Nonlinear Complementarity Problems Smooth Trust Region Algorithm

The nonlinear complementarity problem is an important type of variational inequality. It has wide applications in economics, operations research, control theory, transportation, etc. In recent years, more and more scholars eager to study the numerical methods for nonlinear complementarity problem, which results in an endless stream of various algorithms. Most of these methods apply the line search strategy or nonsmooth trust region method to solve the nonlinear complementarity problem. But they seldom use the smooth trust region algorithm.In view of this, we use the smooth trust region algorithm to solve the nonlinear complementarity problem. This approach should be based on complementary function to determine the smooth approximation function, and then iterative formulation of the smooth factor to ensure the global convergence of the algorithm.In this paper, firstly, we use the smooth approximation function to convert the nonlinear complementarity problem into a optimization problem, next combine the trust region algorithm with the nonmonotonic techniques and the PSO methods, then we propose three new methods for nonlinear complementarity problem.Chapter 3 shows the nonmonotone and completely smooth trust region algorithm for the nonlinear complementary problem. This algorithm interates parameters and the unknown variables as the equivalent variable, and adopts a 'nonmontone ratio' to approximate the reduction of the object functions. When the ratio satisfy certain conditions , we accept this iteration.By using a distinct and smooth approximation function from Chapter3, Chapter 4 presents a new nonmonotone smooth trust region algorithm and adjusts the reference function value of nonmonotonic techniques. A smooth parameter with simple format is constructed. Under certain conditions, this algorithm has global convergence.In chapter 5, based on the algorithm in chapter 4, we obtain the mixed algorithm by introducing the particle swarm algorithm. In the iterative process, we use the particle swarm algorithm to amend the bad points, which makes it get the best convergence.