### Abstract

In this paper, we introduce a class of minimization problems whose objective function is the composite of an isotonic function and finitely many ratios. Examples of an isotonic function include the max-operator, summation, and many others, so it implies a much wider class than the classical fractional programming containing the minimax fractional program as well as the sum-of-ratios problem. Our intention is to develop a generic "Dinkelbach-like" algorithm suitable for all fractional programs of this type. Such an attempt has never been successful before, including an early effort for the sum-of-ratios problem. The difficulty is now overcome by extending the cutting plane method of Barros and Frenk (in J. Optim. Theory Appl. 87:103-120, 1995). Based on different isotonic operators, various cuts can be created respectively to either render a Dinkelbach-like approach for the sum-of-ratios problem or recover the classical Dinkelbach-type algorithm for the min-max fractional programming.

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

Pages (from-to) | 581-601 |

Number of pages | 21 |

Journal | Journal of Optimization Theory and Applications |

Volume | 146 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 May 6 |

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### All Science Journal Classification (ASJC) codes

- Applied Mathematics
- Control and Optimization
- Management Science and Operations Research

### Cite this

*Journal of Optimization Theory and Applications*,

*146*(3), 581-601. https://doi.org/10.1007/s10957-010-9684-3

}

*Journal of Optimization Theory and Applications*, vol. 146, no. 3, pp. 581-601. https://doi.org/10.1007/s10957-010-9684-3

**Minimization of Isotonic Functions Composed of Fractions.** / Lin, J. Y.; Schaible, S.; Sheu, Ruey-Lin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Minimization of Isotonic Functions Composed of Fractions

AU - Lin, J. Y.

AU - Schaible, S.

AU - Sheu, Ruey-Lin

PY - 2010/5/6

Y1 - 2010/5/6

N2 - In this paper, we introduce a class of minimization problems whose objective function is the composite of an isotonic function and finitely many ratios. Examples of an isotonic function include the max-operator, summation, and many others, so it implies a much wider class than the classical fractional programming containing the minimax fractional program as well as the sum-of-ratios problem. Our intention is to develop a generic "Dinkelbach-like" algorithm suitable for all fractional programs of this type. Such an attempt has never been successful before, including an early effort for the sum-of-ratios problem. The difficulty is now overcome by extending the cutting plane method of Barros and Frenk (in J. Optim. Theory Appl. 87:103-120, 1995). Based on different isotonic operators, various cuts can be created respectively to either render a Dinkelbach-like approach for the sum-of-ratios problem or recover the classical Dinkelbach-type algorithm for the min-max fractional programming.

AB - In this paper, we introduce a class of minimization problems whose objective function is the composite of an isotonic function and finitely many ratios. Examples of an isotonic function include the max-operator, summation, and many others, so it implies a much wider class than the classical fractional programming containing the minimax fractional program as well as the sum-of-ratios problem. Our intention is to develop a generic "Dinkelbach-like" algorithm suitable for all fractional programs of this type. Such an attempt has never been successful before, including an early effort for the sum-of-ratios problem. The difficulty is now overcome by extending the cutting plane method of Barros and Frenk (in J. Optim. Theory Appl. 87:103-120, 1995). Based on different isotonic operators, various cuts can be created respectively to either render a Dinkelbach-like approach for the sum-of-ratios problem or recover the classical Dinkelbach-type algorithm for the min-max fractional programming.

UR - http://www.scopus.com/inward/record.url?scp=77956440367&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956440367&partnerID=8YFLogxK

U2 - 10.1007/s10957-010-9684-3

DO - 10.1007/s10957-010-9684-3

M3 - Article

VL - 146

SP - 581

EP - 601

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

IS - 3

ER -