| Nonlinear conjugate gradient methods are efficient for solving unconstrainedoptimization. They are particularly welcome for the solution of large-scale problems due totheir lower storage and good convergence theory. The FR method is a very famous conjugategradient method. It is well-known that the FR conjugate gradient method with exact linesearch is globally and linearly convergent. If some restart strategy is used, the convergencerate of the method can be n-step superlinear/quadratic convergence. Recently, ZHANG,ZHOU and LI proposed a modified FR (MFR) method associated with spectral gradient. Itwas proved that the MFR method is globally convergent if some Armijo type line search isused. The purpose of this paper is to investigate the convergence rate of the MFR methodwith Armijo type inexact line search.In chapter2, we show that the MFR method with Armijo line search is linearlyconvergent. In chapter3, we show that the estimation to the exact line search steplengthderived by LI and TIAN can be accepted by the Armijo inexact line search. We then use it asthe initial steplength in the inexact line search in order to improve the efficiency of the MFRmethod. To improve the convergence rate of the MFR method, we introduce a restart strategyin the MFR method and propose a restart MFR method (called RMFR method). Underappropriate conditions, we show that the RMFR method with an Armijo type inexact linesearch is globally convergent. Moreover, it retains n quadratic convergence property.Finally, we do extensive numerical experiments to test the performance of the proposedRMFR method. We also compare the performance of the RMFR method with that of theordinary MFR method (without use of restart strategy) and CG-DESCENT method, etc. Wecompare them in the CPU time used, the number of function evaluations and the number ofthe gradient evaluations. The results show that the proposed RMFR method outperforms theordinary MFR method, MPRP method and CG-DESCENT method. |
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