Sequential convex relaxation for convex optimization with bilinear matrix equalities

Reinier Doelman; Michel Verhaegen

doi:10.1109/ECC.2016.7810576

Sequential convex relaxation for convex optimization with bilinear matrix equalities

Team Raf Van de Plas

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

15 Citations (Scopus)

543 Downloads (Pure)

Abstract

We consider the use of the nuclear norm operator, and its tendency to produce low rank results, to provide a convex relaxation of Bilinear Matrix Inequalities (BMIs). The BMI is first written as a Linear Matrix Inequality (LMI) subject to a bi-affine equality constraint and subsequently rewritten into an LMI subject to a rank constraint on a matrix affine in the decision variables. The convex nuclear norm operator is used to relax this rank constraint. We provide an algorithm that iteratively improves on the sum of the objective function and the norm of the equality constraint violation. The algorithm is demonstrated on a controller synthesis example.

Original language	English
Title of host publication	Proceedings 2016 European Control Conference (ECC 2016)
Editors	Anders Rantzer, John Bagterp Jørgensen, Jakob Stoustrup
Place of Publication	Piscataway, NJ, USA
Publisher	IEEE
Pages	1946-1951
ISBN (Print)	978-1-5090-2591-6
DOIs	https://doi.org/10.1109/ECC.2016.7810576
Publication status	Published - 2016
Event	2016 European Control Conference, ECC 2016: 15th annual European Control Conference - Aalborg, Denmark Duration: 29 Jun 2016 → 1 Jul 2016 http://www.ecc16.eu/index.shtml

Conference

Conference	2016 European Control Conference, ECC 2016
Abbreviated title	ECC'16
Country/Territory	Denmark
City	Aalborg
Period	29/06/16 → 1/07/16
Internet address	http://www.ecc16.eu/index.shtml

Bibliographical note

Accepted Author Manuscript

Keywords

optimisation
convex programming
linear matrix inequalities

Access to Document

10.1109/ECC.2016.7810576

Doelman, VerhaegenAccepted author manuscript, 413 KB

Cite this

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title = "Sequential convex relaxation for convex optimization with bilinear matrix equalities",

abstract = "We consider the use of the nuclear norm operator, and its tendency to produce low rank results, to provide a convex relaxation of Bilinear Matrix Inequalities (BMIs). The BMI is first written as a Linear Matrix Inequality (LMI) subject to a bi-affine equality constraint and subsequently rewritten into an LMI subject to a rank constraint on a matrix affine in the decision variables. The convex nuclear norm operator is used to relax this rank constraint. We provide an algorithm that iteratively improves on the sum of the objective function and the norm of the equality constraint violation. The algorithm is demonstrated on a controller synthesis example.",

keywords = "optimisation, convex programming, linear matrix inequalities",

author = "Reinier Doelman and Michel Verhaegen",

note = "Accepted Author Manuscript; 2016 European Control Conference, ECC 2016 : 15th annual European Control Conference, ECC'16 ; Conference date: 29-06-2016 Through 01-07-2016",

year = "2016",

doi = "10.1109/ECC.2016.7810576",

language = "English",

isbn = "978-1-5090-2591-6",

pages = "1946--1951",

editor = "Anders Rantzer and {Bagterp J{\o}rgensen}, John and Jakob Stoustrup",

booktitle = "Proceedings 2016 European Control Conference (ECC 2016)",

publisher = "IEEE",

address = "United States",

url = "http://www.ecc16.eu/index.shtml",

}

Doelman, R & Verhaegen, M 2016, Sequential convex relaxation for convex optimization with bilinear matrix equalities. in A Rantzer, J Bagterp Jørgensen & J Stoustrup (eds), Proceedings 2016 European Control Conference (ECC 2016). IEEE, Piscataway, NJ, USA, pp. 1946-1951, 2016 European Control Conference, ECC 2016, Aalborg, Denmark, 29/06/16. https://doi.org/10.1109/ECC.2016.7810576

Sequential convex relaxation for convex optimization with bilinear matrix equalities. / Doelman, Reinier; Verhaegen, Michel.
Proceedings 2016 European Control Conference (ECC 2016). ed. / Anders Rantzer; John Bagterp Jørgensen; Jakob Stoustrup. Piscataway, NJ, USA: IEEE, 2016. p. 1946-1951.

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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T1 - Sequential convex relaxation for convex optimization with bilinear matrix equalities

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AU - Verhaegen, Michel

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N2 - We consider the use of the nuclear norm operator, and its tendency to produce low rank results, to provide a convex relaxation of Bilinear Matrix Inequalities (BMIs). The BMI is first written as a Linear Matrix Inequality (LMI) subject to a bi-affine equality constraint and subsequently rewritten into an LMI subject to a rank constraint on a matrix affine in the decision variables. The convex nuclear norm operator is used to relax this rank constraint. We provide an algorithm that iteratively improves on the sum of the objective function and the norm of the equality constraint violation. The algorithm is demonstrated on a controller synthesis example.

AB - We consider the use of the nuclear norm operator, and its tendency to produce low rank results, to provide a convex relaxation of Bilinear Matrix Inequalities (BMIs). The BMI is first written as a Linear Matrix Inequality (LMI) subject to a bi-affine equality constraint and subsequently rewritten into an LMI subject to a rank constraint on a matrix affine in the decision variables. The convex nuclear norm operator is used to relax this rank constraint. We provide an algorithm that iteratively improves on the sum of the objective function and the norm of the equality constraint violation. The algorithm is demonstrated on a controller synthesis example.

KW - optimisation

KW - convex programming

KW - linear matrix inequalities

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M3 - Conference contribution

SN - 978-1-5090-2591-6

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EP - 1951

BT - Proceedings 2016 European Control Conference (ECC 2016)

A2 - Rantzer, Anders

A2 - Bagterp Jørgensen, John

A2 - Stoustrup, Jakob

PB - IEEE

CY - Piscataway, NJ, USA

T2 - 2016 European Control Conference, ECC 2016

Y2 - 29 June 2016 through 1 July 2016

ER -

Sequential convex relaxation for convex optimization with bilinear matrix equalities

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Bibliographical note

Keywords

Access to Document

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Cite this