The truncated complex $K$-moment problem

Curto, Raúl E.; Fialkow, Lawrence A.

doi:10.1090/s0002-9947-00-02472-7

Cited by 120 publications

(84 citation statements)

References 18 publications

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“…There are various generalizations of this problem. The truncated moment problem asks for necessary and sufficient conditions when only a subset of T is given [13]. This is relevant for the practical problem at hand, where we only compute a subset of correlators.…”

Section: Relation To the Hamburger Moment Problemmentioning

confidence: 99%

Bootstraps to strings: solving random matrix models with positivity

Lin

2020

J. High Energ. Phys.

101

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A new approach to solving random matrix models directly in the large N limit is developed. First, a set of numerical values for some low-pt correlation functions is guessed. The large N loop equations are then used to generate values of higher-pt correlation functions based on this guess. Then one tests whether these higher-pt functions are consistent with positivity requirements, e.g., tr M 2k ≥ 0. If not, the guessed values are systematically ruled out. In this way, one can constrain the correlation functions of random matrices to a tiny subregion which contains (and perhaps converges to) the true solution. This approach is tested on single and multi-matrix models and handily reproduces known solutions. It also produces strong results for multi-matrix models which are not believed to be solvable. A tantalizing possibility is that this method could be used to search for new critical points, or string worldsheet theories.

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Section: Relation To the Hamburger Moment Problemmentioning

confidence: 99%

Bootstraps to strings: solving random matrix models with positivity

Lin

2020

J. High Energ. Phys.

101

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“…, p. This follows from a result of Curto and Fialkow [3, Theor. 1.6] already used in Lasserre [12] to prove finite convergence of SDP-relaxations for 0-1 programs; see also Laurent [14] for a shorter proof of Theorem 1.6 in [3], and related comments.…”

Section: 2mentioning

confidence: 99%

Convergent SDP-Relaxations for Polynomial Optimization with Sparsity

Lasserre

2006

Lecture Notes in Computer Science

112

242

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We consider a polynomial programming problem P on a compact semi-algebraic set K ⊂ R n , described by m polynomial inequalities g j (X) ≥ 0, and with criterion f ∈ R[X]. We propose a hierarchy of semidefinite relaxations in the spirit those of Waki et al. [9]. In particular, the SDP-relaxation of order r has the following two features: (a) The number of variables is O(κ 2r) where κ = max[κ 1 , κ 2 ] witth κ 1 (resp. κ 2) being the maximum number of variables appearing the monomials of f (resp. appearing in a single constraint g j (X) ≥ 0). (b) The largest size of the LMI's (Linear Matrix Inequalities) is O(κ r). This is to compare with the respective number of variables O(n 2r) and LMI size O(n r) in the original SDP-relaxations defined in [11]. Therefore, great computational savings are expected in case of sparsity in the data {g j , f }, i.e. when κ is small, a frequent case in practical applications of interest. The novelty with respect to [9] is that we prove convergence to the global optimum of P when the sparsity pattern satisfies a condition often encountered in large size problems of practical applications, and known as the running intersection property in graph theory. In such cases, and as a by-product, we also obtain a new representation result for polynomials positive on a basic closed semialgebraic set, a sparse version of Putinar's Positivstellensatz [16].

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“…Consider the set Ω defined by the polynomial . 14More details on the method of moments can be found in [10,17].…”

Section: The Moments Approachmentioning

confidence: 99%

“…We need to verify if we have attained an optimal solution. Based on a rank condition of the moment matrix [17], we can test if we have obtained a global optimum at a relaxation order . Also, based on the same rank condition, we can check whether the optimal solution is unique or if it is a convex combination of several minimizers.…”

Section: Semidefinite Programsmentioning

confidence: 99%

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Optimal Control of Switching Topology Networks

Mojica‐Nava

Salgado

Téllez

et al. 2014

Mathematical Problems in Engineering

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We present an extension of a previously proposed approach based on the method of moments for solving the optimal control problem for a switching system considering now a continuous external input. This method is based on the transformation of a nonlinear, nonconvex optimal control problem, into an equivalent optimal control problem with linear and convex structure, which allows us to obtain an equivalent convex formulation more appropriate to be solved by high-performance numerical computing. Finally, the design of optimal logic-based controllers for networked systems with a dynamic topology is presented as an application of this work.

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The truncated complex $K$-moment problem

Cited by 120 publications

References 18 publications

Bootstraps to strings: solving random matrix models with positivity

Bootstraps to strings: solving random matrix models with positivity

Convergent SDP-Relaxations for Polynomial Optimization with Sparsity

Optimal Control of Switching Topology Networks

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