A New Gradient Path In Unconstrained Optimization

The two most important algorithm frameworks to optimization problems are line search method and trust-region method. One of the key parts of these methods are search direction and solving trust-region subproblem.In this paper, we focus on the topics on the nonmonotonic line search method and nonmonotonic trust-region method for unconstrained optimization.In chapter 2, we mainly consider the approximation of Hessian matrix and the application of nonmonotonic technique to line search method. Instead of the monotone sequence, the nonmonotone sequence of function values are employed. The new gradient path algorithm by approximate Hessian matrix for unconstrained optimization problem improves the computational effectiveness. The numerical results show that our algorithm is competitive.In chapter 3, we study trust region algorithms. We solve a trust region sub-problem at each iteration. Among the methods solving the subproblem, the traditional methods would compute B_k and its inversion, so it's very expensive. In this paper, we propose a new gradient path trust region algorithm by approximating Hessian matrix. The algorithm has good convergence properties under commonly used conditions.