Research On Time-Varying Linear Complementarity Problem Based On Zero Neural Network

The linear complementarity problem plays an important role in engineering fields such as fluid problems and economic equilibrium analysis.In practical applications,many linear complementarity problems are constantly changing over time(time-varying linear complementarity problems).However,for a long time,most of the research on linear complementarity problems have focused on situations independent on time.Few people study the time-varying linear complementarity problem,which undoubtedly reduces its scope of application in real life.Therefore,how to solve the time-varying linear complementarity problem faster and more accurately becomes urgent.At the same time,although time-varying linear complementarity problem can describe the physical characteristics of the original problem more naturally and more closely,it is extremely difficult to solve and may even be impossible to find exact solution in real time because time-varying problems often contain time-varying parameters.In recent years,a novel neural network method—zero neural network algorithm has been widely recognized in the field of solving time-varying problems due to its high-speed parallel computing and easy hardware implementation.Based on the above background,this paper uses the zero neural network algorithm to solve a class of time-varying linear complementarity problems.In the research,it is found that the inverse of the time-varying matrix will inevitably occur in the process of solving the time-varying linear complementarity problem,and the inverse of the matrix is often difficult to calculate and solve in real time in high-dimensional operations.Aiming at this problem,this paper first designs a kind of zero neural network to solve the inverse of the time-varying matrix.And on this basis,a novel zero neural network algorithm is constructed to solve the above-mentioned time-varying linear complementarity problem.Finally,by combining two different targeted zero neural network algorithms,a more efficient composite zero neural network algorithm is proposed.At the same time,considering the convergence rate,which is an important indicator of algorithm performance,this paper uses three types of nonlinear activation functions to accelerate the convergence rate,and while maintaining the stability of the algorithm,the convergence rate of the algorithm solution is increased to a finite time convergence.In addition,considering the existence of external noise interference and these noises are objective factors that cannot be ignored,this paper constructs a class of noise-resistant algorithms to solve the time-varying linear complementarity problem by introducing an integral term.Theoretical analysis and numerical examples show that the algorithm can not only maintain stability,but also can efficiently solve the target problem under external noise interference.On the other hand,in engineering applications,when many practical problems are transformed into mathematical problems,the information that can be sampled and the solutions to requirements are all in discrete form.Therefore,it is very necessary to construct a discrete form of network algorithm to solve this problem.In this paper,a high-precision five-step difference formula is constructed through Taylor's formula and used to discretize the aforementioned continuous zero neural network algorithm,and finally the corresponding discrete zero neural network is obtained to solve the time-varying linear complementarity problem.And in the subsequent research,the relevant analysis of the stability of the algorithm and the maximum steady-state error is given.Finally,an example is given to illustrate the effectiveness of the proposed algorithm.