A Couple Of Modified BFGS Algorithms For Large-scale Symmetric And Positive Definite Matrix Largest Eigenvalue Problems

In this thesis,based on Armijo line search,we propose a couple of numerical algorithms for solving the largest eigenvalue problem of large-scale real symmetric and positive definite matrices: the cautious BFGS method and the limited memory BFGS method.Convergence properties of the proposed algorithms are analyzed.Numerical experiments are also reported,which illustrate that the proposed methods are efficient and promising.In Chapter one,we simply introduce the optimal solution of unconstrained problem,the descent direction and the line search rules.We recall the Newton method and the quasi-Newton method for unconstrained minimization.Specially,we list some relevant results for solving the eigenvalue of matrix and optimization models and algorithms for solving the largest eigenvalue of high-dimensional matrices.Finally,we briefly state the main contributions of this thesis,and list some symbols which are used at the context.In Chapter two,based on solving the unconstrained optimization problem,we propose a cautious BFGS method for solving the largest eigenvalue problems of large-scale real symmetric and positive definite matrices.The method effectively avoids the problem of solving the inverse of the large-scale Hession matrix.Then,we prove the global convergence of the algorithm under some appropriate conditions.Finally,we compare our method with EIGS(a matlab implementation for computing the largest eigenvalue of matrix).The numerical experiments show that the proposed method is stable,fast and efficient.In Chapter three,this study aims to present a limited memory BFGS algorithm to solve non-convex minimization problem,and then use it to find the largest eigenvalue of a real symmetric and positive definite matrix.The proposed algorithm is based on the modified quasi-Newton equation and uses an Armijo line search.The proposed algorithm converges to a stable point of the minimization problem without convexity assumption on the objective function.Furthermore,we do extensive experiments to compute the largest eigenvalue of the symmetric and positive definite matrix of order up to 54,929 from the University of Florida(UF)sparse matrix collection,and do performance comparisons with EIGS.Although the proposed algorithm converges to a stable point,rather than a global minimum theoretically,the compared results demonstrate that it can find the largest eigenvalue.In Chapter four,we list some concluding remarks and further research topics.