## Abstract

We study the behavior of subgradient projections algorithms for the quasiconvex feasibility problem of finding a point x* ∈ R^{n} that satisfies the inequalities f_{1} (x*) ≤ 0, f_{2} (x*) ≤ 0,..., f_{m} (x*) ≤ 0, where all functions are continuous and quasiconvex. We consider the consistent case when the solution set is nonempty. Since the Fenchel-Moreau subdifferential might be empty we look at different notions of the subdifferential and determine their suitability for our problem. We also determine conditions on the functions, that are needed for convergence of our algorithms. The quasiconvex functions on the left-hand side of the inequalities need not be differentiable but have to satisfy a Lipschitz or a Hölder condition.

Original language | English |
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Pages (from-to) | 34-50 |

Number of pages | 17 |

Journal | Journal of Computational and Applied Mathematics |

Volume | 185 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2006 |

### Bibliographical note

Funding Information:We are grateful to Prof. Adi Ben-Israel from Rutgers Center for Operations Research, Rutgers, The State University of New Jersey, and Prof. Dan Butnariu from the Department of Mathematics at the University of Haifa for their useful comments. The insightful reports of two anonymous referees helped us to improve the first version of this paper. This research was partially supported by Grant No. 522/04 of the Israel Science Foundation (ISF), by NIH Grant No. HL70472, and by Grant No. 2003275 of the United States-Israel Binational Science Foundation (BSF) and partially done at the Center for Computational Mathematics and Scientific Computation (CCMSC) at the University of Haifa.

## Keywords

- Feasibility problem
- Normal cone
- Quasiconvex function
- Subdifferential

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics