Distributed Optimization Involving Non-Lipschitzian, Linear Composite Functions: Unified Algorithm Design and Convergence Analysis
This paper formulates a generic distributed nonsmooth optimization problem subject to edge-based equality constraints, local inequality constraints characterized by general convex functions, as well as globally coupling constraints including affine equality and nonlinear inequality. The terminal goal is to minimize the sum of local non-differentiable convex functions and proximable nonsmooth convex composite terms attached to linear operators, over networked multi-agent systems. This formulated framework not only applies in complex practical problems but also reduces to existing distributed optimization structures. To cope with technical issues caused by local linear mappings, potential multiple nonsmooth functions and coupling nonlinear constraints, we first derive a new local first-order condition, and further present a novel fully distributed optimization algorithm based on variable decompositions and proximal operators. Specifically, the proposed algorithm allows each agent to update states via local information exchange and through {locally fixed step-sizes}, and to find inexact solutions of subproblems without loss of convergence speed. In contrast to existing tailor-made algorithms for the bounded gradients of objective functions or affine constraints, our algorithm is capable of solving more practical cases involving general convex functions and nonlinear constraints. Under mild assumptions, the convergence rate $o(1/k)$ in terms of the optimality and successive-difference metrics is achieved, by constructing the equivalent variational inequalities of the proposed algorithm. Finally, numerical experiments on practical application problems illustrate the theoretical results.