Moreau Envelope ADMM for Decentralized Weakly Convex Optimization
Reza Mirzaeifard Naveen K. D. Venkategowda Alexander Jung Stefan Werner
This paper proposes a proximal variant of the alternating direction method of
multipliers (ADMM) for distributed optimization. Although the current versions
of ADMM algorithm provide promising numerical results in producing solutions
that are close to optimal for many convex and non-convex optimization problems,
it remains unclear if they can converge to a stationary point for weakly convex
and locally non-smooth functions. Through our analysis using the Moreau
envelope function, we demonstrate that MADM can indeed converge to a stationary
point under mild conditions. Our analysis also includes computing the bounds on
the amount of change in the dual variable update step by relating the gradient
of the Moreau envelope function to the proximal function. Furthermore, the
results of our numerical experiments indicate that our method is faster and
more robust than widely-used approaches.