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145
Global Optimization with Polynomials and the Problem of Moments
 SIAM Journal on Optimization
, 2001
"... We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear mat ..."
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Cited by 320 (33 self)
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We consider the problem of finding the unconstrained global minimum of a realvalued polynomial p(x) : R R, as well as the global minimum of p(x), in a compact set K defined by polynomial inequalities. It is shown that this problem reduces to solving an (often finite) sequence of convex linear matrix inequality (LMI) problems. A notion of KarushKuhnTucker polynomials is introduced in a global optimality condition. Some illustrative examples are provided. Key words. global optimization, theory of moments and positive polynomials, semidefinite programming AMS subject classifications. 90C22, 90C25 PII. S1052623400366802 1.
A comparison of the SheraliAdams, LovászSchrijver and Lasserre relaxations for 01 programming
 Mathematics of Operations Research
, 2001
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Sums of Squares and Semidefinite Programming Relaxations for Polynomial Optimization Problems with Structured Sparsity
 SIAM Journal on Optimization
, 2006
"... Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of ..."
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Cited by 72 (24 self)
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Abstract. Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) relaxations are obtained. Numerical results from various test problems are included to show the improved performance of the SOS and SDP relaxations. Key words.
Optimal inequalities in probability theory: A convex optimization approach
 SIAM Journal of Optimization
"... Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provi ..."
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Cited by 66 (10 self)
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Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provide optimal bounds on P (X ∈ S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and Ω are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite optimization problems of polynomial size in n, for fixed k. We characterize the complexity of the problem of deriving tight moment inequalities. We show that it is NPhard to find tight bounds for k ≥ 4 and Ω = Rn and for k ≥ 2 and Ω = Rn +, when the data in the problem is rational. For k =1andΩ=Rn + we show that we can find tight upper bounds by solving n convex optimization problems when the set S is convex, and we provide a polynomial time algorithm when S and Ω are unions of convex sets, over which linear functions can be optimized efficiently. For the case k =2andΩ=Rn, we present an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently. Key words. optimization probability bounds, Chebyshev inequalities, semidefinite optimization, convex
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 48 (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.
Optimization of polynomials on compact semialgebraic sets
 SIAM J. Optim
"... Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter conve ..."
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Cited by 43 (5 self)
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Abstract. A basic closed semialgebraic subset S of R n is defined by simultaneous polynomial inequalities g1 ≥ 0,..., gm ≥ 0. We give a short introduction to Lasserre’s method for minimizing a polynomial f on a compact set S of this kind. It consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs. We give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ∗ of f on S which is constructive and elementary. In the case where f possesses a unique minimizer x ∗ , we prove that every sequence of “nearly ” optimal solutions of the successive relaxations gives rise to a sequence of points in R n converging to x ∗. 1. Introduction to Lasserre’s method Throughout the paper, we suppose 1 ≤ n ∈ N and abbreviate (X1,..., Xn) by ¯X. We let R [ ¯ X] denote the ring of real polynomials in n indeterminates. Suppose we are given a so called basic closed semialgebraic set, i.e., a set S: = {x ∈ R n  g1(x) ≥ 0,..., gm(x) ≥ 0}
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 37 (7 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.
Feedback Control of Quantum State Reduction
, 2004
"... Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for th ..."
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Cited by 32 (2 self)
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Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
Minimizing polynomials via sum of squares over the gradient ideal
 Math. Program
"... 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 ..."
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Cited by 30 (12 self)
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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 ideal is radical or the polynomial is strictly positive on the gradient variety. This opens up the possibility of solving previously intractable polynomial optimization problems. The related problem of constrained minimization is also considered, and numerical examples are discussed. Experiments show that our method using the gradient variety outperforms prior SOS methods.
A Representation Theorem for Certain Partially Ordered Commutative Rings
"... Let A be a commutative ring with 1, let P A be a preordering of higher level (i.e. 0; 1 2 P , 1 62 P , P + P P , P P P and A 2n P for some n 2 N) and let M A be an archimedean Pmodule (i.e. 1 2 M , 1 62 M , M +M M , P M M and 8 a 2 A 9n 2 N n a 2 M ). We endow X(M) := f' 2 Hom(A;R) j ..."
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Cited by 28 (1 self)
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Let A be a commutative ring with 1, let P A be a preordering of higher level (i.e. 0; 1 2 P , 1 62 P , P + P P , P P P and A 2n P for some n 2 N) and let M A be an archimedean Pmodule (i.e. 1 2 M , 1 62 M , M +M M , P M M and 8 a 2 A 9n 2 N n a 2 M ). We endow X(M) := f' 2 Hom(A;R) j '(M) R+ g with the weak topology with respect to all mappings ba : X(M) ! R, ba(') := '(a) and consider the representation M : A ! C(X(M);R), a 7! ba. We nd that X(M) is a nonempty compact Hausdor space. Further we prove that 1 M (C + (X(M);R)) = fa 2 A j 8n 2 N 9 k 2 N k(1 + na) 2 Mg and ker( M ) = fa 2 A j 8n 2 N 9 k 2 N k(1 na) 2 Mg. (By C + (X(M);R) we denote the set of all continuous functions which do not take negative values.) Introduction Kadison developed in [K] a representation theory for topological algebraic structures. One of his results is a representation theorem for archimedean partially ordered Banach algebras. This theorem was extended by...