Some Researches On The Unconstrained Optimization Problems

Optimization problem is an important subject in mathematical programming. There are many applications in engineering, information technology and economic balance and so on. Quasi-newton method is a kind of very effective method for solv-ing optimization problems. In this thesis, the main work is to investigate numerical method of unconstrained optimization problem.In the Introduction, we introduce some conceptions which will appear in the following chapters, such as, quasi-newton method, non-monotone algorithms, trust region method and conjugate gradient method.In Chapter1, we studied a modified BFGS-trust region algorithm which is based on the non-monotone line search technique. For the method, we used a non-monotone search technology, which lead to reduce the demand for function value is reducing and make the step length selection more flexible. Moreover, the global convergence of the algorithm is proved under proper conditions. Some numerical results are reported in the paper, which confirms the effectiveness of the proposed technique.In Chapter2, we proposed a conjugate gradient method which based on a secant equation satisfied by MBFGS formula. On the basis of the strong Wolfe search technology and MBFGS formula, a new conjugate gradient method is proposed. In addition, the global convergence of the algorithm is proved under proper conditions. Some numerical results show that this method is effective for the support vector machine.In the last chapter, the work of this paper is summarized, and some suggestions are also pointed out for the further study.