# 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.

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.

*Plan*- 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)...

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.- 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:

91B70

Keywords: