Lanczos Path Method To Optimization Problems

Optimization has a very widely applied in many fields, it is a tool which choose the optimalmethod in finite or infinite methods. In this thesis, we propose Lanczos path algorithm for solvinglinear equality constrained optimization and box constrained optimization.For the large-scale sparse symmetric eigenvalue problems: Ax =λx,we solve these problemby Lanczos path method. Through triangulating the matrix A, we obtain a triangular matrix whichwe sign as T. It is important that there can not generate a full sub-matrix in the algorithm process,and the eigenvalues of A much early appear in the triangulating the matrix A process.The conjugate gradient method is the most popular method in optimization. Because it canbe easily programmed and computed, this method has become an important method for solvinglarge-scale problems. Bulteau and Vial formed a conjugate gradient path of unconstrained opti-mization. The path is defined as linear combination of a sequence of conjugate directions whichare obtained by applying standard conjugate direction method to approximate quadratic functionof unconstrained optimization. we can get some properties of the path based on the conjugateproperty of direction sequence. Not only had the path global convergence, but also it has localsuper-linear convergence rate.In this thesis, we combine Lanczos method with conjugate gradient method, and construct anew path ,which has both properties of Lanczos vectors and properties of conjugate gradient path.The main idea of this method is that we employ Lanczos method while employ conjugate gradientmethod for the approximate quadric model of the optimization, we can obtain the sequences ofLanczos vectors and conjugate gradient vectors, then construct Lanczos path with them.The prop-erties of this path are similar to the conjugate gradient path. It is important to analysis the globalconvergence and the local super-linear convergence of the algorithm.The thesis consists of four chapters. In chapter 1, we present some basic concepts aboutoptimizations. The preconditioning Lanczos path for equality constrained optimization is the topicof chapter 2. In chapter 3, we gives an affine scaling interior projected Lanczos path algorithmfor box constrained optimization. The last chapter concludes the main results of this thesis andproposes some further research directions about our work.