Time-Varying Convex Optimization Methods With A Simple Prediction Step

The relationship between optimization and control theory is inseparable and affects each other.With the development of distributed computation techniques,many efficient optimization methods have been proposed to solve some complex control problems.In practical applications,such as signal processing,robot navigation,and finance analysis,the objective function is time-dependent.For time-varying optimization,the optimal solution is a function of time.Therefore,solving time-varying optimization problems is equivalent to tracking the optimal solution over time.In recent years,time-varying optimization theory has been widely deployed.A typical time-varying optimization algorithm is implemented in an online manner.It is particularly suitable for optimizing fast-changing systems.In this paper,the continuous-time system is discretized into discrete-time system,and the online optimization method is used to solve the time-varying unconstrained optimization problem.The convergence analysis of the algorithm is performed by using tools such as optimization,matrix theory and numerical mathematics.The main research contents and contributions of this paper are summarized as follows.First,for the unconstrained optimization problem where the objective function is time-varying,based on the prediction-correction strategy,we propose a gradient method with a simple prediction step and a newton method with a simple prediction step,respectively.The calculation complexity of this estimation step is small,according to the information difference between the information at the current moment and the information at the previous moment,to predict the changing trend of the optimal solution;then,the estimated value obtained through the prediction step is corrected by an iterative algorithm to obtain the estimated value.Compared with the traditional iterative algorithm,the algorithm with the simple estimation step can exhibit better convergence properties.We verify the effectiveness of the algorithm through theoretical analysis and simulation.Additionally,considering that decentralized systems have the advantages offering better system security and autonomous efficiency,we decentralize the newton method with a simple prediction step,proposed a decentralized newton method with a simple prediction step,and give a proof for the convergence of the algorithm.