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

TL;DR

This paper generalizes the proximal point algorithm to higher orders for monotone variational inequality problems, establishing convergence rates and proposing a new higher-order augmented Lagrangian method with numerical validation.

Contribution

It introduces a $p$th-order proximal point algorithm for monotone variational inequalities and develops a corresponding $p$th-order augmented Lagrangian method, extending existing methods.

Findings

Convergence rate of $O(1/k^{p/2})$ for the $p$th-order PPA.

Proposed $p$th-order ALM based on the generalized PPA.

Numerical experiments demonstrating the effectiveness of the $p$th-order ALM.

Abstract

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.