Unconstrained Optimization Problem For A Class Of Nonmonotone Trust Region Algorithm,

In this paper, we mainly study nonmonotonic trust region methods. At present, trust region method and line search method are mainly two typies of numerical algorithms for nonlinear optimization. Compared with line search method, trust region method has new idea, robustness, validity, strongly convergence. In view of the advantage of trust region method, many researchers in nonlinear optimization area focus on structuring the new algorithm by trust region method. However, people found that monotonic algorithm could't pledge the validity for some examples. In 1986, Grippo and others [19, 20] posed a kind of nonmonotonic line search technique and applied it to Newton method and cutted Newton method. In 1993, Nai yang.Deng and others [21] firstly applied nonmonotonic technique to trust region method and proved its global and superlinear convergence under suitable conditions. Numerical experiments showed that the nonmonotonic trust region method was superior to the corresponding monotonic method for some problems. The nonmonotonic technique mentioned above is realized by f_l(k) = , where m(0) = 0, 0 ≤ m(k) ≤ min[m(k - 1) + 1, M], M is the given positive integer. However, for some test function, the numerical results of this nonmonotonic technique relies on the choice of M. Therefor, Xiao wu. Ke and Ji ye.Han [22] proposed that after some step, the function value of the current iterate point should be compared with the maximal function value in some former iterate points(M_k + 1), where the number of iterate points was not fixed, moreover, M_k could be adjusted. Toint [23] proposed the adaptive nonmonotonic trust region algorithm NMTR2, where the parameter M was defied by the algorithm itself and wasn't a prefixed value. Zhang H C and W.W.Hager [26] proposed a new nonmonotonic line-search method, which successfully avoided the preceding shortcoming. Besides, in 1980, Davidon [28] firstly proposed the conic model which is more general than quadratic model. Based on the preceding work, we propose two kinds of new nonmonotonic trustregion algorithms.In chapter 1, firstly, we introduce the development of optimization and some extensive optimality conditions which to decide the optimum solution. Secondly, We review the basic idea of line search methods and trust region methods for unconstrained optimization problems and theirs research works.In chapter 2, we present a quasi-Newton nonmonotonic trust region algorithm. Unlike traditional nonmonotonic trust region algorithms, our algorithm gets the next point by the nonmonotonic Wolfe line search at each iteration. This new algorithm not only does not resolve the subproblem but also satisfies the quasi-Newton condition at each iteration and simultaneously maintains a positive-definite approximation to the Hessian of the objective function. Under mild conditions, we prove the global convergence and Q-quadratic convergence of the algorithm. Numerical results show its efficiency.In chapter 3, we present a nonmonotonic trust region algorithm based on the conic model. The algorithm based on the quadratic model is a special exmaple of our algorithm. Our algorithm overcomes a shortcoming, i.e. the reference function value used to generate non-monotonicity relys on some positive integer M. When the trial step is not accepted, we employ a nonmonotomic line search to reduce the cost. Under mild conditions, we prove the global convergence and Q-quadratic convergence of the algorithm. Numerical results show its efficiency.