Global Solver based on the Sperner-Lemma and Mazurkewicz-Knaster-Kuratowski-Lemma based proof of the Brouwer Fixed-Point Theorem

TL;DR

This paper introduces a gradient-free, global fixed-point solver for simplex mappings, leveraging topological proofs from Sperner's and Mazurkewicz-Knaster-Kuratowski lemmas, requiring only d function evaluations.

Contribution

It presents a novel fixed-point solver based on topological lemmas, offering a new constructive approach with optimal evaluation complexity.

Findings

Solver is gradient-free and globally convergent.

Requires only d function evaluations, where d is the dimension.

Based on topological proofs, ensuring rigorous correctness.

Abstract

In this paper a fixed-point solver for mappings from a Simplex into itself that is gradient-free, global and requires $d$ function evaluations for halvening the error is presented, where $d$ is the dimension. It is based on topological arguments and uses the constructive proof of the Mazurkewicz-Knaster-Kuratowski lemma as used as part of the proof for Brouwers Fixed-Point theorem.