Unconstrained Optimization Methods Based On Negative Curvature Directions And Nonmonotone Techniques

The unconstrained optimization problem min x∈R~n f(x) is considered in this thesis, where f(x), the objective function, is a real-valued twice continuously differentiable function, which we assume is bounded below. This class of problems are the most basic and important optimization problems.When the Hessian matrix of f(x) is indefinite, for the line search method or the curvilinear search method to solve the above unconstrained optimization problem, a direction of negative curvature, should be exploited. On the other hand, the nonmonotone technique is proved to be very effective and more and more popular. In this thesis, we engage in studying the algorithms which combine the negative curvature direction and nonmonotone technique to solve the unconstrained optimization.Chapter 1 of this thesis introduces the background of our research and some notations and definitions that will be used in our work. In Chapter 2, we mainly discuss three nonmonotone second order steplength rules and an algorithm which uses these rules. The convergence results of this algorithm are described. A new adaptive nonmonotone line search method is presented in Chapter 3. The iterates produced by this new algorithm can converge to a second order stationary point. Numerical results show the efficiency of the algorithm. And in Chapter 4, we present two modifications of the nonmonotone preconditioned modified gradient path algorithm which was proposed in [31]. The global and superlinear convergence results of the new algorithms are established. Numerical tests show that our modifications are efficient and robust. Finally, some conclusions and perspectives are presented in Chapter 5.