On the sample complexity of uncertain linear and bilinear matrix inequalities

Mohammadreza Chamanbaz; Fabrizio Dabbene; Roberto Tempo; Venkatakrishnan Venkataramanan; Qing Guo Wang

doi:10.1109/CDC.2013.6760140

On the sample complexity of uncertain linear and bilinear matrix inequalities

Mohammadreza Chamanbaz, Fabrizio Dabbene, Roberto Tempo, Venkatakrishnan Venkataramanan, Qing Guo Wang

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

2 Citations (Scopus)

Abstract

In this paper, we consider uncertain linear and bilinear matrix inequalities which depend in a possibly non-linear way on a vector of uncertain parameters. Motivated by recent results in statistical learning, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chevonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of the problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality.

Original language	English
Title of host publication	2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013
Publisher	Institute of Electrical and Electronics Engineers Inc.
Pages	1780-1785
Number of pages	6
ISBN (Print)	9781467357173
DOIs	https://doi.org/10.1109/CDC.2013.6760140
Publication status	Published - 2013
Externally published	Yes
Event	52nd IEEE Conference on Decision and Control, CDC 2013 - Florence, Italy Duration: 10 Dec 2013 → 13 Dec 2013

Publication series

Name	Proceedings of the IEEE Conference on Decision and Control
ISSN (Print)	0743-1546
ISSN (Electronic)	2576-2370

Conference

Conference	52nd IEEE Conference on Decision and Control, CDC 2013
Country/Territory	Italy
City	Florence
Period	10/12/13 → 13/12/13

ASJC Scopus subject areas

Control and Systems Engineering
Modeling and Simulation
Control and Optimization

Access to Document

10.1109/CDC.2013.6760140

Cite this

Chamanbaz, M., Dabbene, F., Tempo, R., Venkataramanan, V., & Wang, Q. G. (2013). On the sample complexity of uncertain linear and bilinear matrix inequalities. In 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013 (pp. 1780-1785). Article 6760140 (Proceedings of the IEEE Conference on Decision and Control). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2013.6760140

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title = "On the sample complexity of uncertain linear and bilinear matrix inequalities",

abstract = "In this paper, we consider uncertain linear and bilinear matrix inequalities which depend in a possibly non-linear way on a vector of uncertain parameters. Motivated by recent results in statistical learning, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chevonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of the problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality.",

author = "Mohammadreza Chamanbaz and Fabrizio Dabbene and Roberto Tempo and Venkatakrishnan Venkataramanan and Wang, {Qing Guo}",

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Chamanbaz, M, Dabbene, F, Tempo, R, Venkataramanan, V & Wang, QG 2013, On the sample complexity of uncertain linear and bilinear matrix inequalities. in 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013., 6760140, Proceedings of the IEEE Conference on Decision and Control, Institute of Electrical and Electronics Engineers Inc., pp. 1780-1785, 52nd IEEE Conference on Decision and Control, CDC 2013, Florence, Italy, 10/12/13. https://doi.org/10.1109/CDC.2013.6760140

On the sample complexity of uncertain linear and bilinear matrix inequalities. / Chamanbaz, Mohammadreza; Dabbene, Fabrizio; Tempo, Roberto et al.
2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013. Institute of Electrical and Electronics Engineers Inc., 2013. p. 1780-1785 6760140 (Proceedings of the IEEE Conference on Decision and Control).

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

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AB - In this paper, we consider uncertain linear and bilinear matrix inequalities which depend in a possibly non-linear way on a vector of uncertain parameters. Motivated by recent results in statistical learning, we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms. In particular, we show that the Vapnik-Chevonenkis dimension (VC-dimension) of the two problems is finite, and we compute upper bounds on it. In turn, these bounds allow us to derive explicitly the sample complexity of the problems. Using these bounds, in the second part of the paper, we derive a sequential scheme, based on a sequence of optimization and validation steps. The algorithm is on the same lines of recent schemes proposed for similar problems, but improves both in terms of complexity and generality.

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Chamanbaz M, Dabbene F, Tempo R, Venkataramanan V, Wang QG. On the sample complexity of uncertain linear and bilinear matrix inequalities. In 2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013. Institute of Electrical and Electronics Engineers Inc. 2013. p. 1780-1785. 6760140. (Proceedings of the IEEE Conference on Decision and Control). doi: 10.1109/CDC.2013.6760140

On the sample complexity of uncertain linear and bilinear matrix inequalities

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