Comments on "Generalization of the gradient method with fractional order gradient direction"

TL;DR

This paper critiques a previous work on fractional gradient methods, identifying a significant mathematical mistake in its convergence proof and questioning the validity of its subsequent theorems.

Contribution

It provides a critical analysis highlighting a fundamental flaw in the original proof of convergence in the fractional gradient method paper.

Findings

Theorem 1's proof contains a mathematical mistake.

Subsequent theorems rely on the flawed proof without proper validation.

The critique questions the validity of the original convergence claims.

Abstract

In this paper, a detrimental mathematical mistake is pointed out in the proof of \textit{Theorem 1} presented in the paper\textit{ [Generalization of the gradient method with fractional order gradient direction, J. Franklin Inst., 357 (2020) 2514-2532]}. It is highlighted that the way the authors prove the convergence of the fractional extreme points of a real valued function to its integer order extreme points lacks correct and valid mathematical argument. Rest of the theorems contained in the paper are mostly announced without any proof relaying on that of Theorem 1.