### Abstract

In this paper, we combine a branch and bound algorithm with SOS programming in order to obtain arbitrarily accurate solutions to Global Polynomial Optimization (GPO) problems with bounded feasible sets. These arbitrarily accurate solutions are then fed into local gradient descent algorithms to obtain the true global optimizer. The algorithm successively bisects the feasible set and uses SOS to compute a Greatest Lower Bound (GLB) over each feasible set. For any desired accuracy, ϵ, we prove that the algorithm will return a point x such that |x-y| ≤ϵ for some point with objective value |f(y)-f(x∗)| ≤ϵ where x∗ is the global optimizer. To achieve this point, x, the algorithm sequentially solves O(\log(1/ϵ)) GLB problems, each of identical polynomial-time complexity. The point x, can then be used as an accurate initial value for gradient descent algorithms. We illustrate this approach using a numerical example with several local optima and demonstrate that the proposed algorithm dramatically increases the effectiveness of standard global optimization solvers.

Original language | English (US) |
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Title of host publication | 2018 Annual American Control Conference, ACC 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 2190-2197 |

Number of pages | 8 |

Volume | 2018-June |

ISBN (Print) | 9781538654286 |

DOIs | |

State | Published - Aug 9 2018 |

Event | 2018 Annual American Control Conference, ACC 2018 - Milwauke, United States Duration: Jun 27 2018 → Jun 29 2018 |

### Other

Other | 2018 Annual American Control Conference, ACC 2018 |
---|---|

Country | United States |

City | Milwauke |

Period | 6/27/18 → 6/29/18 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*2018 Annual American Control Conference, ACC 2018*(Vol. 2018-June, pp. 2190-2197). [8431203] Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.23919/ACC.2018.8431203

**Combining SOS with Branch and Bound to Isolate Global Solutions of Polynomial Optimization Problems.** / Colbert, Brendon; Mohammadi, Hesameddin; Peet, Matthew.

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

*2018 Annual American Control Conference, ACC 2018.*vol. 2018-June, 8431203, Institute of Electrical and Electronics Engineers Inc., pp. 2190-2197, 2018 Annual American Control Conference, ACC 2018, Milwauke, United States, 6/27/18. https://doi.org/10.23919/ACC.2018.8431203

}

TY - GEN

T1 - Combining SOS with Branch and Bound to Isolate Global Solutions of Polynomial Optimization Problems

AU - Colbert, Brendon

AU - Mohammadi, Hesameddin

AU - Peet, Matthew

PY - 2018/8/9

Y1 - 2018/8/9

N2 - In this paper, we combine a branch and bound algorithm with SOS programming in order to obtain arbitrarily accurate solutions to Global Polynomial Optimization (GPO) problems with bounded feasible sets. These arbitrarily accurate solutions are then fed into local gradient descent algorithms to obtain the true global optimizer. The algorithm successively bisects the feasible set and uses SOS to compute a Greatest Lower Bound (GLB) over each feasible set. For any desired accuracy, ϵ, we prove that the algorithm will return a point x such that |x-y| ≤ϵ for some point with objective value |f(y)-f(x∗)| ≤ϵ where x∗ is the global optimizer. To achieve this point, x, the algorithm sequentially solves O(\log(1/ϵ)) GLB problems, each of identical polynomial-time complexity. The point x, can then be used as an accurate initial value for gradient descent algorithms. We illustrate this approach using a numerical example with several local optima and demonstrate that the proposed algorithm dramatically increases the effectiveness of standard global optimization solvers.

AB - In this paper, we combine a branch and bound algorithm with SOS programming in order to obtain arbitrarily accurate solutions to Global Polynomial Optimization (GPO) problems with bounded feasible sets. These arbitrarily accurate solutions are then fed into local gradient descent algorithms to obtain the true global optimizer. The algorithm successively bisects the feasible set and uses SOS to compute a Greatest Lower Bound (GLB) over each feasible set. For any desired accuracy, ϵ, we prove that the algorithm will return a point x such that |x-y| ≤ϵ for some point with objective value |f(y)-f(x∗)| ≤ϵ where x∗ is the global optimizer. To achieve this point, x, the algorithm sequentially solves O(\log(1/ϵ)) GLB problems, each of identical polynomial-time complexity. The point x, can then be used as an accurate initial value for gradient descent algorithms. We illustrate this approach using a numerical example with several local optima and demonstrate that the proposed algorithm dramatically increases the effectiveness of standard global optimization solvers.

UR - http://www.scopus.com/inward/record.url?scp=85052575576&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052575576&partnerID=8YFLogxK

U2 - 10.23919/ACC.2018.8431203

DO - 10.23919/ACC.2018.8431203

M3 - Conference contribution

AN - SCOPUS:85052575576

SN - 9781538654286

VL - 2018-June

SP - 2190

EP - 2197

BT - 2018 Annual American Control Conference, ACC 2018

PB - Institute of Electrical and Electronics Engineers Inc.

ER -