Theory of generating spaces of convex sets and their applications to solvability of convex programs in Banach spaces

TL;DR

This paper introduces generating spaces related to nonsupport points in convex sets within Banach spaces, enabling the application of classical optimization techniques to infinite-dimensional problems with convex subsets that have empty interiors.

Contribution

It develops a novel method to replace original spaces with generating spaces, characterized as isometric to $L_$ or its subspaces, facilitating optimization in infinite-dimensional spaces.

Findings

Generating spaces are linearly isometric to $L_$ or their subspaces.

The method applies to Penalty principle, Lagrange duality, and scalarization functions.

Enables classical techniques to be used in general infinite-dimensional convex optimization problems.

Abstract

When optimization theorists consider optimization problems in infinite dimensional spaces, they need to deal with closed convex subsets(usually cones) which mostly have empty interior. These subsets often prevent optimization theorists from applying powerful techniques to study these optimization problems. In this paper, by nonsupport point, we present generating spaces which are relative to a Banach space and a nonsupport point of its convex closed subset. Then for optimization problems in infinite dimensional spaces, in some general cases, we replace original spaces by generating spaces while containing solutions. Thus this method enable us to apply powerful classical techniques to optimization problems in very general class of infinite dimensional spaces. Based on functional analysis, from classical Banach spaces to separable Banach spaces, from Banach lattice to latticization, we give characterizations of generating spaces and conclude that they are actually linearly isometric to $L_\infty$($\ell _\infty$) or their closed subspaces. Thus continuous linear functional involved in these techniques could be chosen from $L_\infty^*$($\ell_\infty^*$). After that, applications in Penalty principle, Lagrange duality and scalarization function are further studied by this method.