Nonmonotone Smoothing Newton Methods For Systems Of Nonlinear Equalities And Inequalities

As an important mathematical structure,the systems of equalities and inequalities have been widely used in many fields.The discussion on how to effectively solve the systems of equalities and inequalities has attracted much attention from scholars at home and abroad.At present,there are many excellent research results,but as the systems of nonlinear equalities and inequalities,there are still many problems to be improved and solved.In this paper,the smoothing Newton's algorithms for solving the systems of nonlinear equalities and inequalities are mainly studied and improved.Firstly,for the systems of nonlinear inequalities,by constructing a new smoothing function,the problem is approximated via a family of parameterized smooth equations A smoothing Newton method is developed by adopting a new nonmonotone line search scheme.By solving the system of smooth equations,a feasible solution of the problem is found.Under mild assumptions,our algorithm is shown to possess global and local quadratic convergence properties.Finally,numerical experiments are performed.The numerical results show that the algorithm is feasible.Secondly,this paper extends the research scope to the systems of nonlinear equalities and inequalities,which are more general than the systems of nonlinear inequalities.For the mixed systems of nonlinear equalities and inequalities,by introducing a new smoothing function,the problem is approximated via a family of parameterized smooth equations.A new smoothing Newton method is developed by adopting a modified nonmontone line search technique.By solving the system of smooth equations,a feasible solution of the problem is found.And the global and local superlinear convergence of the algorithm are proved under mild assumptions.Finally,numerical experiments are performed.The numerical results show that the algorithm is feasible.