Iteration-complexity of a Rockafellar's proximal method of multipliers for convex programming based on second-order approximations

TL;DR

This paper introduces a new primal-dual algorithm combining Rockafellar's proximal method of multipliers with the relaxed hybrid proximal extragradient method, achieving improved iteration-complexity for convex programming with inequality constraints.

Contribution

It develops a novel hybrid algorithm that integrates second-order approximations and extragradient steps, enhancing convergence analysis for convex optimization.

Findings

Establishes iteration-complexity bounds for the proposed method.

Demonstrates the effectiveness of the hybrid approach in convex programming.

Provides theoretical convergence guarantees for the combined algorithm.

Abstract

This paper studies the iteration-complexity of a new primal-dual algorithm based on Rockafellar's proximal method of multipliers (PMM) for solving smooth convex programming problems with inequality constraints. In each step, either a step of Rockafellar's PMM for a second-order model of the problem is computed or a relaxed extragradient step is performed. The resulting algorithm is a (large-step) relaxed hybrid proximal extragradient (r-HPE) method of multipliers, which combines Rockafellar's PMM with the r-HPE method.