asymptotically get a flat extension of y by solving the hierarchy {(SDR) k } ∞ k=1 ; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of y by solving (SDR) k for some k; iii) the obtained flat extensions admit a ratomic K-measure with r ≤ |A|. The decomposition problems for completely positive matrices and sums of even powers of real linear forms, and the standard truncated K-moment problems, are special cases of A-TKMPs. They can be solved numerically by this algorithm. R A (K) := {y ∈ R A : meas(y, K) = ∅}. 1991 Mathematics Subject Classification. 44A60, 47A57, 90C22, 90C90. Key words and phrases. A-truncated multisequence, A-truncated K-moment problem, completely positive matrices, flat extension, moment matrix, localizing matrix, representing measure, semidefinite program, sums of even powers. 1 For each J ⊆ {1, . . . , m 2 + 1}, denoteThe set of critical points of (A.1) with the active set J is contained in V J , which is called a critical variety of (A.1). We show that if the coefficients of f hom , h hom , g hom J satisfy some discriminantal inequalities, then V J is finite. We refer to [33, Section 3]) for the definition of discriminants ∆. Proposition A.1. Let f, h i (i ∈ [m 1 ]), g j (j ∈ [m 2 + 1]) ∈ R[x], and V J be defined as above. For any J ⊆ [m 2 + 1], if ∆(f hom , h hom , g hom J ) = 0, ∆(h hom , g hom J ) = 0, (A.3) rank Jac(f hom , h hom , g hom J )| x ≤ m 1 + |J|, h hom (x) = g hom J (x) = 0. Since ∆(h hom , g hom J ) = 0, rank Jac(h hom , g hom J )| x = m 1 + |J| for all 0 = x ∈ V C (h hom , g hom J ). The rank condition in (A.3) implies that f hom (v) = 0 (cf. [33, Section 3]). Hence, v is a nonzero singular solution to f hom (x) = h hom (x) = g hom J (x) = 0, which contradicts ∆(f hom , h hom , g hom J ) = 0 (cf. [33]). So, V J must be finite.

Abstract. Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.

Given polynomials f (x), g i (x), h j (x), we study how to minimize f (x) on the setLet f min be the minimum of f on S. Suppose S is nonsingular and f min is achievable on S, which are true generically. This paper proposes a new type semidefinite programming (SDP) relaxation which is the first one for solving this problem exactly. First, we construct new polynomials ϕ 1 , . . . , ϕ r , by using the Jacobian of f, h i , g j , such that the above problem is equivalent toSecond, we prove that for all N big enough, the standard N -th order Lasserre's SDP relaxation is exact for solving this equivalent problem, that is, its optimal value is equal to f min . Some variations and examples are also shown.

Abstract. This paper studies the problem of finding best rank-1 approximations for both symmetric and nonsymmetric tensors. For symmetric tensors, this is equivalent to optimizing homogeneous polynomials over unit spheres; for nonsymmetric tensors, this is equivalent to optimizing multi-quadratic forms over multi-spheres. We propose semidefinite relaxations, based on sum of squares representations, to solve these polynomial optimization problems. Their special properties and structures are studied. In applications, the resulting semidefinite programs are often large scale. The recent Newton-CG augmented Lagrangian method by Zhao, Sun and Toh [32] is suitable for solving these semidefinite relaxations. Extensive numerical experiments are presented to show that this approach is efficient in getting best rank-1 approximations.