The Study Of Unconstrained Optimization Problems With Conjugate Gradient Method

Conjugate gradient method is an important method in optimization computation method. It had forty histories since the last century between 1960s and 1970s. In recent years, with the rapid development of computer and a grate deal of large-scale optimization problems make the research in conjugate gradient method revive. In many areas, such as electricity distribution, oil exploration, economic management and weather forecasts, it has many applications. In the thesis, we mainly pose a modified LS conjugate gradient algorithm, a hybrid conjugate gradient method and study conjugate gradient method in the modeling of financial time series.Firstly, we introduce five classical conjugate gradient algorithms respectively which are FR method, PRP method, HS method, CD descent method and DY method. Meanwhile, we give the corresponding conclusions of their global convergence property and descent conditions. Then we also give some basic theories on heteroscedastic time series.Secondly, we propose a modified LS conjugate gradient algorithm, a three terms conjugate gradient algorithm of Beale, and also give the proof of the decline property and convergence property. Finally the numerical experiments demonstrated the good effect of the algorithm.Furthermore, we propose a hybrid conjugate gradient algorithm-NLSDY algorithm. The new algorithm makes the Liu-Storey algorithm and Dai-Yuan method combining, which takes advantages of the two methods. Then we prove the global convergence of the new algorithm with strong Wolfe line search, numerical experiments also show that the new algorithm has good computational performance.Finally, we modify a heteroscedastic time series model based on conjugate gradient method, which makes the error smallest minimization and conjugate gradient method, then we use SAS statistical analysis software SAS to solve the economic non-stationary time series model fitting. In the last we give a fitting model of the Reserve Bank of Australia 2-year monthly rate securities examples of data. The final fitting result diagram show the model fits successfully.