Unconstrained Conjugate Gradient Optimization Methods and TheirInterrelationships

摘要

Connections between conjugate gradient algorithms and the corresponding two-dimensional Newton methods are investigated. All the conjugate gradient algorithms were implemented and used the same line search routine. The line searches were quite exact with at least one cubic interpolation, forced at each iteration. The algorithms were used on nine test functions, such as the extended Rosenbrock, Beale, Wood and Powell functions, the trigonometric function, the penalty function, the tridiagonal function, the extended Powell singular function and the matrix square root 2 function. The assumption, that the nearer a conjugate gradient algorithm is to its two-dimensional Newton counter part, the greater its efficiency will be, is verified by numerical experimentation. Geometrical interpretations and a generalization of the concepts of conjugacy are used to bring out connections between the studied methods.