Optimization of Mathematical Functions Using Fractional Steepest Descent Method with Self-Adaptive Order

Resumo

Fractional calculus represents a mathematical framework that has numerous applications. In the context of optimization, it can be used to increase the performance of gradient-based methods. However, when the direction of the first order integer gradient is generalized using a fractional order, this approach may converge to a different solution from the one obtained by the classical Steepest Descent Method, making the application of this type of methodology complex. To solve this convergence issue, this contribution aims to propose an adaptive approach in which the fractional order is updated along the iterations so that at the end of the optimization process the fractional order is equal to one. For this purpose, the adaptive fractional order is defined from a new parameter, namely, the reduction rate. The results obtained with the optimization of two mathematical functions demonstrate the potential of the proposed methodology.