Bounded Variable Constrained Nonlinear Optimization Problems Affine Conjugate Gradient Path Method And Its Applications

In this paper, we propose a new approach of affine scaling interior discrete conjugate gradientpath for solving bound constrained nonlinear optimization, and its applications to unconstrainednonlinear systems.Conjugate gradient method, which can be easily computed and merely requires the first or-der information with small storage, is one of the most popular method for solving large scaleoptimization problems. Bulteau and Vial [1] formed a conjugate gradient path of unconstrainedoptimization. The path is defined as linear combination of a sequence of conjugate directionswhich are obtained by applying standard conjugate direction method to approximate quadraticmodel of unconstrained optimization. This path is continuous with respect to the parameterτin the conjugate gradient path. But the continuous gradient path needs to form the entire pathfirst, and then to search, which will increase the total computation and may result in the difficultyof constructing the path. In this paper, discrete path will be used for solving bound constrainednonlinear optimization to avoid the difficulty and reduce the computation. Theoretically, we onlyneeds to construct part of the conjugate gradient path to solve the approximate quadratic functionat every iteration such that the efficiency of the algorithm is greatly improved. It has relative su-periority especially to the large scale optimization problem.Coleman and Li [3] proposed the double-trust region method for the nonlinear minimizationsubject to bounds, and overcome the difficulty imposed by the bound constraints via construct-ing an affine scaling matrix. Following this idea, an affine scaling matrix is introduced in orderto transform the bound constrained optimization to unconstrained optimization, consequently weget the Newton iteration of this unconstrained problem. We get the iterative direction by solv-ing quadratic model via constructing preconditioned conjugate gradient path. Combining interiorbacktracting line search, we obtain the next iteration. Global convergence and local superlinearconvergence rate of the proposed algorithm are established on some reasonable conditions.Ortega and Rheinboldt introduced the iterative solution of nonlinear equations in several vari-ables in [13] systematically. In this paper, we construct the merit function of nonlinear equations,and consider the approximate linear model of each iterative point. According to the idea of thealgorithm designed in chapter 2, we consider its application to nonlinear equations, and proposethe conjugate gradient path algorithm combining the nonmonotone line search technique. Globalconvergence and local superlinear convergence rate of the proposed algorithm are established onsome reasonable conditions. Numerical experiments show the effectiveness of the algorithm.