An algorithm for quadratic optimization with one quadratic constraint and bounds on the variables

Fehmers, GC; Kamp, Leon Lpj; Sluijter, FW Frans

doi:10.1088/0266-5611/14/4/009

Cited by 19 publications

(29 citation statements)

References 7 publications

(7 reference statements)

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“…To begin with, for definite feasible QCQP there always exists a global solution x * . [24]) For a definite feasible QCQP (2), there exist x * ∈ R n , λ * ≥ 0 such that the conditions (8) hold.…”

Section: Definite Feasible Qcqp: Strictly Feasible and Definitementioning

confidence: 99%

“…Similarly, if A 0, takingλ = 0 shows the pencil is definite, so such QCQP is definite feasible as long as it is strictly feasible. Indeed a number of studies focus on such cases [8,9].…”

Section: Theorem 2 (Morémentioning

confidence: 99%

“…In view of the third condition in (8), the main goal is to find λ * such that γ (λ * ) = 0. This argument apparently dismisses the cases where A + λ * B is singular (then x(λ) is not well-defined) or λ * = 0 (then γ (λ) = 0 is unnecessary).…”

Section: Preparationsmentioning

confidence: 99%

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Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Adachi

Nakatsukasa

2017

Math. Program.

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A nonconvex quadratically constrained quadratic programming (QCQP) with one constraint is usually solved via a dual SDP problem, or Moré's algorithm based on iteratively solving linear systems. In this work we introduce an algorithm for QCQP that requires finding just one eigenpair of a generalized eigenvalue problem, and involves no outer iterations other than the (usually black-box) iterations for computing the eigenpair. Numerical experiments illustrate the efficiency and accuracy of our algorithm. We also analyze the QCQP solution extensively, including difficult cases, and show that the canonical form of a matrix pair gives a complete classification of the QCQP in terms of boundedness and attainability, and explain how to obtain a global solution whenever it exists.Keywords QCQP · Generalized eigenvalue problem · Canonical form for symmetric matrix pair Mathematics Subject Classification 49M37 · 65K05 · 90C20 · 90C30

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Section: Definite Feasible Qcqp: Strictly Feasible and Definitementioning

confidence: 99%

“…Similarly, if A 0, takingλ = 0 shows the pencil is definite, so such QCQP is definite feasible as long as it is strictly feasible. Indeed a number of studies focus on such cases [8,9].…”

Section: Theorem 2 (Morémentioning

confidence: 99%

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Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Adachi

Nakatsukasa

2017

Math. Program.

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“…30. This algorithm is based on the active set approach from quadratic programming and can be effectively combined with the matrix decompositions presented in the next section.…”

Section: Minimum Fisher Informationmentioning

confidence: 99%

Optimized tomography methods for plasma emissivity reconstruction at the ASDEX Upgrade tokamak

Odstrčil

Pütterich

Odstrčil

et al. 2016

Review of Scientific Instruments

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The soft X-ray (SXR) emission provides valuable insight into processes happening inside of high-temperature plasmas. A standard method for deriving the local emissivity profiles of the plasma from the line-of-sight integrals measured by pinhole cameras is the tomographic inversion. Such an inversion is challenging due to its ill-conditioned nature and because the reconstructed profiles depend not only on the quality of the measurements, but also on the inversion algorithm used. This paper provides a detailed description of several tomography algorithms, which solve the inversion problem of Tikhonov regularization with linear computational complexity in the number of basis functions. The feasibility of combining these methods with the Minimum Fisher Information Regularization is demonstrated, and various statistical methods for the optimal choice of the regularization parameter are investigated with emphasis on their reliability and robustness. Finally, the accuracy and the capability of the methods are demonstrated by reconstructions of experimental SXR profiles, featuring poloidal asymmetric impurity distributions as measured at the ASDEX Upgrade tokamak.

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“…Application of a statistical approach to atmospheric remote sensing can be found in Rodgers (2000). A constrained optimization method -where a smoothness function is used as quadratic constraint -is used successfully by Fehmers et al (1998) in the field of the tomography of the ionosphere. If good a priori information is available these inversion techniques are useful to include such information.…”

Section: Overview Over Inversion Techniquesmentioning

confidence: 99%

Longpath DOAS tomography on a motorway exhaust gas plume: numerical studies and application to data from the BAB II campaign

et al. 2004

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Abstract. This paper presents a procedure for performing and optimizing inversions for DOAS tomography and its application to measurement data. DOAS tomography is a new technique to determine 2-and 3-dimensional concentration fields of air pollutants or other trace gases by combining differential optical absorption spectroscopy (DOAS) with tomographic inversion techniques. Due to the limited amount of measured data, the resulting concentration fields are sensitive to the inversion process. Therefore detailed error estimations are needed to determine the quality of the reconstruction. In this paper we compare different row acting methods for the inversion, present a procedure for optimizing the parameters of the reconstruction process and propose a way to estimate the error-fields by numerical studies. The procedure was applied to data from the motorway emission campaign BAB II. Two dimensional NO 2 cross sections at right angles to the motorway could be reconstructed qualitatively well at different meteorological situations. Additionally we present error fields for the reconstructions which show the problems and skills of the used measurement setup. Numerical studies on an improved setup for future motorway campaigns show, that DOAS tomography is able to produce high quality concentration maps.

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An algorithm for quadratic optimization with one quadratic constraint and bounds on the variables

Cited by 19 publications

References 7 publications

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Eigenvalue-based algorithm and analysis for nonconvex QCQP with one constraint

Optimized tomography methods for plasma emissivity reconstruction at the ASDEX Upgrade tokamak

Longpath DOAS tomography on a motorway exhaust gas plume: numerical studies and application to data from the BAB II campaign

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