A Modified Method For Solving Nonlinear Systems Of Inequalities

The research on numerical methods of solving the system of inequalities is at a very important position in the field of numerical calculation,and this thesis is on the numerical optimization algorithms of solving inequality systems.The systems of in-equalities in this thesis include over-derdetermined inequality systems,which have no legal solution,and standard inequality systems,which have at least one solution.When some of inequalities become equations,the system therefore becomes a hybrid system of inequalities and equations.It is worth mentioning that the system of equations is also a special case of the systems of inequalities,since each equation can be written in the form of two inequalities,whose directions of the unequal signs are opposite.The most special case takes place when all the unequal signs in the inequality system become equal signs.In this thesis,solving the system of inequalities is converted into solving the system of equalities.This thesis engages in solving the optimization problems of equality systems in stead of solving the inequality ones by converting the form to which,in the end,be-comes least-square problems.By calculation,the error estimate which generates from converting the form is obtained.Based on the most well-known traditional methods in least-square optimization,Gauss-Newton algorithm and Newton algorithm,a tensor-form of Newton-class algorithm is used in this thesis.The algorithm constructs the quartic form of the iteration step s.In each iteration step,the algorithm needs to solve a subproblem to obtain the iteration step s.When solving the subproblem,a hybrid algo-rithm is used to achieve the purpose of convergence.At the same time,by monitoring the function value and the norm of Jacobian matrix,an algorithm switching mechanism is set up in the main loop of iterations.Therefore,an algorithm named GLTN is pro-posed for the purpose of making the algorithm more efficient,so that it can complete in fewer iterations.When the iteration point is close enough to the optimal solution,that is,when the iteration point is in a closed ball with the optimal value as the center,and the function is convex in the closed ball,the regularization parameter ? vanishes so as to reduce the error caused by the regularization parameter in the last steps,and thus reduce the number of iterations near the optimal value.Later,this thesis analyzes the convergence of the algorithm,and compares the performance of the four algorithms in20 cases,each with five different initial values by numerical experiments,which veri-fies the advantages of GLTN algorithm.Numerical experiments show that the success rate of GLTN algorithm is 1.92 times that of Newton method and 1.57 times that of tensor Newton method.