Perturbed Spectal-Scaling BFGS Method And Its Convergence

A great deal of numerical results have shown that the BFGS method performed better. It has become the most welcome quasi-Newton method. However, when used to solve the unconstrained optimization problem with nonconvex objective function, the standard BFGS method may not be globally convergent. To overcome his shortcoming, Li and Fukusima proposed a modified BFGS (MBFGS). Under certain condition, the MBFGS has been proved to be globally and superlinearly convergent. However, the MBFGS method does not possess the affine invariant of the BFGS method. In order to overcome this defect of the MBFGS method, Liu and Li proposed a perturbed BFGS method. That method is also globally and superlinear convergent for nonconvex minimization problems. Moreover, it retains the affine invariant of the standard BFGS method.In BFGS method and its modified forms, the size of the condition number of quasi-Newton matrix affects the performance of the methods. To improve the condition number of quasi-Newton matrix, recently, Cheng and Li proposed a spectral-scaling BFGS method, called SSBFGS method. The basic idea of SSBFGS method is to introduce spectral-scaling factor in the update formula of the BFGS method. This strategy can improve the condition number of quasi-Newton matrix. By estending such strategy to MBFGS method, Li and Qiao proposed a spectral-scaling MBFGS method, namely SSMBFGS method. Under certain condition, the SSMBFGS method is globally and R-linear convergent for nonconvex minmization problems.In view of the advantages of perturbed factor and the spectral-scaling technology, in this thesis, by combining the perturbation strategy with the spectral scaling technology, we propose a perturbed pectral-scaling BFGS method. We call it the PSSBFGS method. We prove that under appropriate conditions, the PSSBFGS method is also globally and at least R-linear convergent. At last,we do some numerical experiments to test the proposed methods. The results show that the proposed method is numerically efficient. In particular, when used to solve large scale problems, the PSSBFGS method outperforms the MBFGS method and the perturbed BFGS method much.