### Abstract

We consider a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. On one hand, (P) can be solved under some technical conditions by the Dinkelbach iterative method [W. Dinkelbach, On nonlinear fractional programming, Manag. Sci. 13 (1967), pp. 492–498] which has dominated the development of the area for nearly half a century. On the other hand, some special case of (P), typically the one in Beck and Teboulle [A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid, Math. Program. Ser. A 118 (2009), pp. 13–35], could be directly solved via an exact semi-definite reformulation, rather than iteratively. In this paper, by a recent breakthrough of Xia et al. [S-Lemma with equality and its applications. Available at http://arxiv.org/abs/1403.2816] on the S-lemma with equality, we propose to analyse (P) with three cases and show that each of them admits an exact SDP relaxation. As a result, (P) can be completely solved in polynomial time without any condition. Finally, the paper is presented with many interesting examples to illustrate the idea of our approach and to visualize the structure of the problem.

Original language | English |
---|---|

Pages (from-to) | 701-719 |

Number of pages | 19 |

Journal | Optimization Methods and Software |

Volume | 31 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 Jul 3 |

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### All Science Journal Classification (ASJC) codes

- Software
- Control and Optimization
- Applied Mathematics

### Cite this

*Optimization Methods and Software*,

*31*(4), 701-719. https://doi.org/10.1080/10556788.2015.1029575

}

*Optimization Methods and Software*, vol. 31, no. 4, pp. 701-719. https://doi.org/10.1080/10556788.2015.1029575

**An SDP approach for quadratic fractional problems with a two-sided quadratic constraint.** / Nguyen, Van Bong; Sheu, Ruey-Lin; Xia, Yong.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An SDP approach for quadratic fractional problems with a two-sided quadratic constraint

AU - Nguyen, Van Bong

AU - Sheu, Ruey-Lin

AU - Xia, Yong

PY - 2016/7/3

Y1 - 2016/7/3

N2 - We consider a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. On one hand, (P) can be solved under some technical conditions by the Dinkelbach iterative method [W. Dinkelbach, On nonlinear fractional programming, Manag. Sci. 13 (1967), pp. 492–498] which has dominated the development of the area for nearly half a century. On the other hand, some special case of (P), typically the one in Beck and Teboulle [A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid, Math. Program. Ser. A 118 (2009), pp. 13–35], could be directly solved via an exact semi-definite reformulation, rather than iteratively. In this paper, by a recent breakthrough of Xia et al. [S-Lemma with equality and its applications. Available at http://arxiv.org/abs/1403.2816] on the S-lemma with equality, we propose to analyse (P) with three cases and show that each of them admits an exact SDP relaxation. As a result, (P) can be completely solved in polynomial time without any condition. Finally, the paper is presented with many interesting examples to illustrate the idea of our approach and to visualize the structure of the problem.

AB - We consider a fractional programming problem (P) which minimizes a ratio of quadratic functions subject to a two-sided quadratic constraint. On one hand, (P) can be solved under some technical conditions by the Dinkelbach iterative method [W. Dinkelbach, On nonlinear fractional programming, Manag. Sci. 13 (1967), pp. 492–498] which has dominated the development of the area for nearly half a century. On the other hand, some special case of (P), typically the one in Beck and Teboulle [A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid, Math. Program. Ser. A 118 (2009), pp. 13–35], could be directly solved via an exact semi-definite reformulation, rather than iteratively. In this paper, by a recent breakthrough of Xia et al. [S-Lemma with equality and its applications. Available at http://arxiv.org/abs/1403.2816] on the S-lemma with equality, we propose to analyse (P) with three cases and show that each of them admits an exact SDP relaxation. As a result, (P) can be completely solved in polynomial time without any condition. Finally, the paper is presented with many interesting examples to illustrate the idea of our approach and to visualize the structure of the problem.

UR - http://www.scopus.com/inward/record.url?scp=84928386408&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84928386408&partnerID=8YFLogxK

U2 - 10.1080/10556788.2015.1029575

DO - 10.1080/10556788.2015.1029575

M3 - Article

AN - SCOPUS:84928386408

VL - 31

SP - 701

EP - 719

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 4

ER -