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

TL;DR

The paper tackles the problem of finding fixed points for mappings $F$ from a simplex $\mathcal{S}$ to itself in a gradient-free, global manner, achieving error halving with each $d$-factor refinement. It builds a constructive solver based on topological arguments, leveraging the Sperner Lemma and the Knaster–Kuratowski–Mazurkiewicz–Lemma to locate a point in $\bigcap_i C_i$ where $C_i=\{u: \lambda_i(F(u)) \le \lambda_i(u)\}$. The contributions include a detailed Knaster solver implementation, convergence analysis, discussion of Sperner simplex explosion, and a strategy to reduce complexity by unifying Ci memberships; the approach also extends to general convex, compact domains via a mapping to a simplex. The discussion covers optimization use, the explorative nature of the method, potential enhancements, and integration with other solvers, anchored by Mapping Degree Theory. Practically, the method provides a globally exploring, topology-backed alternative to local Newton-type methods and offers a framework for design-of-experiment style data generation during zero-search.

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.

Figures (12)

• Figure 1: Right: Sets $C_i$ of points that are mapped not closer to corner $\boldsymbol{v}_i$ by some mapping (left). The intersection of all $C_i$ are fixed points.
• Figure 2: Left: The fixed point is inside the Sperner simplex and will be in such after refinement. Middle: The fixed point is not inside the Sperner simplex, but in the non-Sperner above. Right: A simplex is non-Sperner and contains a pair number of fixed points.
• Figure 3: The $2-d$ unit simplex refined twice.
• Figure 4: The $3-d$ unit simplex refined three times.
• Figure 5: Above left: The $C_i$ of example in Eq.\ref{['eq:x/2']}: $(x,y)^\intercal \mapsto \frac{1}{2} (x,y)^\intercal$. All $\boldsymbol{x}$ are mapped farther away from $(1,0)^\intercal$, so in $C_1$ (orange), same with $C_2$ (yellow). Only the origin is mapped not closer to $(0,0)^\intercal$, so in $C_0$ (black), and is the fixed point. Right and below: Knaster algorithms refinement steps 2, 4 and 5 for example \ref{['eq:x/2']}. The coloured markers show the $C_i$ memberships: All points are in $C_1$ and $C_2$, so orange and yellow, only the origin is in $C_0$ as well and black.

Theorems & Definitions (11)

• Remark 2.1
• Theorem 2.1
• proof
• Theorem 2.2
• Theorem B.1: Sperner's Lemma
• proof
• Theorem B.2: The Lemma of Knaster, Kuratowski and Mazurkewicz
• proof
• Theorem B.3: Brouwer's Fixed Point theorem
• proof