Dwindling Filter Methods For Unconstrained Optimization

Many important problems can be expressed in terms of nonconvex nonlinear multivariate unconstrained optimization problems. Two broad classes of algorithms for the unconstrained optimization problems are line-search algorithms and trust-region algorithms. Since filter technique was introduced by Fletcher and Leyffer in 1997 and subsequently published as [13], filter technique has been studied and applied in many areas, and it performs well.We proposed a new dwindling filter technique for large-scale unconstrained optimization problems. Generally, for the purpose of global convergence, the multidimensional filter [21,25. 27] is covered by a fixed envelope. Since a fixed envelope is not suitable for backtracking line-search process, we further study it in this thesis and put forward a conception of dwindling filter technique. The main idea is that the dwindling filter envelope is not fixed, but becomes thinner and thinner as the step-length of the backtracking line-search becomes smaller and smaller.Naturally, combination of the dwindling filter technique and the second-order line-search method is a very significant algorithm for unconstrained optimization. The new algorithm is shown to be globally convergent to one second-order stationary point at least. Preliminary numerical experiments on a set of CUTEr test problems indicate that the new algorithm is very competitive with some classical line-search algorithms. We show that the linked-list construction in the FORTRAN language works well and that the storage of the dwindling filter is medium.And then, we notice that the trust-region method is always more effective than line-search method. This inspire us to apply the dwindling filter technique in trust-region line-search framework to give another new algorithm. We prove that, under reasonable assumptions, the new algorithm converges to one second-order stationary point at least. Some preliminary numerical results on a set of CUTEr test problems are reported, which show that this new algorithm has a significant performance and that the storage of the dwindling filter is medium as well.