A comparison of eigenvalue-based algorithms and the generalized Lanczos trust-region algorithm for Solving the trust-region subproblem Article Swipe
Zhongxiao Jia
,
Fa Wang
·
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2102.09693
YOU?
·
· 2021
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2102.09693
Solving the trust-region subproblem (TRS) plays a key role in numerical optimization and many other applications. Based on a fundamental result that the solution of TRS of size $n$ is mathematically equivalent to finding the rightmost eigenpair of a certain matrix pair of size $2n$, eigenvalue-based methods are promising due to their simplicity. For $n$ large, the implicitly restarted Arnoldi (IRA) and refined Arnoldi (IRRA) algorithms are well suited for this eigenproblem. For a reasonable comparison of overall efficiency of the algorithms for solving TRS directly and eigenvalue-based algorithms, a vital premise is that the two kinds of algorithms must compute the approximate solutions of TRS with (almost) the same accuracy, but such premise has been ignored in the literature. To this end, we establish close relationships between the two kinds of residual norms, so that, given a stopping tolerance for IRA and IRRA, we are able to determine a reliable one that GLTR should use so as to ensure that GLTR and IRA, IRRA deliver the converged approximate solutions with similar accuracy. We also make a convergence analysis on the residual norms by the Generalized Lanczos Trust-Region (GLTR) algorithm for solving TRS directly, the Arnoldi method and the refined Arnoldi method for the equivalent eigenproblem. A number of numerical experiments are reported to illustrate that IRA and IRRA are competitive with GLTR and IRRA outperforms IRA.
Related Topics
Concepts
Premise
Residual
Lanczos resampling
Algorithm
Eigenvalues and eigenvectors
Mathematics
Convergence (economics)
Trust region
Generalized minimal residual method
Arnoldi iteration
Lanczos algorithm
Computer science
Applied mathematics
Mathematical optimization
Linguistics
Physics
Economic growth
Computer security
RADIUS
Economics
Philosophy
Quantum mechanics
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2102.09693
- https://arxiv.org/pdf/2102.09693
- OA Status
- green
- References
- 32
- Related Works
- 10
- OpenAlex ID
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All OpenAlex metadata
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https://openalex.org/W3129699271Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2102.09693Digital Object Identifier
- Title
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A comparison of eigenvalue-based algorithms and the generalized Lanczos trust-region algorithm for Solving the trust-region subproblemWork title
- Type
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2021Year of publication
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2021-02-19Full publication date if available
- Authors
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Zhongxiao Jia, Fa WangList of authors in order
- Landing page
-
https://arxiv.org/abs/2102.09693Publisher landing page
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https://arxiv.org/pdf/2102.09693Direct link to full text PDF
- Open access
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2102.09693Direct OA link when available
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Premise, Residual, Lanczos resampling, Algorithm, Eigenvalues and eigenvectors, Mathematics, Convergence (economics), Trust region, Generalized minimal residual method, Arnoldi iteration, Lanczos algorithm, Computer science, Applied mathematics, Mathematical optimization, Linguistics, Physics, Economic growth, Computer security, RADIUS, Economics, Philosophy, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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0Total citation count in OpenAlex
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32Number of works referenced by this work
- Related works (count)
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.make | 175 |
| abstract_inverted_index.many | 13 |
| abstract_inverted_index.must | 99 |
| abstract_inverted_index.pair | 41 |
