Algorithm And Application Of Optimization Problems With Orthogonal Constraints

In this thesis,we study the optimization problems with orthogonal constraints because it is a representative non-convex nonlinear problem and has a wide range of practical applications.For instance,it applies to electronic structure calculation in density functional theory,face recognition,computer vision,signal processing and clustering etc.Therefore,the orthogonal constrained optimization problem has become an active research area.In this thesis,we mainly study the optimization problems with orthogonal constraints.We study three algorithms for solving such problems and prove the convergence of the given algorithms on the basis of the existing algorithms.Finally,the effectiveness of the algorithms is verified by several numerical experiments.The main contributions of this thesis are as follows:(1)We study a projection-based non-monotone line search algorithm to solve the optimization problem on Stiefel manifold.The non-monotone line search method has great advantages to solve optimization problems with orthogonal constraints.We propose a new search direction,then we give a projection-based non-monotone line search method.And we prove the convergence of the proposed algorithm.Finally,the numerical effects of several projection-based non-monotone line search methods are studied by numerical experiments,and the effectiveness of the algorithm is verified.The experimental results show that the proposed algorithm has better performance in some numerical values in the existing methods.(2)In order to solve the optimization problems with linear equality constraints and orthogonal constraints,we give a proximal augmented Lagrangian method and a parallel column-wise block computing method.It is well known that using the augmented Lagrangian method to solve nonlinear optimization problems has some disadvantages.When using the augmented Lagrangian method to solve the optimization problem with linear equality constraints and orthogonal constraints,we use the proximal methods and the parallel computing method to calculate the matrix variables,simultaneously using a symmetric Lagrangian multiplier.Then we give two algorithms.And we demonstrate the validity of the given algorithms by several numerical experiments.The experimental results show that the given algorithm has obvious advantages in some numerical values compared with several existing algorithms.