A Hessian-based zeroing neurodynamic approach for quaternion-variable time-varying constrained optimization problems

Abstract

In this paper, a quaternion-variable zeroing neurodynamic (ZND) is designed, in order to address a class of quaternion-variable time-varying convex optimization problems (TCOP) with equality constraints and general inequality constraints. Through utilizing the Karush–Kuhn–Tucker (KKT) condition, the TCOP is transformed into a time-varying equation system for solving. During this process, the introduction of two approaches distinguishes our works in solving TCOPs. On the one hand, the penalty approach is utilized to reduce the influence of time-varying inequality constraints, and thus, makes the ZND more suitable for TCOPs with general and complex inequality constraints. On the other hand, the approach to simplify the KKT condition significantly simplifies the structure of the proposed ZND by reducing the neurons’ number. The proposed ZND realizes efficient convergence while keeping the structure as simple as possible, making it easy to apply in engineering tasks. Besides, to improve its efficiency further, a nonlinear activated function is utilized to achieve a faster convergent rate, by which the convergent rate of the ZND is accelerated to finite-time convergence. Apart from that, the ZND here also owes superior robustness. To the best of our knowledge, this is the first time to extend the ZND for solving TCOP to the quaternion domain. Finally, numerical examples and a robot manipulator control experiment further explain the effectiveness of the proposed ZND.