Methods For Solving Systems Of Time-Varying Constrained Nonlinear Equations

A large number of systems of constrained nonlinear equations exist in chemical engineering,power engineering,and mechanical engineering.In practice,the requirements for time-varying and accuracy of problem solving are increasing.The operating environment often changes due to the influence of time.Therefore,it is not enough to consider only static issues for real-life applications.So we consider the systems of time-varying constrained nonlinear equations and explore their numerical solutions.Artificial neural networks（ANNs）,also known as neural networks（NNs）,have highly parallel structure and parallel implementation capability,and are highly robust and faulttolerant.In particular,Zhang neural networks（ZNN）,developed from recurrent neural networks（RNN）,have been applied to solve a variety of different time-varying problems.In this paper,we systematically study the systems of time-varying constrained nonlinear equations problem from three aspects: the analysis of ZNN model,the construction of discrete-time ZNN model and the extension of the methods for static problems.（1）The ZNN model is used to solve the time-varying constrained nonlinear system of equations.Firstly,the systems of time-varying constrained nonlinear equations are transformed into the time-varying optimization problems.Then,the original problems are transformed into the time-varying unconstrained nonlinear system of equations based on the KKT conditions.The ZNN model is further used to solve the transformed time-varying problem.The corresponding convergence theorems are then given.Finally,the validity of the method is verified by numerical experiments.（2）The generalized discrete-time ZNN model is constructed.We analyze the structure of the continuous-time ZNN model.Based on the construction of linear multi-step methods in ordinary differential equations,we compare the coefficients by Taylor’s expansion.Combined with the convergence theorems,two algorithms are proposed for constructing discrete-time ZNN models with any number of steps and any order.Further we analysis the orders of the models.Finally,comparative numerical experiments are used to verify this analysis.（3）The method of solving static problems is extended to solve time-varying problems.For the classical penalty function method,the penalty functions are not differentiable due to the adding of the penalty terms.In order to overcome the nondifferentiability,the classical penalty functions are first smoothened to obtain the modified penalty function methods.Then,the time interval of the time-varying problem is partitioned equidistantly,and the modified penalty function method is used to solve the problem at each time node.By using the iterative solution of the previous time node as the initial value of the current time node,the computation time can be greatly reduced.Numerical experiments show that this method is effective in solving this type of time-varying problems.