Solving the Trust-Region Subproblem By a Generalized Eigenvalue Problem

Adachi, Sadao; Iwata, Satoru; Nakatsukasa, Yuji; Takeda, Akiko

doi:10.1137/16m1058200

Cited by 69 publications

(147 citation statements)

References 35 publications

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“…This algorithm is of an iterative nature, and is not matrix-free (a property desirable for dealing with large-sparse problems). Many other iterative algorithms have been proposed for TRS, but as indicated in the experiments in [1] for TRS, a one-step algorithm based on eigenvalues can significantly outperform such algorithms. Another approach [9,16] is to note the Lagrange dual problem can be expressed equivalently as …”

Section: Introductionmentioning

confidence: 99%

“…1 The running time is O(n 3 ) when the matrices A, B are dense, and it can be significantly faster if the matrices are sparse. The algorithm requires (i) finding aλ ≥ 0 such that A +λB is positive definite, and (ii) computing an extremal eigenpair of an (2n + 1) × (2n + 1) generalized eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

“…We present experiments that illustrate the efficiency and accuracy of our algorithm compared with the SDP-based approach. Our algorithm is based on the framework established in [1,11,17] of formulating the KKT conditions as an eigenvalue problem.…”

Section: Introductionmentioning

confidence: 99%

“…To our knowledge the earliest reference is Gander, Golub and von Matt [11] for TRS. This algorithm was revisited and further developed recently in [1] to illustrate its high efficiency, which was extended in [34] to deal with an additional linear constraint. This paper is largely an outgrowth of [1], extending the scope from TRS to QCQP, relaxing the convexity assumption in the constraint, and fully analyzing the degenerate cases.…”

Section: Introductionmentioning

confidence: 99%

“…This algorithm was revisited and further developed recently in [1] to illustrate its high efficiency, which was extended in [34] to deal with an additional linear constraint. This paper is largely an outgrowth of [1], extending the scope from TRS to QCQP, relaxing the convexity assumption in the constraint, and fully analyzing the degenerate cases. We also note [33], which solves QCQP with an additional ball constraint (generalized CDT problem; GCDT) via a two-parameter eigenvalue problem.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

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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Convergence of the block Lanczos method for the trust‐region subproblem in the hard case

Feng,

2024

Numerical Linear Algebra App

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SummaryThe trust‐region subproblem (TRS) plays a vital role in numerical optimization, numerical linear algebra, and many other applications. It is known that the TRS may have multiple optimal solutions in the hard case. In [Carmon and Duchi, SIAM Rev., 62 (2020), pp. 395–436], a block Lanczos method was proposed to solve the TRS in the hard case, and the convergence of the optimal objective value was established. However, the convergence of the KKT error as well as that of the approximate solution are still unknown for this method. In this paper, we give a more detailed convergence analysis on the block Lanczos method for the TRS in the hard case. First, we improve the convergence speed of the approximate objective value. Second, we derive the speed of the KKT error tends to zero. Third, we establish the convergence of the approximation solution, and show theoretically that the projected TRS obtained from the block Lanczos method will be close to the hard case more and more as the block Lanczos process proceeds. Numerical experiments illustrate the effectiveness of our theoretical results.

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Resolution of protein hydrogen/deuterium exchange by fitting amide exchange probabilities to the peptide isotopic envelopes

Babić

Kazazić

Smith

2019

Rapid Comm Mass Spectrometry

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Rationale: Mass spectra processing in protein hydrogen/deuterium (H/D) exchange has been remarkably improved by the introduction of fitting of the amide exchange probabilities to peptide isotopic envelope intensities (Kan et al., 2013), in contrast to methods in which only the peptide deuterium uptakes (centroid shifts of isotopic envelopes) are used. However, the known implementations are based on the general fitting routines that use only the objective function values. Besides, applicability of more than one fitting method makes necessary their comparative evaluation.Methods: Two fitting methods were considered: the common least squares and the fitting of the multinomial distribution representing the number of deuterium atoms exchanged in the individual peptides. Both methods were applied either directly to the isotopic envelope data or to the deuterium distributions obtained by envelope deconvolution (i.e. de-isotoping).Results: An autonomous Matlab script was prepared, based on the exact expressions for the gradient and Hessian of the objective function, with the trust-region algorithm implemented in the compact analytical form recently made available. The least-squares fitting to the envelope data produced the best results, with the greatest precision and good coverage of exact values by the confidence intervals. The deuterium distributions were sensitive to the (simulated) experimental error whose progression by envelope deconvolution caused degradation in accuracy. The multinomial distribution fitting exhibited poor performance due to inadequate representation of the experimental error and missing of the appropriate weight parameters. Some specific peptide arrangement details were discussed as potential sources of ambiguity in the fitting results. Conclusions:The method of fitting to peptide isotopic envelopes has been improved by using the exact gradient and Hessian of the objective function. The fitting should be repeated with different initial guesses in order to find not only the global minimum, but also the local minima with similar depths which may exist due to eventual ambiguity of the fitting results.

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Solving the Trust-Region Subproblem By a Generalized Eigenvalue Problem

Cited by 69 publications

References 35 publications

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

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

Convergence of the block Lanczos method for the trust‐region subproblem in the hard case

Resolution of protein hydrogen/deuterium exchange by fitting amide exchange probabilities to the peptide isotopic envelopes

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