# A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality

## Abstract

In this paper, we study the extended trust region subproblem (eTRS) in which the trust region intersects the Euclidean ball with a single linear inequality constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter linear eigenvalue problem, we state a necessary and sufficient condition for its strong duality in terms of an optimal solution of a linearly constrained bivariate concave maximization problem. This results in an efficient algorithm for solving eTRS of large size whenever the strong duality is detected. Finally, some numerical experiments are given to show the effectiveness of the proposed method.

- Authors:

- University of Guilan, Faculty of Mathematical Sciences (Iran, Islamic Republic of)

- Publication Date:

- OSTI Identifier:
- 22769384

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 1; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; EIGENVALUES; EUCLIDEAN SPACE; LAGRANGIAN FUNCTION; MATHEMATICAL SOLUTIONS; RELAXATION

### Citation Formats

```
Salahi, M., E-mail: salahim@guilan.ac.ir, and Taati, A..
```*A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality*. United States: N. p., 2018.
Web. doi:10.1007/S40314-016-0347-3.

```
Salahi, M., E-mail: salahim@guilan.ac.ir, & Taati, A..
```*A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality*. United States. doi:10.1007/S40314-016-0347-3.

```
Salahi, M., E-mail: salahim@guilan.ac.ir, and Taati, A.. Thu .
"A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality". United States. doi:10.1007/S40314-016-0347-3.
```

```
@article{osti_22769384,
```

title = {A fast eigenvalue approach for solving the trust region subproblem with an additional linear inequality},

author = {Salahi, M., E-mail: salahim@guilan.ac.ir and Taati, A.},

abstractNote = {In this paper, we study the extended trust region subproblem (eTRS) in which the trust region intersects the Euclidean ball with a single linear inequality constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter linear eigenvalue problem, we state a necessary and sufficient condition for its strong duality in terms of an optimal solution of a linearly constrained bivariate concave maximization problem. This results in an efficient algorithm for solving eTRS of large size whenever the strong duality is detected. Finally, some numerical experiments are given to show the effectiveness of the proposed method.},

doi = {10.1007/S40314-016-0347-3},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 1,

volume = 37,

place = {United States},

year = {2018},

month = {3}

}

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