We consider the problem of minimizing a polynomial over a semi-algebraic set defined by polynomial equalities and inequalities. When the polynomial equalities have a finite number of complex solutions and define a radical ideal we can reformulate this problem as a semidefinite programming problem. This semidefinite program involves combinatorial moment matrices, which are matrices indexed by a basis of the quotient vector space R[x 1 , . . . , x n ]/I. Our arguments are elementary and extend known facts for the grid case including 0/1 and polynomial programming. They also relate to known algebraic tools for solving polynomial systems of equations with finitely many complex solutions. Semidefinite approximations can be constructed by considering truncated combinatorial moment matrices; rank conditions are given (in a grid case) that ensure that the approximation solves the original problem at optimality.

We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

Abstract. We consider the problem of minimizing a polynomial function on R n, known to be hard even for degree 4 polynomials. Therefore approximation algorithms are of interest. Lasserre [15] and Parrilo [23] have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We propose here a method for computing tight upper bounds based on perturbing the original polynomial and using semidefinite programming. The method is applied to several examples. We consider the problem:

We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...

real radical ideals

Abstract. In this note we prove a generalization of the flat extension theorem of Curto and Fialkow [4] for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.

Abstract. Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β (2n) = {βij}i,j≥0,i+j≤2n, with β00> 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure µ, supported in Q(x, y) = 0, such that βij = ∫ yixj dµ (0 ≤ i + j ≤ 2n). We prove that β admits a Q-representing measure µ (as above) if and only if the associated moment matrix M(n)(β) is positive semidefinite, recursively generated, has a column relation Q(X, Y) = 0, and the algebraic variety V(β) associated to β satisfies card V(β) ≥ rank M(n)(β). In this case, rank M(n) ≤ 2n + 1; if rank M(n) ≤ 2n, then β admits a rank M(n)-atomic (minimal) Q-representing measure; if rank M(n) = 2n + 1, then β admits a Q-representing measure µ satisfying 2n + 1 ≤ card supp µ ≤ 2n + 2. 1.

Abstract. Let β ≡ β (2n) be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rank-preserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, which is r-atomic, with suppµ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an r-atomic (minimal) representing measure supported in a semi-algebraic set KQ subordinate to a family Q ≡ {qi} m i=1 ⊆ R[t1,..., tN] if and only if M(n) is positive semidefinite and admits a rank-preserving extension M(n + 1) for which the associated localizing matrices Mqi (n + [ 1+deg qi]) are positive semidefinite (1 ≤ i ≤ m); in this case, µ (as 2 above) satisfies supp µ ⊆ KQ, and µ has precisely rank M(n) − rank Mqi (n + [ 1+deg qi]) atoms in 2 Z(qi) ≡ { t ∈ R N: qi(t) = 0} , 1 ≤ i ≤ m.

Abstract. In this paper we propose a unified methodology for computing the set VK(I) of complex (K = C) or real (K = R) roots of an ideal I ⊆ R[x], assuming VK(I) is finite. We show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety VR(I), as shown in the authors ’ previous work, but also the complex variety VC(I), thus leading to a unified treatment of the algebraic and real algebraic problem. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related. Key words. Polynomial ideal, zero-dimensional ideal, complex roots, real roots, numerical linear algebra.

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