## Abstract

We focus on the problem of minimizing a finite sum f(x) = \sum\nolimits_{i = 1}^n {{f_i}(x)} of n functions fi, where fi are convex and available only locally to an agent i. The n agents are connected in a directed network \mathcal{G} (V, E). In this article, we present the Directed-Distributed Alternating Direction Method of Multiplier (D-DistADMM) Algorithm, which is an Alternating Direction Method of Multiplier (ADMM) based scheme and utilizes a finite-time approximate consensus method to solve the above optimization problem distributively. At each iteration of the proposed scheme the agents solve their local optimization problem and utilize an approximate consensus protocol to update a local estimate of the global optimization variable. We show that for convex and not-necessarily differentiable objective functions the proposed D-DistADMM method converges at a rate O(1/k), where k is the iteration counter, in terms the difference between the Lagrangian function evaluated at any iteration k of the D-DistADMM algorithm and the optimal solution. We further demonstrate the features of our algorithm by solving a distributed least squares problem.

Original language | English (US) |
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Title of host publication | 2020 59th IEEE Conference on Decision and Control, CDC 2020 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 2992-2997 |

Number of pages | 6 |

ISBN (Electronic) | 9781728174471 |

DOIs | |

State | Published - Dec 14 2020 |

Event | 59th IEEE Conference on Decision and Control, CDC 2020 - Virtual, Jeju Island, Korea, Republic of Duration: Dec 14 2020 → Dec 18 2020 |

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
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Volume | 2020-December |

ISSN (Print) | 0743-1546 |

ISSN (Electronic) | 2576-2370 |

### Conference

Conference | 59th IEEE Conference on Decision and Control, CDC 2020 |
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Country/Territory | Korea, Republic of |

City | Virtual, Jeju Island |

Period | 12/14/20 → 12/18/20 |

### Bibliographical note

Funding Information:This work is supported by Advanced Research Projects Agency-Energy OPEN through the project titled "Rapidly Viable Sustained Grid" via grant no. DE-AR0001016.

Publisher Copyright:

© 2020 IEEE.

## Keywords

- Distributed optimization
- distributed gradient descent
- finite-time consensus
- multi-agent networks