Bounded Variable Constraints Of Nonlinear Equations Of The Affine Conjugate Gradient Path Method

Optimization theroy and method, which makes research on how to find the optimal solu-tion among many feasible plans, is very popular and useful subject. Optimization technology iswidely applied in many fields such as defense, industrial and agricultural production, transporta-tion, financial, trade, management, scientific reach. With the development of the computers,optimization theroy and method is playing an increasing role in practical application.Conjugate gradient method, which can be easily computed and merely requires the first orderinformation with small storage, is one of the most popular method for solving large scale opti-mization problems. Bulteau and Vial [1] 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 model ofunconstrained optimization. In the thesis, by employing a affine scaling matrix and constructing asuitable quadratic model, we propose a approach of affine scaling interior discrete conjugate gra-dient path for solving nonlinear equality systems subject to bounds on variable. This method onlyneeds to construct part of the conjugate gradient path to solve the approximate quadratic functionat every iteration such that to reduce the computation and improve the efficiency of the algorithmgreatly. Global convergence and local super linear convergence rate of the proposed algorithmare established on some reasonable conditions. The numerical results of the proposed algorithmindicate to be effective.For solving the constrained nonlinear equations, we propose a method of inexact Newtonaffine scaling interior discrete conjugate gradient path. A classical algorithm for the problem isNewton's method. The method is attractive because it converges rapidly for any sufficiently goodinitial point. However, solving a system of linear equations exactly at each stage can be expensiveif the number of unknowns is large. But the drawbacks can be overcomed to some extent if wesolved linear equations inexactly. According to this characteristic, we combine it with discreteconjugate gradient path. we can obtain a sequence of conjugate direction by the preconditionedconjugate gradient method. Checked by the inexact Newton method , we will get a descent direc-tion . Combining with the non-monotone technique, we will find an acceptable trial step lengthalong this direction which is strictly feasible and make the objective function decreasing. Undersome reasonable assumption, the feasibility of the algorithm is proved.Finally, the last chapter concludes the main results of this paper and proposes some furtherresearch directions.