This paper presents and studies the iteration-complexity of two new inexact variants of Rockafellar’s proximal method of multipliers (PMM) for solving convex programming (CP) problems with a fi nite number of functional inequality constraints. In contrast to the first variant which solves convex quadratic programming (QP) subproblems at every iteration, the second one solves convex constrained quadratic QP subproblems. Their complexity analysis are performed by: a) viewing the original CP problem as a monotone inclusion problem (MIP); b) proposing a largestep inexact higher-order proximal extragradient framework for MIPs, and; c) showing that the above two PMM variants are just instances of this framework.