Linear Constrained Optimization Affine Interior Point Conjugate Gradient Path Method And Its Application

Optimization methods are widely used in many fields. This paper is devoted to a new ap-proach of affine scaling interior conjugate gradient path for solving nonlinear optimization with linear equality and inequality constraints, and its applications to unconstrained nonlinear systems.Conjugate gradient method, which can be easily computed and merely requires the first order information with small storage, is one of the most popular methods for solving large scale opti-mization problems.Bulteau and Vial formed a continuous conjugate gradient path of unconstrained optimiza-tion. The path is defined as linear combination of a sequence of conjugate directions which are obtained by applying standard conjugate direction method to approximate quadratic model of un-constrained optimization. But the continuous gradient path needs to form the entire path first, and then to search, which will increase the total computations. In order to improve the computation efficiency, discrete path will be used for solving bound constrained nonlinear optimization. The-oretically, we only need to construct part of the conjugate gradient path to solve the approximate quadratic function at every iteration such that the efficiency of the algorithm is greatly improved.From another perspective, linear equality and inequality constraints are involved. So the sub-problem is equivalent to an unconstrained optimization problem in the null space. The path is defined as linear combination of a sequence of conjugate directions which are obtained by apply-ing discrete conjugate direction method to the approximate quadratic function in the null space.In this paper, we propose an approach of affine scaling interior discrete conjugate gradient path for solving nonlinear optimization with linear equality and inequality constraints. We obtain the direction by solving quadratic model via constructing discrete conjugate gradient path. By combining interior backtracking line search, we obtain the iteration. Based on some reasonable conditions, the global convergence and local super linear convergence rate of the proposed algo-rithm are established. Finally, we calculate some standard test examples by Matlab to illustrate the effectiveness of the proposed algorithm.The thesis consists of five parts. In Chapter 1, we summarize the basic concepts of optimiza-tion technique, conjugate gradient method and the properties. In Chapter 2, we analyze related theories after giving the core algorithm. In Chapter 3, global convergence and local convergence rate of the proposed method are established under some reasonable conditions. Finally, in Chapter 4, we conclude the main results of this thesis and propose some further research directions about our work.