A Galerkin approach to optimization in the space of convex and compact subsets of $\R^d$

TL;DR

This paper develops a Galerkin approximation framework for the space of convex and compact subsets of , enabling both theoretical insights and practical solutions for optimization problems in this geometric space.

Contribution

It introduces a novel Galerkin approach tailored for convex and compact sets, bridging theory and computation for optimization in geometric spaces.

Findings

Galerkin spaces are effectively constructed and analyzed.

The approach enables approximate solutions to convex set optimization problems.

The method shows promising computational properties.

Abstract

The aim of this paper is to establish a theory of Galerkin approximations to the space of convex and compact subsets of $\R^d$ with favorable properties, both from a theoretical and from a computational perspective. These Galerkin spaces are first explored in depth and then used to solve optimization problems in the space of convex and compact subsets of $\R^d$ approximately.