Adaptive Nonmonotone Trust Region Method

The trust region method has a strong convergence.And the initial point does not need near the optimized point in the prove of the convergence and no need much more suppose of the function,also no need the Hessen Matrix positive defined.So many people are studding in this field.The key of the trust region method for unconstrained optimization is the acceptable level of agreement and the trust region radius.A new nonmonotone line search algorithm proposed by Hongchao Zhang is used in this article for adjust the acceptable level of agreement between the quadratic model and the objective function. After that,we adjust the trust region radius by a nonnegative integer P and the radius isα_k =c~p‖g_k‖~γ,because all the adaptive trust region radius is adjust by constant C,gradient and two times differential information.But numerical results show that C is insensitive.And gradient include in two times differential information.So we can adjust the radius only by gradient information.For the trust region methods,the choice of the initial radius is very important,since it affects the running efficiency of the methods.In traditional methods it is no general rule for the choice of the initial trust region radius.So it is best to use adaptive method.In sectionⅠ,we describe the development of usual trust region method,solution of the subproblem,the nonmonotone line search and the adaptive trust region method.In sectionⅡ,we proposed the algorithm and proved the global convergence and the local quadratic convergence;In sectionⅢ,numerical results show that the algorithm is efficiency.