The Modified Lanczos Method For Solving Large-Scale Nonlinear Sparse Optimization Problem

This thesis discusses the methods for solving large-scale nonlinear sparse optimization problems and derives four modified algorithms for dealing with different cases based on the truncated Newton algorithm.First of all, we propose a new algorithm, named modified Lanczos method (SML) for solving large-scale unconstrained optimization problems on the basis of the truncated Newton algorithm. Because the Lanczos method is adaptive to the indefinite and definite linear systems, the algorithm has not only nice convergence property, but also good stabilization.Secondly, we apply the Modified Lanczos method to the large-scale bound constrained problems. Based on an efficient technique of active set, we propose a new algorithm named active set SML method. The algorithm has not only nice convergence, but also good stabilization.At last, we analyze the Modified Lanczos mehod again and we find that there are some valuable information not made full use of when we are solving indefinite linear systems. Therefore we propose two algorithms. One is the SML method with curvilinear search technique (SML-CS). The other one is the SML method with adaptive linear search technique (SML-ALS).We have given theoretical analysis on each new algorithm and done many numerical experiments and comparisons. Theoretical results and numerical experiments show that new modified algorithms have better practical performance significantly less computational costs, less CPU time and better exaction.