## Abstract

We present an algorithm for the variational inequality problem on convex sets with nonempty interior. The use of Bregman functions whose zone is the convex set allows for the generation of a sequence contained in the interior, without taking explicitly into account the constraints which define the convex set. We establish full convergence to a solution with minimal conditions upon the monotone operator F, weaker than strong monotonicity or Lipschitz continuity, for instance, and including cases where the solution needs not be unique. We apply our algorithm to several relevant classes of convex sets, including orthants, boxes, polyhedra and balls, for which Bregman functions are presented which give rise to explicit iteration formulae, up to the determination of two scalar stepsizes, which can be found through finite search procedures.

Original language | English |
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Pages (from-to) | 373-400 |

Number of pages | 28 |

Journal | Mathematical Programming |

Volume | 81 |

Issue number | 3 |

DOIs | |

State | Published - 1 May 1998 |

### Bibliographical note

Funding Information:The work of Y. Censor was partially supported by NIH grant HL-28438 while visiting the Medical Image Processing Group (MIPG), Department of Radiology, and by NSF grant CCR-91-04042 while visiting the HERMES Laboratory for Financial Modeling and Simulation, The Wharton School, both at the University of Pennsylvania, Philadelphia, PA, USA. The research of A. Iusem was partially supported by CNPq grant no. 301280/86. The work of S.A. Zenios was funded in part by NSF grant CCR-91-04042.

## Keywords

- Convex programming
- Generalized distances
- Monotone operators
- Paramonotone operators
- Variational inequalities

## ASJC Scopus subject areas

- Software
- Mathematics (all)