### Abstract

The focus of this paper is on the set intersection problem for closed convex sets admitting projection operation in a closed form. The objective is to investigate algorithms that would converge (in some sense) if and only if the problem has a solution. To do so, we view the set intersection problem as a stochastic optimization problem of minimizing the "average" residual error of the set collection. We consider a stochastic gradient method as a main tool for investigating the properties of the stochastic optimization problem. We show that the stochastic optimization problem has a solution if and only if the stochastic gradient method is convergent almost surely. We then consider a special case of the method, namely the random projection method, and we analyze its convergence. We show that a solution of the intersection problem exists if and only if the random projection method exhibits certain convergence behavior almost surely. In addition, we provide convergence rate results for the expected residual error.

Original language | English (US) |
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Title of host publication | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |

Pages | 7655-7660 |

Number of pages | 6 |

DOIs | |

State | Published - Dec 1 2010 |

Externally published | Yes |

Event | 2010 49th IEEE Conference on Decision and Control, CDC 2010 - Atlanta, GA, United States Duration: Dec 15 2010 → Dec 17 2010 |

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
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ISSN (Print) | 0191-2216 |

### Other

Other | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |
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Country | United States |

City | Atlanta, GA |

Period | 12/15/10 → 12/17/10 |

### Keywords

- Convex sets
- Intersection problem
- Random projection

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

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## Cite this

*2010 49th IEEE Conference on Decision and Control, CDC 2010*(pp. 7655-7660). [5717734] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2010.5717734