A Convergence Note on Inertial Proximal Admm for Two-Block Structured Convex Optimization

This paper considers a special class of linearly constrained convex optimization problems, where the variables can be divided into two coupled blocks, and the separable objective functions are coupled through a linear consistency constraint. Such problems often appear in large-scale scenarios but may be sensitive to the stability of the algorithm.

To address this, we develop an inertial proximal variant of the alternating direction method of multipliers that incorporates controlled extrapolation together with proximal stabilization in the primal updates. The proposed scheme is analyzed through a Lyapunov-type construction that captures both the augmented Lagrangian structure and the effect of inertial corrections.

Under standard convexity and well-posedness conditions, a descent mechanism for the associated energy sequence is derived, while the generated sequence is proved to be bounded with asymptotic regularity. It is also proved that all limit points of the sequence satisfy the primal-dual optimality conditions, which ensure the convergence of the sequence.

A simple illustrative example has been included to demonstrate how the theoretical framework applies in a concrete constrained optimization setting.