### Abstract

We present a fast distributed gradient method for a convex optimization problem with linear inequalities, with a particular focus on the network utility maximization (NUM) problem. Most existing works in the literature use (sub)gradient methods for solving the dual of this problem which can be implemented in a distributed manner. However, these (sub)gradient methods suffer from an O(1/√k) rate of convergence (where k is the number of iterations). In this paper, we assume that the utility functions are strongly concave, an assumption satisfied by most standard utility functions considered in the literature. We develop a completely distributed fast gradient method for solving the dual of the NUM problem. We show that the generated primal sequences converge to the unique optimal solution of the NUM problem at rate O(1/k).

Original language | English (US) |
---|---|

Article number | 6756941 |

Pages (from-to) | 64-73 |

Number of pages | 10 |

Journal | IEEE Transactions on Control of Network Systems |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2014 |

Externally published | Yes |

### Fingerprint

### Keywords

- convex functions
- Gradient methods
- network utility maximization

### ASJC Scopus subject areas

- Control and Systems Engineering
- Signal Processing
- Computer Networks and Communications
- Control and Optimization

### Cite this

*IEEE Transactions on Control of Network Systems*,

*1*(1), 64-73. [6756941]. https://doi.org/10.1109/TCNS.2014.2309751

**An O(1/k) gradient method for network resource allocation problems.** / Beck, Amir; Nedich, Angelia; Ozdaglar, Asuman; Teboulle, Marc.

Research output: Contribution to journal › Article

*IEEE Transactions on Control of Network Systems*, vol. 1, no. 1, 6756941, pp. 64-73. https://doi.org/10.1109/TCNS.2014.2309751

}

TY - JOUR

T1 - An O(1/k) gradient method for network resource allocation problems

AU - Beck, Amir

AU - Nedich, Angelia

AU - Ozdaglar, Asuman

AU - Teboulle, Marc

PY - 2014/3/1

Y1 - 2014/3/1

N2 - We present a fast distributed gradient method for a convex optimization problem with linear inequalities, with a particular focus on the network utility maximization (NUM) problem. Most existing works in the literature use (sub)gradient methods for solving the dual of this problem which can be implemented in a distributed manner. However, these (sub)gradient methods suffer from an O(1/√k) rate of convergence (where k is the number of iterations). In this paper, we assume that the utility functions are strongly concave, an assumption satisfied by most standard utility functions considered in the literature. We develop a completely distributed fast gradient method for solving the dual of the NUM problem. We show that the generated primal sequences converge to the unique optimal solution of the NUM problem at rate O(1/k).

AB - We present a fast distributed gradient method for a convex optimization problem with linear inequalities, with a particular focus on the network utility maximization (NUM) problem. Most existing works in the literature use (sub)gradient methods for solving the dual of this problem which can be implemented in a distributed manner. However, these (sub)gradient methods suffer from an O(1/√k) rate of convergence (where k is the number of iterations). In this paper, we assume that the utility functions are strongly concave, an assumption satisfied by most standard utility functions considered in the literature. We develop a completely distributed fast gradient method for solving the dual of the NUM problem. We show that the generated primal sequences converge to the unique optimal solution of the NUM problem at rate O(1/k).

KW - convex functions

KW - Gradient methods

KW - network utility maximization

UR - http://www.scopus.com/inward/record.url?scp=84930245436&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84930245436&partnerID=8YFLogxK

U2 - 10.1109/TCNS.2014.2309751

DO - 10.1109/TCNS.2014.2309751

M3 - Article

AN - SCOPUS:84930245436

VL - 1

SP - 64

EP - 73

JO - IEEE Transactions on Control of Network Systems

JF - IEEE Transactions on Control of Network Systems

SN - 2325-5870

IS - 1

M1 - 6756941

ER -