Hessian-Free Fixed-/Predefined-Time Algorithms for Distributed Time-Varying Optimization

Abstract

This article proposes distributed algorithms free of Hessian for both time-invariant and time-varying optimization (TVO) problems. To this end, a subsystem is introduced to estimate the system’s gradient-sum in a distributed average tracking manner, based on which a distributed protocol is designed by coupling the gradient-sum descent method and state consensus scheme. Additionally, in our TVO method, a norm-normalized signum function is introduced to compensate for the internal drift of the system using its discontinuity. These methods are interesting as they can achieve the optimization goal within a specific time independent of system’s initial states, i.e., satisfy fixed-/predefined-time convergence. Moreover, a fully distributed adaptive gain method is proposed to avoid obtaining some global information. The numerical simulation and case study are provided to corroborate the effectiveness of proposed algorithms.