The Properties of Average Gradient in Local Region

TL;DR

This paper investigates the properties of average gradients in local regions, introduces a homogenization function, and demonstrates its effectiveness in identifying global and local extrema using gradient algorithms.

Contribution

It presents a novel approach to analyze local and global extrema through homogenization functions and gradient methods, enhancing optimization techniques.

Findings

Homogenization function has good properties at local and global extrema.

Using homogenization functions improves the effectiveness of gradient algorithms.

Method helps in sifting out local extreme points of functions.

Abstract

This paper studies the average gradient over the local region of a function and constructs the homogenization function of a function. It is found that there are some good properties about the local extreme points and the global extreme points of the function. By using the gradient algorithm, it is more effective to use the homogenization function to find the extreme values of the function. This method implies a method of sifting out the local extreme points of a function.