Nomonotone Trust Region Algorithms For Unconstrained Optimization

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 researchIn Chapter 2, for the unconstrained optimization, we propose a nonmonotonic BFGS-trust-region algorithm, applying nonmonotonic algorithm to solve the problem of trust-region, people have acquired large achievements, the key of this paper is to propose a new BFGS formula. The advantage of this algorithm is that the subproblem of trust-region method ensures the update matrix is positive, that is the subproblem is a strictly convex quadratic programming, we combine correlation to prove the algorithm possesses global convergence under suitable conditions.In Chapter 3 , a nonmonotonic trust region algorithm of unconstrained optimization is given, this algorithm is the improvement and the promotion compared to the algorithm which was proposed in [20], in this paper , the algorithm enlargesthe value scope of (?) when r_k < 0. Then r_k may increase such that the algorithmhas a quicker iteration rate. On the other hand, in order to prove its convergence and the rate of convergence, we demonstrate that the algorithm is globally convergent and the rate of convergence is superlinear without the use of assumption. This may enlarge the applications of nonmonotomic trust region method .In Chapter 4, change the trust region subproblem into equivalent subproblembased on Bunch-Parlett decompose of, contruct the conjugate gradient path, then find the optimal solution of the problem by the path using nonmonotomic trust region method. The method has no positive contrations on Hessn metric. And further prove that the algorithm has global convergence and quadratic convergence properties.In Chapter 5, combine the new trust region subproblem proposed in[17] withthe nonmonotone technique to propose a new algorithm for unconstrained optimization-the nonmonotone adaptive trust region method. The global convergence properties of the algorithm is proved.