On the complexity of randomized approximations of nonconvex sets

Fabrizio Dabbene, Constantino Lagoa, Pavel Shcherbakov

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

    4 Scopus citations

    Abstract

    We consider a nonconvex set X ∈ ℝn endowed with a probability measure P, and assume that random samples taken according to this measure are available. For ε ∈ (0; 1), we say that the set A ∈ ℝn is an ε-probabilistic approximation of X if the probability of a point to belong to X and not to A is less than ε. In this paper, we show that a simple randomized algorithm returning with high probability convex ε-approximations of X can be easily devised. In particular, we consider different possible shapes for the set A, namely ellipsoids, hyperrectangles, and parallelotopes, and show how specific approximations based on such shapes can be constructed. Moreover, we derive explicit bounds on the complexity of such approximations, in terms of the number of samples needed in the different cases. It turns out that, while the complexity of ellipsoidal and parallelotopic approximations grow quadratically with respect to the dimension n, in the case of orthotopes one obtains linear dependence. The second part of the paper is devoted to the study of tighter approximations, based on recent results on the connection between chanceconstrained problems and scenario problems with discarded constraints. There are numerous applications of this result in systems and control theory; a specific motivation for the research conducted in this paper is the characterization of the reachable sets of nonlinear systems, and the related problem of nonlinear filtering.

    Original languageEnglish (US)
    Title of host publication2010 IEEE International Symposium on Computer-Aided Control System Design, CACSD 2010
    Pages1564-1569
    Number of pages6
    DOIs
    StatePublished - Dec 13 2010
    Event2010 IEEE International Symposium on Computer-Aided Control System Design, CACSD 2010 - Yokohama, Japan
    Duration: Sep 8 2010Sep 10 2010

    Publication series

    NameProceedings of the IEEE International Symposium on Computer-Aided Control System Design

    Other

    Other2010 IEEE International Symposium on Computer-Aided Control System Design, CACSD 2010
    Country/TerritoryJapan
    CityYokohama
    Period9/8/109/10/10

    All Science Journal Classification (ASJC) codes

    • Control and Systems Engineering
    • Computer Science Applications
    • Control and Optimization

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