Bounded Constrained Nonlinear Indeterminate Equations Of Interior Affine Scaling Trust Region Methods

In this paper, we combine nonmonotone interior backtracking line technique, affine scaling method and trust region algorithm so as to develop, analyze and solve the underdetermined equa-tions and underdetermined equations and inequalities. Among the trust region algorithms known to us, how to solve the subproblem is still worth studying, especially the affine scaling trust region algorithm with interior backtracking technique. And in the most papers, we can read the strict complementarity assumption to ensure the global and local convergence. But in reality, when solving the underdetermined equations, we can't avoid that the strict complementarity condition doesnot hold at the solution, which restrict the application of relevant algorithm. And in this paper, we transform the problem and construct a trust region subproblem. Meanwhile, we introduce the identification function and its relevant set, and with the given affine scaling and without the strict complementarity condition, we can get global and local convergence which therefore strengthen the application field of our algorithm.And, line search and trust region are the two basic ways to ensure the global convergence. We also apply the two ideas in our algorithm to ensure global convergence and speed up the convergence progress in the contours of objective function with large curvature.