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# Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints

## Abstract

We propose two limited-memory BFGS (L-BFGS) trust-region methods for large-scale optimization with linear equality constraints. Here, the methods are intended for problems where the number of equality constraints is small. By exploiting the structure of the quasi-Newton compact representation, both proposed methods solve the trust-region subproblems nearly exactly, even for large problems. We derive theoretical global convergence results of the proposed algorithms, and compare their numerical effectiveness and performance on a variety of large-scale problems.

- Authors:

- Argonne National Lab. (ANL), Lemont, IL (United States)
- Univ. of California, Merced, CA (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1575872

- Report Number(s):
- LLNL-JRNL-755231

Journal ID: ISSN 0926-6003; 942036

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computational Optimization and Applications

- Additional Journal Information:
- Journal Volume: 74; Journal Issue: 3; Journal ID: ISSN 0926-6003

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Linear equality constraints; Quasi-Newton; L-BFGS; Trust-region algorithm; Compact representation; Eigendecomposition; Shape-changing norm

### Citation Formats

```
Brust, Johannes J., Marcia, Roummel F., and Petra, Cosmin G. Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints. United States: N. p., 2019.
Web. doi:10.1007/s10589-019-00127-4.
```

```
Brust, Johannes J., Marcia, Roummel F., & Petra, Cosmin G. Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints. United States. doi:10.1007/s10589-019-00127-4.
```

```
Brust, Johannes J., Marcia, Roummel F., and Petra, Cosmin G. Thu .
"Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints". United States. doi:10.1007/s10589-019-00127-4.
```

```
@article{osti_1575872,
```

title = {Large-scale quasi-Newton trust-region methods with low-dimensional linear equality constraints},

author = {Brust, Johannes J. and Marcia, Roummel F. and Petra, Cosmin G.},

abstractNote = {We propose two limited-memory BFGS (L-BFGS) trust-region methods for large-scale optimization with linear equality constraints. Here, the methods are intended for problems where the number of equality constraints is small. By exploiting the structure of the quasi-Newton compact representation, both proposed methods solve the trust-region subproblems nearly exactly, even for large problems. We derive theoretical global convergence results of the proposed algorithms, and compare their numerical effectiveness and performance on a variety of large-scale problems.},

doi = {10.1007/s10589-019-00127-4},

journal = {Computational Optimization and Applications},

number = 3,

volume = 74,

place = {United States},

year = {2019},

month = {9}

}

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