### Abstract

We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to non-negative constraints x ≥ 0. The algorithms are based on an entropy-like second-order homogeneous distance function. Under the assumption that the global minimizer set is nonempty and bounded, we prove the full convergence of the sequence generated by the algorithms, and furthermore, obtain two important convergence results through imposing certain conditions on the proximal parameters. One is that the sequence generated will converge to a stationary point if the proximal parameters are bounded and the problem is continuously differentiable, and the other is that the sequence generated will converge to a solution of the problem if the proximal parameters approach to zero. Numerical experiments are done for a class of quasi-convex optimization problems where the function f(x) is a composition of a quadratic convex function from R^{n} to R and a continuously differentiable increasing function from R to R, and computational results indicate that these algorithms are very promising in finding a global optimal solution to these quasi-convex problems.

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

Pages (from-to) | 555-575 |

Number of pages | 21 |

Journal | Journal of Global Optimization |

Volume | 39 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2007 Dec 1 |

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### Keywords

- Entropy-like distance
- Proximal-like method
- Quasi-convex programming

### ASJC Scopus subject areas

- Applied Mathematics
- Control and Optimization
- Management Science and Operations Research
- Global and Planetary Change

### Cite this

**Entropy-like proximal algorithms based on a second-order homogeneous distance function for quasi-convex programming.** / Pan, Shaohua; Chen, Jein-Shan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Entropy-like proximal algorithms based on a second-order homogeneous distance function for quasi-convex programming

AU - Pan, Shaohua

AU - Chen, Jein-Shan

PY - 2007/12/1

Y1 - 2007/12/1

N2 - We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to non-negative constraints x ≥ 0. The algorithms are based on an entropy-like second-order homogeneous distance function. Under the assumption that the global minimizer set is nonempty and bounded, we prove the full convergence of the sequence generated by the algorithms, and furthermore, obtain two important convergence results through imposing certain conditions on the proximal parameters. One is that the sequence generated will converge to a stationary point if the proximal parameters are bounded and the problem is continuously differentiable, and the other is that the sequence generated will converge to a solution of the problem if the proximal parameters approach to zero. Numerical experiments are done for a class of quasi-convex optimization problems where the function f(x) is a composition of a quadratic convex function from Rn to R and a continuously differentiable increasing function from R to R, and computational results indicate that these algorithms are very promising in finding a global optimal solution to these quasi-convex problems.

AB - We consider two classes of proximal-like algorithms for minimizing a proper lower semicontinuous quasi-convex function f(x) subject to non-negative constraints x ≥ 0. The algorithms are based on an entropy-like second-order homogeneous distance function. Under the assumption that the global minimizer set is nonempty and bounded, we prove the full convergence of the sequence generated by the algorithms, and furthermore, obtain two important convergence results through imposing certain conditions on the proximal parameters. One is that the sequence generated will converge to a stationary point if the proximal parameters are bounded and the problem is continuously differentiable, and the other is that the sequence generated will converge to a solution of the problem if the proximal parameters approach to zero. Numerical experiments are done for a class of quasi-convex optimization problems where the function f(x) is a composition of a quadratic convex function from Rn to R and a continuously differentiable increasing function from R to R, and computational results indicate that these algorithms are very promising in finding a global optimal solution to these quasi-convex problems.

KW - Entropy-like distance

KW - Proximal-like method

KW - Quasi-convex programming

UR - http://www.scopus.com/inward/record.url?scp=35748949858&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35748949858&partnerID=8YFLogxK

U2 - 10.1007/s10898-007-9156-y

DO - 10.1007/s10898-007-9156-y

M3 - Article

AN - SCOPUS:35748949858

VL - 39

SP - 555

EP - 575

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

IS - 4

ER -