Two Algorithms For Solving Unconstrained Optimization

Recently, there are many kinds of effective methods and their corresponding convergence analyses for the following unconstrained optimization problemwhere f : R~n → R. Quasi-Newton methods and trust region methods are two kinds of these effective methods. Both of them use the value of objective function and the information of its first-order derivation to construct approximate Hessian matrix instead of calculating Hessian matrix of objective function at every iteration, they also have the advantage of fast convergence rate. In this thesis, we first prove the global convergence of a modified DFP method, then propose a nonmonotone adaptive trust region method for nonsmooth convex optimization. The thesis is organized as follows:In Chapter 1, we briefly introduce quasi-Newton methods and trust region methods for unconstrained optimization, then give the preliminary knowledge for nonsmooth convex optimization.In Chapter 2, Wei et al. proposed modified quasi-Newton formulas in [11] , and established the global convergence and the superlinear convergence for the modified BFGS method in [11] and [12], respectively. However, the global convergence of the modified DFP method was not given. In this chapter, under suitable conditions we prove that the modified DFP method is globally convergent with weak Wolfe-Powell line search . Primary numerical results show that this method is promising.In Chapter 3, trust region methods have strong convergence features, not only globally converge but also superlinearly converge. Nonmonotone technique can enhance the likelihood of finding a global optimum, furthermore, it can improve convergence rate when the function computed is forced to creep along the bottom of a narrow curved valley. En-Iighened by [24] and [25], in this chapter we propose a nonmonotone adaptive trust region method for nonsmooth convex optimization. Under suitable conditions, we show that the...