Adaptive Nonmonotone Trust Region Methods

Nonmonotone trust region method applies nonmonotone technique to trust region method, which does not restrict a monotone decrease of the function value at each iteration. So it is useful for the convergence rate of the minimization process, especially in the presence of steep-sided valleys. Because of its strong convergence theory and good numerical performance, it is regarded as a class of important numerical methods and has been paid great attention for many years by researchers in unconstrained optimization.In chapter 1, we briefly introduce the achievement of trust region method and our main research.In chapter 2, we give a new reference function value which can modify itself adap-tively and present an adaptive nonmonotone trust region method. With some conditions satisfied, a smaller one is chosen as our reference function value among candidate reference function values, which can avoid that the reference function value is much larger than the function value, and can improve the rate of convergence. Under mild conditions, we prove that our algorithm has the properties of global convergence and quadratic convergence. Primary numerical results are encouraging.In chapter 3, on the basis of the new reference function value in chapter 2, a hybrid method combining nonmonotone trust region and line search is presented. At each iteration, the method gives priority to nonmonotone trust region method. But if a trial step is not accepted, a new direction satisfying given conditions is found, then the method turns to the line search method and gets a new iterative point. So, a large quantity of computation by solving the trust region subproblem repeatedly is avoided. Under suitable conditions, global convergence and quadratic convergence are well proved. Primary numerical results show that the method is efficient.