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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
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.
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 19 (4 self)
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We revisit two results of Curto and Fialkow on moment matrices. The first result asserts that every sequence...
Solving the Truncated Moment Problem Solves the Full Moment Problem
, 1999
"... It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem. ..."
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Cited by 4 (0 self)
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It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem.
Moment Problems via Semidefinite Programming: Applications in Probability and Finance
, 2000
"... We address the problem of deriving optimal inequalities for E[(X)], for a multivariate random variable X that has a given collection of general moments E[f i (X)] = q i . The goal of this paper is twofold: First, to present the beautiful interplay of optimization and moment inequalities, from a mode ..."
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Cited by 1 (0 self)
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We address the problem of deriving optimal inequalities for E[(X)], for a multivariate random variable X that has a given collection of general moments E[f i (X)] = q i . The goal of this paper is twofold: First, to present the beautiful interplay of optimization and moment inequalities, from a modern perspective, motivated by problems in probability and nance. Second, to characterize the complexity of deriving tight moment inequalities, search for ecient algorithms in a general framework, and, when possible, derive simple closedform bounds. We use semidenite and convex optimization methods to derive optimal bounds on the probability that a multivariate random variable belongs in a given set, when some of the moments of the random variable are known. In the nance context, we use the same approach to nd optimal bounds for option prices with general payo given only moments of underlying asset prices, and without assuming any model for the underlying price dynamics. Department o...
Weighted Moments Based Identification of ContinuousTime Systems
"... Abstract: In this paper we present an algorithm for continuoustime model identification from sample data using the weighted power moments of the output signal of a linear, timeinvariant system. While most of the latest methods used in identification utilize a discretetime model, the moments meth ..."
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Abstract: In this paper we present an algorithm for continuoustime model identification from sample data using the weighted power moments of the output signal of a linear, timeinvariant system. While most of the latest methods used in identification utilize a discretetime model, the moments method is an alternative approach to directly identify a continuoustime model from discretetime data. The method defines a set of relationships between the power series coefficients of a stable transfer function and the power moments of the output signal of this system. Based on these relations, an algorithm for offline parameter identification is developed. The method is applied to identify the parameters of a real experimental platform. KeyWords: offline identification, weighted power moments, sampled data 1