The q-gradient method for global optimization

TL;DR

The paper introduces the q-gradient method, a novel global optimization technique based on Jackson's derivative, which effectively escapes local minima and outperforms genetic algorithms on multimodal functions.

Contribution

It presents the q-gradient method as a new optimization approach that generalizes steepest descent using Jackson's derivative for better global search capabilities.

Findings

Outperformed genetic algorithms on multimodal test functions.

Achieved higher accuracy with fewer function evaluations.

Effectively escapes local minima during optimization.

Abstract

The q-gradient is an extension of the classical gradient vector based on the concept of Jackson's derivative. Here we introduce a preliminary version of the q-gradient method for unconstrained global optimization. The main idea behind our approach is the use of the negative of the q-gradient of the objective function as the search direction. In this sense, the method here proposed is a generalization of the well-known steepest descent method. The use of Jackson's derivative has shown to be an effective mechanism for escaping from local minima. The q-gradient method is complemented with strategies to generate the parameter q and to compute the step length in a way that the search process gradually shifts from global in the beginning to almost local search in the end. For testing this new approach, we considered six commonly used test functions and compared our results with three Genetic Algorithms (GAs) considered effective in optimizing multidimensional unimodal and multimodal functions. For the multimodal test functions, the q-gradient method outperformed the GAs, reaching the minimum with a better accuracy and with less function evaluations.