Modified L-BFGS Methods For Large-scale Unconstrained Optimization

In this thesis, we first propose a modified limited memory BFGS method for solving non-convex minimization problem, then we give a compact limited memory BFGS method for convex optimization. The thesis is structured as follows:First of all, we give a brief introduction for unconstrained optimization and emphasize the research and development of the BFGS method and limited memory BFGS method.In chapter 2, based on the Modified BFGS method proposed by Li and Fukushima we propose a modified limited memory BFGS method (L-BFGS) for solving large-scale unconstrained minimization problems. A remarkable feature of the proposed method is that it possesses the global convergence property without convexity assumption on the objective function. Under some suitable conditions, the global convergence of the proposed method is established. Numerical test on some CUTEr test problems indicate that the proposed method is competitive with standard L-BFGS method.In chapter 3, a compact limited memory method for solving large-scale unconstrained optimization problems is proposed. The proposed method is based on the Modified BFGS update proposed by Yuan. In additionally, the global convergence and R-linear convergence rate are proved.