  In this paper, we study some optimization problems in uniformly convex and
uniformly smooth Bochner spaces. We consider four cases of the underlying
subsets: closed and convex subsets, closed and convex cones, closed subspaces
and closed balls. In each case, we study the existence problems and the inverse
image properties. We think that the results and the analytic methods in this
paper could be applied to the theories of stochastic optimization, stochastic
variational inequality and stochastic fixed point.
