A $q$-spectral Polak-Ribi{\'e}re-Polyak conjugate gradient method for unconstrained optimization problems with motion control application

Abstract

Frank Hilton Jackson generalized the classical notion of derivative and developed the $q$-derivative, widely known as Jackson's derivative. To solve large-scale unconstrained optimization problems, we propose a $q$-spectral Polak-Ribi{\'e}re-Polyak (PRP) conjugate gradient method. The proposed method can be viewed as a generalization of the spectral PRP method, which replaces the classical gradient with the $q$-gradient vector that uses the first-order partial $q$-derivatives obtained from Jackson's derivative. Furthermore, as the value of $q$ approaches one, the proposed method reduces to the classical version. Numerical experiments are performed and compared with the existing method to show the improvement of the proposed method. Furthermore, we have solved the motion control problem.