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Let S = {x ∈ R n : g 1 (x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials g i (x). Assume S is convex, compact and has nonempty interior. Let S i = {x ∈ R n : g i (x) ≥ 0}, and ∂S (resp. ∂S i) be the boundary of S (resp. S i). This paper, as does the subject of semidefinite programming (SDP), concerns Linear Matrix(More)
Let S = {x ∈ R n | g 1 (x) ≥ 0,. .. , gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials g i. Putinar's Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = P m i=0 σ i g i where g 0 := 1 and each σ i is a sum of squares of(More)
Given a 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(More)
This paper studies the so-called bi-quadratic optimization over unit spheres min x∈R n ,y∈R m subject to x = 1, y = 1. We show that this problem is NP-hard and there is no polynomial time algorithm returning a positive relative approximation bound. After that, we present various approximation methods based on semidefinite programming (SDP) relaxations. Our(More)
A method is proposed for finding the global minimum of a multivariate polynomial via sum of squares (SOS) relaxation over its gradient variety. That variety consists of all points where the gradient is zero and it need not be finite. A polynomial which is nonnegative on its gradient variety is shown to be SOS modulo its gradient ideal, provided the gradient(More)
A set S ⊆ R n is called to be Semidefinite (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI). Clearly, if S is SDP representable, then S must be convex and semialgebraic (it is describable by conjunctions and disjunctions of polynomial equalities or inequalities).(More)
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 multiquadratic forms over multispheres. We propose semidefinite relaxations, based on(More)
Given polynomials f (x), g i (x), h j (x), we study how to minimize f (x) on the set S = {x ∈ R n : h 1 (x) = · · · = h m 1 (x) = 0, g 1 (x) ≥ 0,. .. , g m 2 (x) ≥ 0}. Let 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(More)