On the Asymptotics of Convex Stochastic Optimization
By CHARLES J. GEYER
University of Minnesota
Abstract
The asymptotics of minimizers of a sequence of random convex functions
on a finite-dimensional Euclidean space are described using very weak
regularity conditions. If random convex functions [g(n)] finite on some
open set converge in law pointwise on a dense set to a random function
[g] that almost surely is finite on some nonempty open set and possesses
a unique minimizer, then the minimizers of the [g(n)] converge in law
to the minimizer of [g]. Under the same conditions, confidence sets
constructed as level sets of the [g(n)] converge in law to the corresponding
level sets of [g] in the metric of Painleve-Kuratowski set convergence.
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