Similarly, the q-gradient of f is the q-derivative that has also a straightforward geometric interpretation as the slope of the secant line passing through the points and. Geometrically, it is the slope of the tangent line at a given point x see Fig. The gradient of one-variable function f( x) is simply the derivative. The q-gradient is calculated based on q-derivatives, or Jackson’s derivatives, and requires a dilation parameter q that controls the balance between global and local search. The main idea behind this new method is the use of the negative of the q-gradient of the objective function as the search direction. Recently, based on Jackson’s derivative, a q-version of the classical Steepest Descent method, called the q-Gradient ( q-G) method, has been proposed for solving unconstrained continuous global optimization problems (Soterroni et al. His work gave rise to generalizations of special numbers, series, functions and, more importantly, to the concepts of the q-derivative (Jackson 1908), or Jackson’s derivative, and the q-integral (Jackson 1910). The history of q-calculus dates back to the beginning of the previous century when, based on the pioneering works of Euler and Heine, the English reverend Frank Hilton Jackson developed q-calculus in a systematic way (Chaundy 1962 Ernst 2000 Kac and Cheung 2002). Solving multimodal optimization problems. Our results show that the q-G method is competitive and has a great potential for We evaluated the q-G method on 34 test functions, and compared its performance with 34 optimization algorithms, including derivative-free algorithms and the Steepest Descent method. The algorithm has three free parameters and it is implemented so that the search process gradually shifts from global exploration in the beginning to local exploitation in the end. The q-G method reduces to the Steepest Descent method when the parameter q tends to 1. Its use provides the algorithmĪn effective mechanism for escaping from local minima. The q-gradient vector, or simply the q-gradient, is a generalization of the classical gradient vector based on the concept of Jackson’s derivative from the q-calculus. The main idea behind the q-G method is the use of the negative of the q-gradient vector of the objective function as the search direction. In this work, the q-Gradient ( q-G) method, a q-version of the Steepest Descent method, is presented.
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