A novel approach to find the minimum of real functions and an anomalous test function

TL;DR

This paper introduces a geometric-based algorithm for finding minima of strictly convex functions by intersecting the function with decreasing horizontal lines, and discusses its limitations with an anomalous test function.

Contribution

The paper presents a novel geometric approach to minimize strictly convex functions and extends it to a broader class, including identifying cases where the method fails.

Findings

The algorithm effectively finds minima for certain convex functions.

The approach can be generalized to wider function classes.

An anomalous function demonstrates the algorithm's limitations.

Abstract

The aim of this paper is to present an original approach that takes advantage from the geometric features of strictly convex functions to tackle the problem of finding the minimum from another perspective. The general idea is that near the point of minimum, the function is 'v-shaped' and so we can reduce the interval where the minimum lies by finding the intersection between the function and a proper horizontal line whose levels decrease step by step. This idea, under some appropriate assumptions, led us to formalise an algorithm that is able to determine the minimum point sought. Furthermore, we see that this approach can be generalized to a wider class of functions. In the last part of this paper we provide the construction of an anomalous function for which the algorithm cannot be used.