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The proximal point algorithm (PPA) has been developed to solve the monotonevariational inequality problem. It provides a theoretical foundation for somemethods, such as the augmented Lagrangian method (ALM) and the alternatingdirection 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 basedon the $p$th-order PPA. Some numerical experiments are presented to demonstratethe performance of the $p$th-order ALM.

high-order proximal point algorithm for the monotone variational inequality problem and its application