## Abstract

We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: minxαRn+f(x) s.t. Ax ≥ 1, where f:Rn+ → R+ is a monotone convex function, and A is an m×n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: maxyαRm+ Σ mj=1 yj - g(AT y), where g:Rn+→R+ is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

Original language | English |
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Title of host publication | Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 |

Publisher | IEEE Computer Society |

Pages | 148-157 |

Number of pages | 10 |

ISBN (Electronic) | 9781509039333 |

DOIs | |

State | Published - 14 Dec 2016 |

Externally published | Yes |

Event | 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States Duration: 9 Oct 2016 → 11 Oct 2016 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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Volume | 2016-December |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 |
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Country/Territory | United States |

City | New Brunswick |

Period | 9/10/16 → 11/10/16 |

### Bibliographical note

Publisher Copyright:© 2016 IEEE.

## Keywords

- Convex optimization
- Online algorithm
- Primal-dual algorithm