Detecting Global Optimality and Extracting Solutions in GloptiPoly

For an ideal I ⊆ R [x] given by a set of generators, a new semidefinite characterization of its real radical I(V R (I)) is presented, provided it is zero-dimensional (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 V R (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.

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“…[20]). The present paper builds on this approach and shows how it can be applied to finding the real radical of a zero-dimensional ideal.…”

Section: Laas-cnrs Andmentioning

confidence: 99%

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

2007

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“…In a basis of monomials of degree up to 3 there is no Gram matrix X solving problem (48). However, it was observed in [HL05] and later on shown theoretically in [Las06] that the perturbed polynomial…”

Section: Motzkin's Polynomialmentioning

confidence: 99%

Projection Methods in Conic Optimization

Henrion

Malick

2011

International Series in Operations Research &Amp; Management Science

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There exist efficient algorithms to project a point onto the intersection of a convex cone and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques.

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“…see [15], [16]. By solving this minimization problem we obtain two values for the position and the velocity of the target, viz.…”

Section: An Algorithm To Estimate the Position And The Velocity Omentioning

confidence: 99%

Doppler Shift Target Localization

Shames

Bishop

Smith³

et al. 2013

IEEE Trans. Aerosp. Electron. Syst.

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This is an author produced version of a paper published in IEEE Transactions on Aerospace and Electronic Systems.This paper has been peer-reviewed but does not include the final publisher proofcorrections or proceedings pagination.© 2013 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Abstract-This paper outlines the problem of doppler-based target position and velocity estimation using a sensor network. The minimum number of doppler shift measurements at distinct generic sensor positions to have a finite number of solutions, and later, a unique solution for the unknown target position and velocity is stated analytically. Furthermore, we study the same problem where not only doppler shift measurements are collected, but also other types of measurements are available, e.g. bearing or distance to the target from each of the sensors. Later, we study the Cramer-Rao inequality associated with the doppler-shift measurements to a target in a sensor network, and use the Cramer-Rao bound to illustrate some results on optimal placements of the sensors when the goal is to estimate the velocity of the target. Some simulation results are presented in the end.

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Detecting Global Optimality and Extracting Solutions in GloptiPoly

Cited by 254 publications

References 16 publications

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

Projection Methods in Conic Optimization

Doppler Shift Target Localization

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