Distributed Optimization of Clique-Wise Coupled Problems via Three-Operator Splitting

This study explores distributed optimization problems with clique-wise coupling via operator splitting and how we can utilize this framework for performance analysis and enhancement. This framework extends beyond conventional pairwise coupled problems (e.g., consensus optimization) and is applicable to broader examples. To this end, we first introduce a new distributed optimization algorithm by leveraging a clique-based matrix and the Davis-Yin splitting (DYS), a versatile three-operator splitting method. We then demonstrate that this approach sheds new light on conventional algorithms in the following way: (i) Existing algorithms (NIDS, Exact diffusion, diffusion, and our previous work) can be derived from our proposed method; (ii) We present a new mixing matrix based on clique-wise coupling, which surfaces when deriving the NIDS. We prove its preferable distribution of eigenvalues, enabling fast consensus; (iii) These observations yield a new linear convergence rate for the NIDS with non-smooth objective functions. Remarkably our linear rate is first established for the general DYS with a projection for a subspace. This case is not covered by any prior results, to our knowledge. Finally, numerical examples showcase the efficacy of our proposed approach.