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

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.