A geometric simplex method in infinite-dimensional spaces

TL;DR

This paper extends the geometric simplex method to infinite-dimensional locally convex topological vector spaces, establishing convergence conditions and analyzing polytope structures in such spaces.

Contribution

It generalizes the simplex method to broader infinite-dimensional settings without relying on pivoting, connecting basic feasible solutions to extreme points.

Findings

The method converges in value to the optimal solution under certain conditions.

Polytopes in this setting have exposed extreme points connected by edge paths.

The approach applies to optimization over the Hilbert cube, a complex infinite-dimensional object.

Abstract

We expand the basic geometric elements of the simplex method to linear programs in locally convex topological vector spaces and provide conditions under which the method converges in value to optimality. This setting generalizes many previous investigations of the simplex method, which are restricted to Hilbert spaces or otherwise specially structured instances. Our generality is obtained by avoiding the ``algebraic'' machinery of pivoting via column operations, which has required stronger topological conditions in establishing a connection between basic feasible solutions and extreme point structure. We show that our definition of polytopes captures optimization over the Hilbert cube, a quintessential object in infinite-dimensional spaces known for its surprisingly complicated properties. Moreover, all polytopes (under our definition) have exposed extreme points connected by edge paths.