It is good to read that you are interested in stochastic
optimizations and ready for research. Come and see me in my office in
September, I have more than enough problems for you.
Here is a problem which does not look stochastic but it is related to it.
Consider the following 2^n points in the n-dimensional space:
$(\pm 1, \pm 1, ... \pm 1)$ (these are the vertices of the
n-dimensional standard cube). Given the radius r, find the maximal
number m= m(n, r) of
vertices you can cover by a sphere of radius r (the center of the
sphere varies).