Research On The Algorithm Of Trust Region With Conic Model

Modern economic conditions are characterized by strong turbulence and environment complexity, as well as by the deficiency of resources, time and information in an enterprise. Because of that, the problems of economic systems management are becoming more complex. The application of scientific methods in studying and solving the management problems has been an obvious necessity. Then a number of very suitable and effective methods for making optimal solutions of the economic problems was formulated. These methods are known as operations research, they are of great importance in the process of decisionmaking.Optimization is an important branch of operations research, which is theoretically important as well as practically urgent in many fields. In recent years, with the development of modernization production, optimization theory and methods are finding wider and wider application.Trust region method is a class of important numerical method for solving nonlinear optimization problems. Because of its remarkable numerical reliability and strong convergent theory, researchers in nonlinear optimization area have paid great attention to it for recent years. At present, trust region methods have applications in constrained optimizations, large-scale optimizations, nonsmooth optimizations and many optimization problems. The problem of minimizing a function of several variables is usually attacked by trust region method based upon modeling the function locally by a quadratic function. The conic model is a generalization of the quadratic model and can take into account more information from previous iterations. However, if the objective function has strong non-quadratic behavior or its curvature changes severely, the quadratic model methods often produce a poor prediction of the minimizer of the function. Moreover, many researches indicate that conic functions may serve well as a local model for a general function. During the past twenty years, the conic model has attracted the attention of more and more researchers.The key is to solve the subproblems by trust region algorithms. The new trust region subproblem with conic model proposed in 2005 cancelled the limitation on the level vector. A new feasible set of the subproblem was given, and the subproblem was divided into three different cases and three trust region subproblems with new conic model were established. At the same time, necessary and sufficient optimality conditions for the solutions of all different cases are given. The purpose of this dissertation is to give a series of new algorithms which are based on the trust region subproblem with new conic model for solving nonlinear optimization problem. The dissertation consists of eigtht chapters. Chapter 1 is introductory. The purposes, significance, contents and its research status of this dissertation were discussed. Chapter 2 contains preparatory material for the conic model trust region method. It consists of mainly fundamental theory on the conic model trust region method, in particular, the necessary and sufficient optimality conditions for the trust region subproblem with new conic model. Chapters 3-7 are our main contribution. Chapter 3 analyzes the monotonicity of conic function in the steepest descent direction, and an expression of the steepest descent point of new conic model subproblems is given. Then an estimate of the decrease in conic model function achieved by the steepest descent point is obtained, and it is proved that this point satisfies the descent condition which guarantees the global convergence of the conic model trust region method for solving unconstrained optimization problem. Chapter 4 discusses the properties of conic function of new subproblem, and analyzes the monotonicity of conic function in the gradient or in the line section between gradient and conic quasi-Newton's direction. We derive a general expression of the solution of new conic model subproblems. Then a dogleg algorithm for solving new trust region subproblems of conic model is proposed. Based on the results of Chapter 4, in Chapter 5, we restrict the parameter?0 of new subproblems, a modified dogleg method for solving the new conic model subproblems is proposed, and a quasi-Newton algorithm based on the modified dogleg method is developed. Chapter 6 covers a conic model trust-region algorithm based on conjugate gradient. Firstly the monotonicity of conic function in the conic conjugate directions is discussed, and the descent properties in the conic conjugate direction are proved. And a conjugate gradient algorithm for solving new conic model trust-region subproblems is present. Then we develop a conic model trust-region algorithm based on conjugate gradient for solving unconstrained optimization problem. Chapter 7 proposes a trust region method with new conic model for linearly constrained optimization problems. With null space technique, linear equality constraints are deleted in the trust region subproblem. The modified dogleg method is used to solve the transformed subproblems.In the dissertation, the global convergence of all proposed methods are established and proved under some reasonable conditions. Numerical experiments for all these methods are widely carried out. Results show that these methods are very significant and worth to study further.