Some Researches On The Algorithms For Solving A Class Of Nonlinear Equations And Inequalities Problems

In this thesis, the algorithms for solving symmetric nonlinear equations and nonlinear inequalities problems are discussed.In chapter one, a trust-region-based modified Newton method for solving sym-metric nonlinear equations is studied. At first, the symmetric nonlinear equations are reformulated as a kind of unconstrained optimization problem. Then the trust-region-based modified Newton method is presented for solving symmetric nonlinear equations. In this algorithm, the searching direction by solving nonlinear equations. Besides, the iteration sequences generated by this algorithm are included in a level set which is bounded. The sequences generated by this algorithm globally converge to the solution of symmetric nonlinear equations under some additional assump-tions. What's more, this algorithm has the property of local quadratic convergence. Two examples are quoted as the numerical examples which are computed under different dimensions of the problems including 10,100,200,300,400 and 500 with five sets of initial values, which powerfully illustrate the efficiency of the algorithm presented.In chapter two, an adaptive simulated annealing floating point genetic algo-rithm for solving nonlinear inequalities problem is studied. At first, the nonlinear inequalities problems are reformulated as a set of nonsmoothing equations, then by making use of the smoothing auxiliary function the problem was equally converted as the smoothing unconstrained optimization problems. Based on this transformula-tion, the adaptive simulated annealing floating point genetic algorithm is presented, which not only posses the global searching capability of adaptive simulated an-nealing algorithm and genetic algorithm, but also make up the shortage of genetic algorithm. In the end, the numerical experiment powerfully illustrate the efficiency of the algorithm presented.In chapter three, the whole thesis is generated. Besides the unsolved problems in this field and some suggestions for our further study are also pointed out.