| abstract_inverted_index.role | 8 |
| abstract_inverted_index.same | 109 |
| abstract_inverted_index.size | 27, 43 |
| abstract_inverted_index.such | 112 |
| abstract_inverted_index.that | 21, 93, 152, 160, 215 |
| abstract_inverted_index.this | 70, 121 |
| abstract_inverted_index.well | 67 |
| abstract_inverted_index.with | 106, 170, 221 |
| abstract_inverted_index.$2n$, | 44 |
| abstract_inverted_index.(IRA) | 60 |
| abstract_inverted_index.(TRS) | 4 |
| abstract_inverted_index.Based | 16 |
| abstract_inverted_index.IRRA, | 143 |
| abstract_inverted_index.close | 125 |
| abstract_inverted_index.given | 136 |
| abstract_inverted_index.kinds | 96, 130 |
| abstract_inverted_index.norms | 182 |
| abstract_inverted_index.other | 14 |
| abstract_inverted_index.plays | 5 |
| abstract_inverted_index.that, | 135 |
| abstract_inverted_index.their | 51 |
| abstract_inverted_index.vital | 90 |
| abstract_inverted_index.(GLTR) | 188 |
| abstract_inverted_index.(IRRA) | 64 |
| abstract_inverted_index.ensure | 159 |
| abstract_inverted_index.large, | 55 |
| abstract_inverted_index.matrix | 40 |
| abstract_inverted_index.method | 196, 201 |
| abstract_inverted_index.norms, | 133 |
| abstract_inverted_index.number | 207 |
| abstract_inverted_index.result | 20 |
| abstract_inverted_index.should | 154 |
| abstract_inverted_index.suited | 68 |
| abstract_inverted_index.Arnoldi | 59, 63, 195, 200 |
| abstract_inverted_index.Lanczos | 186 |
| abstract_inverted_index.Solving | 0 |
| abstract_inverted_index.between | 127 |
| abstract_inverted_index.certain | 39 |
| abstract_inverted_index.compute | 100 |
| abstract_inverted_index.deliver | 165 |
| abstract_inverted_index.finding | 33 |
| abstract_inverted_index.ignored | 116 |
| abstract_inverted_index.methods | 46 |
| abstract_inverted_index.overall | 77 |
| abstract_inverted_index.premise | 91, 113 |
| abstract_inverted_index.refined | 62, 199 |
| abstract_inverted_index.similar | 171 |
| abstract_inverted_index.solving | 83, 191 |
| abstract_inverted_index.(almost) | 107 |
| abstract_inverted_index.analysis | 178 |
| abstract_inverted_index.directly | 85 |
| abstract_inverted_index.reliable | 150 |
| abstract_inverted_index.reported | 212 |
| abstract_inverted_index.residual | 132, 181 |
| abstract_inverted_index.solution | 23 |
| abstract_inverted_index.stopping | 138 |
| abstract_inverted_index.accuracy, | 110 |
| abstract_inverted_index.accuracy. | 172 |
| abstract_inverted_index.algorithm | 189 |
| abstract_inverted_index.converged | 167 |
| abstract_inverted_index.determine | 148 |
| abstract_inverted_index.directly, | 193 |
| abstract_inverted_index.eigenpair | 36 |
| abstract_inverted_index.establish | 124 |
| abstract_inverted_index.numerical | 10, 209 |
| abstract_inverted_index.promising | 48 |
| abstract_inverted_index.restarted | 58 |
| abstract_inverted_index.rightmost | 35 |
| abstract_inverted_index.solutions | 103, 169 |
| abstract_inverted_index.tolerance | 139 |
| abstract_inverted_index.algorithms | 65, 81, 98 |
| abstract_inverted_index.comparison | 75 |
| abstract_inverted_index.efficiency | 78 |
| abstract_inverted_index.equivalent | 31, 204 |
| abstract_inverted_index.illustrate | 214 |
| abstract_inverted_index.implicitly | 57 |
| abstract_inverted_index.reasonable | 74 |
| abstract_inverted_index.subproblem | 3 |
| abstract_inverted_index.Generalized | 185 |
| abstract_inverted_index.algorithms, | 88 |
| abstract_inverted_index.approximate | 102, 168 |
| abstract_inverted_index.competitive | 220 |
| abstract_inverted_index.convergence | 177 |
| abstract_inverted_index.experiments | 210 |
| abstract_inverted_index.fundamental | 19 |
| abstract_inverted_index.literature. | 119 |
| abstract_inverted_index.outperforms | 225 |
| abstract_inverted_index.simplicity. | 52 |
| abstract_inverted_index.Trust-Region | 187 |
| abstract_inverted_index.optimization | 11 |
| abstract_inverted_index.trust-region | 2 |
| abstract_inverted_index.applications. | 15 |
| abstract_inverted_index.eigenproblem. | 71, 205 |
| abstract_inverted_index.relationships | 126 |
| abstract_inverted_index.mathematically | 30 |
| abstract_inverted_index.eigenvalue-based | 45, 87 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |