Derivative-Free Global Minimization in One Dimension: Relaxation, Monte Carlo, and Sampling

TL;DR

This paper presents a new derivative-free global optimization algorithm for one-dimensional functions that combines relaxation, Monte Carlo, and sampling techniques to efficiently find minima with fewer function evaluations.

Contribution

The paper introduces a novel algorithm that approximates gradient flow through relaxation and sampling methods, improving efficiency over existing approaches for one-dimensional optimization.

Findings

Algorithm converges reliably as proven mathematically.

Significantly reduces function evaluations compared to traditional methods.

Demonstrates strong performance on diverse benchmark problems.

Abstract

We introduce a derivative-free global optimization algorithm that efficiently computes minima for various classes of one-dimensional functions, including non-convex, and non-smooth functions.This algorithm numerically approximates the gradient flow of a relaxed functional, integrating strategies such as Monte Carlos methods, rejection sampling, and adaptive techniques. These strategies enhance performance in solving a diverse range of optimization problems while significantly reducing the number of required function evaluations compared to established methods. We present a proof of the convergence of the algorithm and illustrate its performance by comprehensive benchmarking. The proposed algorithm offers a substantial potential for real-world models. It is particularly advantageous in situations requiring computationally intensive objective function evaluations.