Generalization of the Brodsky-Lepage-Mackenzie optimization within the $\{\beta\}$-expansion and the Principle of Maximal Conformality

We discuss generalizations of the BLM optimization procedure for renormalization group invariant quantities. In this respect, we discuss in detail the features and construction of the $\{\beta\}$--expansion representation instead of the standard perturbative series with regards to the Adler $D$-function and Bjorken polarized sum rules obtained in order of ${\cal O}(\alpha_s^4)$. Based on the $\{\beta\}$--expansion we analyse different schemes of optimization, including the corrected Principle of Maximal Conformality, numerically illustrating their results. We suggest our scheme for the series optimization and apply it to both the above quantities.