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

Brendon Colbert, Hesameddin Mohammadi, Matthew Peet

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publication2018 Annual American Control Conference, ACC 2018
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages2190-2197
Number of pages8
Volume2018-June
ISBN (Print)9781538654286
DOIs
StatePublished - Aug 9 2018
Event2018 Annual American Control Conference, ACC 2018 - Milwauke, United States
Duration: Jun 27 2018Jun 29 2018

Other

Other2018 Annual American Control Conference, ACC 2018
CountryUnited States
CityMilwauke
Period6/27/186/29/18

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Polynomials
Global optimization

ASJC Scopus subject areas

  • Electrical and Electronic Engineering

Cite this

Colbert, B., Mohammadi, H., & Peet, M. (2018). Combining SOS with Branch and Bound to Isolate Global Solutions of Polynomial Optimization Problems. In 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.

2018 Annual American Control Conference, ACC 2018. Vol. 2018-June Institute of Electrical and Electronics Engineers Inc., 2018. p. 2190-2197 8431203.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Colbert, B, Mohammadi, H & Peet, M 2018, Combining SOS with Branch and Bound to Isolate Global Solutions of Polynomial Optimization Problems. in 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
Colbert B, Mohammadi H, Peet M. Combining SOS with Branch and Bound to Isolate Global Solutions of Polynomial Optimization Problems. In 2018 Annual American Control Conference, ACC 2018. Vol. 2018-June. Institute of Electrical and Electronics Engineers Inc. 2018. p. 2190-2197. 8431203 https://doi.org/10.23919/ACC.2018.8431203
Colbert, Brendon ; Mohammadi, Hesameddin ; Peet, Matthew. / Combining SOS with Branch and Bound to Isolate Global Solutions of Polynomial Optimization Problems. 2018 Annual American Control Conference, ACC 2018. Vol. 2018-June Institute of Electrical and Electronics Engineers Inc., 2018. pp. 2190-2197
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