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39
Sums of squares, moment matrices and optimization over polynomials
, 2008
"... We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory ..."
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Cited by 59 (8 self)
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We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NPhard 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 zerodimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.
The truncated complex Kmoment problem
 Trans. Amer. Math. Soc
"... on the occasion of his eightyseventh birthday Abstract. Let γ ≡ γ (2n) denote a sequence of complex numbers γ00,γ01,γ10,...,γ0,2n,...,γ2n,0 (γ00> 0,γij = ¯γji), and let K denote a closed subset of the complex plane C. The Truncated Complex KMoment Problem for γ entails determining whether there ex ..."
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Cited by 47 (5 self)
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on the occasion of his eightyseventh birthday Abstract. Let γ ≡ γ (2n) denote a sequence of complex numbers γ00,γ01,γ10,...,γ0,2n,...,γ2n,0 (γ00> 0,γij = ¯γji), and let K denote a closed subset of the complex plane C. The Truncated Complex KMoment Problem for γ entails determining whether there exists a positive Borel measure µ on C such that γij = ∫ ¯z izj dµ (0 ≤ i + j ≤ 2n) and supp µ ⊆ K. For K ≡ KP a semialgebraic set determined by a collection of complex polynomials P = {pi (z, ¯z)} m i=1, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix M (n)(γ) and the localizing matrices Mp i. We prove that there exists a rank M (n)atomic representing measure for γ (2n) supported in KP if and only if M (n) ≥ 0andthereissomerankpreserving extension M (n +1)forwhichMp i (n + ki) ≥ 0, where deg pi =2ki or 2ki − 1(1 ≤ i ≤ m). 1.
Semidefinite Representations for Finite Varieties
 MATHEMATICAL PROGRAMMING
, 2002
"... We consider the problem of minimizing a polynomial over a semialgebraic 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 prob ..."
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Cited by 36 (6 self)
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We consider the problem of minimizing a polynomial over a semialgebraic 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.
Revisiting Two Theorems of Curto and Fialkow on Moment Matrices
, 2004
"... We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence... ..."
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Cited by 18 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
Semidefinite approximations for global unconstrained polynomial optimization
 SIAM J. Optim
"... 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 ..."
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Cited by 15 (0 self)
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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:
Semidefinite characterization and computation of zerodimensional real radical ideals
, 2007
"... real radical ideals ..."
Truncated Kmoment problems in several variables
 J. Operator Theory
"... Abstract. Let β ≡ β (2n) be an Ndimensional real multisequence 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 rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing meas ..."
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Cited by 8 (4 self)
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Abstract. Let β ≡ β (2n) be an Ndimensional real multisequence 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 rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, which is ratomic, with suppµ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an ratomic (minimal) representing measure supported in a semialgebraic 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 rankpreserving 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.
Semidefinite characterization and computation of real radical ideals
 Foundations of Computational Mathematics
"... For an ideal I ⊆ R[x] given by a set of generators, a new semidefinite characterization of its real radical I(VR(I)) is presented, provided it is zerodimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute ..."
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Cited by 6 (5 self)
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For an ideal I ⊆ R[x] given by a set of generators, a new semidefinite characterization of its real radical I(VR(I)) is presented, provided it is zerodimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety VR(I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components. AMS: 14P05 13P10 12E12 12D10 90C22 1
NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials
 Optimization Methods and Software
, 2011
"... Abstract. NCSOStools is a Matlab toolbox for • symbolic computation with polynomials in noncommuting variables; • constructing and solving sum of hermitian squares (with commutators) programs for polynomials in noncommuting variables. It can be used in combination with semidefinite programming softw ..."
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Cited by 6 (4 self)
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Abstract. NCSOStools is a Matlab toolbox for • symbolic computation with polynomials in noncommuting variables; • constructing and solving sum of hermitian squares (with commutators) programs for polynomials in noncommuting variables. It can be used in combination with semidefinite programming software, such as SeDuMi, SDPA or SDPT3 to solve these constructed programs. This paper provides an overview of the theoretical underpinning of these sum of hermitian squares (with commutators) programs, and provides a gentle introduction to the primary features of NCSOStools. 1.