Abstract: We proposed an improved approximate optimal gradient method to solve the unconstrained objective function in the graph partition problem. We first used  the modified BFGS updating formula and selected the linear combination of BB class step sizes as scalar matrices to obtain  the approximate optimal step sizes, then we introduced parameters to improve the classical Zhang-Hager line search form, construced the algorithm framework  and gave the proof of R-linear convergence. The experimental results show that the improved algorithm improves the performance of the original algorithm.

Key words: modified BFGS updating formula,  , approximate optimal step size, Zhang-Hager line search, R-linear convergence, graph partition problem