### 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) |
---|---|

Title of host publication | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |

Pages | 7655-7660 |

Number of pages | 6 |

DOIs | |

State | Published - 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 |

### Other

Other | 2010 49th IEEE Conference on Decision and Control, CDC 2010 |
---|---|

Country | United States |

City | Atlanta, GA |

Period | 12/15/10 → 12/17/10 |

### Fingerprint

### Keywords

- Convex sets
- Intersection problem
- Random projection

### ASJC Scopus subject areas

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

### Cite this

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

**Random projection algorithms for convex set intersection problems.** / Nedich, Angelia.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2010 49th IEEE Conference on Decision and Control, CDC 2010.*, 5717734, pp. 7655-7660, 2010 49th IEEE Conference on Decision and Control, CDC 2010, Atlanta, GA, United States, 12/15/10. https://doi.org/10.1109/CDC.2010.5717734

}

TY - GEN

T1 - Random projection algorithms for convex set intersection problems

AU - Nedich, Angelia

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

KW - Convex sets

KW - Intersection problem

KW - Random projection

UR - http://www.scopus.com/inward/record.url?scp=79953141950&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953141950&partnerID=8YFLogxK

U2 - 10.1109/CDC.2010.5717734

DO - 10.1109/CDC.2010.5717734

M3 - Conference contribution

AN - SCOPUS:79953141950

SN - 9781424477456

SP - 7655

EP - 7660

BT - 2010 49th IEEE Conference on Decision and Control, CDC 2010

ER -