Stochastic models with application to approximation of<br/><br/>optimization problems

Monday, October 18, 2010 - 2:00pm - 3:00pm
Keller 3-180
Christian Hess (Université de Paris IX (Paris-Dauphine))
In this lecture it will be shown how basic concepts of
Probability Theory, such as distribution, independence,
(conditional) expectation, can be extended to the case of
random sets and random (lower semi-continuous)
functions. Then, some convergence results for sequences of
random sets and random functions, already known for
sequences or real-valued random variables, will be presented.
It will be also shown how these results give rise to
various applications to the convergence or approximation of
some optimization problems.


  1. Review on convergence of sequences of sets and functions in
    the deterministic case.

    Painleve-Kuratowski's Convergence, epi-convergence, variational
    properties of epi-convergence.

    Convex Analysis : conjugate of an extended real-valued
    function, epi-sum (alias inf-convolution)...

  2. Convergence of sequences of sets and functions in a
    stochastic context

    Random sets and random functions : denition, notion of
    equi-distribution and independence, set-valued integral.

    Strong laws of large numbers, Birkhoś Ergodic Theorem.

    Conditional expectation and martingales of random sets and
    random functions, almost sure convergence.

    Set-valued versions of Fatou's Lemma.

  3. Application to the approximation of optimization problems

    Convergence of discrete epi-sums to continuous epi-sum.

    Almost sure convergence of estimators.

    Convergence of integral functionals.

MSC Code: