The q-G method : A q-version of the Steepest Descent method for global optimization

Area of Science:

• Optimization Theory
• Numerical Analysis
• Applied Mathematics

Background:

• Classical optimization methods can get trapped in local minima.
• Gradient-based methods are fundamental but have limitations in complex landscapes.
• Q-calculus offers novel mathematical tools for extending classical concepts.

Purpose of the Study:

• Introduce the q-Gradient (q-G) method, a novel optimization algorithm.
• Generalize the Steepest Descent method using q-calculus.
• Develop an algorithm capable of escaping local minima.

Main Methods:

• The q-G method utilizes the negative q-gradient vector as the search direction.
• The q-gradient is a generalization of the classical gradient based on Jackson's derivative.
• The algorithm incorporates three free parameters for adaptive exploration and exploitation.

Main Results:

• The q-G method demonstrated competitive performance against 34 other optimization algorithms.
• It showed effectiveness in escaping local minima, a common challenge in optimization.
• Performance was evaluated on 34 diverse test functions.

Conclusions:

• The q-Gradient method offers a promising alternative for solving multimodal optimization problems.
• It effectively combines global exploration with local exploitation.
• The method converges to the Steepest Descent method as q approaches 1.