Optimization in Bochner Spaces

TL;DR

This paper investigates optimization problems within specific types of Bochner spaces, analyzing existence and inverse image properties across various subset types, with potential applications in stochastic optimization and related fields.

Contribution

It provides new theoretical results on optimization in Bochner spaces, focusing on existence and inverse image properties for different subset configurations.

Findings

Established existence results for optimization problems in Bochner spaces.

Analyzed inverse image properties for various subset types.

Potential applications to stochastic optimization and variational inequalities.

Abstract

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.