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Research Papers

Contractors and Linear Matrix Inequalities

[+] Author and Article Information
Jeremy Nicola

Lab STICC, ENSTA Bretagne, 2 Rue Francois Verny, 29806 Brest Cedex 9, France e-mail: jeremy.nicola@ensta-bretagne.org

Luc Jaulin

Lab STICC, ENSTA Bretagne, 2 Rue Francois Verny, 29806 Brest Cedex 9, France e-mail: luc.jaulin@ensta-bretagne.org

Manuscript received October 25, 2014; final manuscript received June 3, 2015; published online July 1, 2015. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 1(3), 031004 (Jul 01, 2015) (6 pages) Paper No: RISK-14-1076; doi: 10.1115/1.4030781 History: Received October 25, 2014; Accepted June 04, 2015; Online July 01, 2015

Linear matrix inequalities (LMIs) comprise a large class of convex constraints. Boxes, ellipsoids, and linear constraints can be represented by LMIs. The intersection of LMIs are also classified as LMIs. Interior-point methods are able to minimize or maximize any linear criterion of LMIs with complexity, which is polynomial regarding to the number of variables. As a consequence, as shown in this paper, it is possible to build optimal contractors for sets represented by LMIs. When solving a set of nonlinear constraints, one may extract from all constraints that are LMIs in order to build a single optimal LMI contractor. A combination of all contractors obtained for other non-LMI constraints can thus be performed up to the fixed point. The resulting propogation is shown to be more efficient than other conventional contractor-based approaches.

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Figures

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Fig. 1

Characterization of [S1∩S2]: (a) maximization of x1, (b) minimization of x1, (c) minimization of x2, and (d) maximization of x2

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Fig. 2

Illustration of a minimal and nonminimal contractor for the set X: C2 is minimal and C1 is not

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Fig. 5

Characterization of S (a) with forward-backward contractions only and (b) with forward-backward contractors and LMI contractors

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Fig. 3

Subpaving of F (a) with forward-backward contractors only and (b) with a forward-backward and a LMI contractor

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Fig. 4

Characterization of E (a) with forward-backward contractions only and (b) with a forward-backward contractor and the LMI contractor

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