  The proximal point algorithm (PPA) has been developed to solve the monotone
variational inequality problem. It provides a theoretical foundation for some
methods, such as the augmented Lagrangian method (ALM) and the alternating
direction method of multipliers (ADMM). This paper generalizes the PPA to the
$p$th-order ($p\geq 1$) and proves its convergence rate $O
\left(1/k^{p/2}\right)$ . Additionally, the $p$th-order ALM is proposed based
on the $p$th-order PPA. Some numerical experiments are presented to demonstrate
the performance of the $p$th-order ALM.
