Research On Spectral Conjugate Gradient Methods For Unconstrained Optimization Problems

The spectral conjugate gradient method,which combines advantages of the conjugate gradient method and the spectral method,is one of the most common methods to solve unconstrained optimization problems.Based on the selection of classical conjugate coefficients,it is a hot research topic to modify the conjugate coefficients and select the appropriate spectral coefficients in spectral conjugate gradient method.In addition,it is worth to explore the appropriate line search strategy.Based on the existing researches,some more effective methods are presented in this paper.Firstly,we modify the non-monotone line search strategy by Hu et al and propose a class of modified non-monotone spectral conjugate gradient method.On one hand,with the modified strategy,when the search direction is determined,the restarting technique is adopted to ensure the descending and convergence of the modified algorithm.On the other hand,we propose another non-monotone spectral conjugate gradient method by using new conjugate coefficient.The convergence of the algorithm is established.The new method is applied to non-negative matrix factorization and numerical experiments illustrate that the method is effective.Secondly,a new class of spectral conjugate gradient methods is proposed,in which the spectral parameters are selected by applying the quasi-Newton equation and a modified quasi-Newton equation to the three-term modified PRP conjugate gradient methods by Sun et al.The search direction generated by this method is a descent direction,independent of the employed line search techniques.The global convergence of the developed algorithm with the strong Wolfe line search strategy is proved.Numerical results show that the new algorithm is more effective than the three-term modified PRP conjugate gradient method.