A new zero-order 1-D optimization algorithm: trichotomy method

TL;DR

The paper introduces the trichotomy method, a novel zero-order 1D optimization algorithm that divides the interval into six parts, reducing the interval three times per iteration, and demonstrates superior efficiency and accuracy over existing methods.

Contribution

It presents a new 1D search algorithm based on dividing the interval into six parts, improving upon the interval halving method and other classical algorithms.

Findings

Requires fewer function evaluations than classical methods.

Achieves higher accuracy in locating the minimum point.

Outperforms interval halving, golden section, and Fibonacci search methods.

Abstract

A new 1D search method is proposed for minimizing an arbitrary real valued function. The algorithm is a modification of the interval halving method which is based on dividing the interval of uncertainty by three points into four equal parts. The trichotomy method is based on dividing the interval by five points into six equal parts and provides the interval reducing exactly three times in every iteration. New algorithm's performance has been extensively tested and compared to well-known 1D search algorithms: the interval halving method, golden section method, Fibonacci search. The results show that the trichotomy method usually require less calculations of the values of the minimized function to determine the minimum point with the given accuracy. Also it has better accuracy when finding the minimum point after using N calculations of the function values.