Non-monotonic Automatically Determines The Trust Region Radius Of Trust Region Methods

Regarding the trust region method for unconstrained optimization problem, its main computation is the solution of the trust region subproblem. And the selection of the trust region radius is very important to the solution because the trust region radius decides the current iterative direction and the length of step. It is well known that the traditional trust region radius△_k is independent of g_k,B_k. So, at each iterate point x_k which is far from the optimum x^*, we do not know whether the quasi-Newton setp -B_k^-1g_k is feasible; even the test condition of the merit function is satisfied. This situation would decresase the efficiency of these methods. Fourthermore , choice of△₀ also affects the efficiency of these methods, but there does not exist any gereral rule on choosing△₀.The self-adpative trust region method is an effective method for unconstrained optimization problem. In this paper, we present a new self-adaptive trust region method, in which the trust region radius depends on the gradient g_k and the ratio‖y_k-1‖/‖δ_k-1‖with y_k-1=g_k-g_k-1 so that the self-adpative trust region method proposed by zhang et.al.[26] can be improve. In addition, the new self-adaptive trust region method with the nonmonotone technique proposed by zhao[37] is presented and analyzed. Acrodding to the ideas of the hybird dogleg method proposed by Zhang[36], we solve the trust region subproblem basing on the improved hybird dogleg path method which applys safely positive definite matrix B_k. Based on some appropriate hypothesis, we analyz the improved hybird dogleg pathΓ_oplsatisfy both properties (P1) and (P2). The global and local superlinear convengence results of the algorithm are established. Finally, some numerical results are reported show the new nonmonotonic self-adpative trust region method is efficient.