Analysis Of A Neural Dynamics Method For Time-varying Optimization Problems

In practical application,the information provided by the problems and the required resolutions are always real-time,such as power scheduling,optimal controller design,robot motion generation and so on.When these problems are transformed into mathematical problems,the objective functions and constraints vary with time-varying,which are termed as time-varying optimization problems.Particularly,the time-varying convex optimization problems with linear equality and affine inequality constraints are a class of comprehensive problems,which are of great significance to study.In recent years,a class of neural dynamic named Zhang neural dynamic(ZND)is highly favored by scholars because of great performance on time-varying problems.For this reason,in this paper,Zhang neural dynamic for solving time-varying convex optimization problems with equality and affine inequality constraints is studied.Owing to the existence of equality constraints and inequality constraints,there are extra Lagrange multipliers.Therefore,the neural dynamic models always own more neurons when abovementioned problems are studied,which will give rise to the computation burden in high-dimension computation.Besides,though simplified models are proposed by researchers,the models after simplifying depend on a real-time matrix inverse in the process of calculation,which is impractical to solve in high-dimensional computation.Therefore,a class of ZND model is designed to solve real-time matrix inverse in the first place.Based on this,a class of ZND model with lower dimensions is constructed aiming at time-varying convex optimization problems with linear equality and affine inequality constraints.In the last,by combining abovementioned models,a class of compound ZND methods which are able to solve real-time matrix inverse effectively and own less neurons is constructed.In the meanwhile,convergence rate is an important indicator to measure the performance of neural dynamic models.Therefore,three kinds of activate functions are introduced and analyzed in order to accelerate convergence rate.Considering the external disturbance,a class of ZND model with robustness is also proposed in order to solve time-varying optimization problems accurately with external disturbance.Finally,numerical examples and an application on robot inverse kinematic problem are given to illustrate the superiority and effectiveness of the proposed model.On the other hand,in engineer and scientific fields,the provided information is always discrete when the practical problems are transferred into mathematical problems.As a result,it's necessary to construct discrete ZND models corresponding to continuous ZND models.In this paper,a class of discrete ZND model aiming at time-varying convex optimization problems with equality and affine inequality constraints is constructed.And numerical examples are given to illustrate effectiveness of the proposed model.