### Abstract

In this paper, we extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve minmax fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous minmax problems, P(α_{k})=min _{x∈X}(max_{t∈T}[f_{t}(x)- α_{k}g_{t}(x)]), until {α_{k}} converges to the root of P(α)=0. The solution of P(α _{k} ) is used to generate α _{k+1}. However, calculating the exact optimal solution of P(α_{k} ) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P(α _{k} ) incompletely, generating an approximate sequence {α̃_{k}}, while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.

Original language | English |
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Pages (from-to) | 323-343 |

Number of pages | 21 |

Journal | Journal of Optimization Theory and Applications |

Volume | 126 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2005 Aug 1 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*Journal of Optimization Theory and Applications*, vol. 126, no. 2, pp. 323-343. https://doi.org/10.1007/s10957-005-4717-z

**Modified Dinkelbach-type algorithm for generalized fractional programs with infinitely many ratios.** / Lin, J. Y.; Sheu, Ruey-Lin.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Modified Dinkelbach-type algorithm for generalized fractional programs with infinitely many ratios

AU - Lin, J. Y.

AU - Sheu, Ruey-Lin

PY - 2005/8/1

Y1 - 2005/8/1

N2 - In this paper, we extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve minmax fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous minmax problems, P(αk)=min x∈X(maxt∈T[ft(x)- αkgt(x)]), until {αk} converges to the root of P(α)=0. The solution of P(α k ) is used to generate α k+1. However, calculating the exact optimal solution of P(αk ) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P(α k ) incompletely, generating an approximate sequence {α̃k}, while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.

AB - In this paper, we extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve minmax fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous minmax problems, P(αk)=min x∈X(maxt∈T[ft(x)- αkgt(x)]), until {αk} converges to the root of P(α)=0. The solution of P(α k ) is used to generate α k+1. However, calculating the exact optimal solution of P(αk ) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P(α k ) incompletely, generating an approximate sequence {α̃k}, while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.

UR - http://www.scopus.com/inward/record.url?scp=23844477256&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23844477256&partnerID=8YFLogxK

U2 - 10.1007/s10957-005-4717-z

DO - 10.1007/s10957-005-4717-z

M3 - Article

VL - 126

SP - 323

EP - 343

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

IS - 2

ER -