Distributed Methods for Constrained Nonconvex Multi-Agent Optimization-Part I: Theory

In this two-part paper, we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints. The algorithm solves a sequence of (separable) strongly convex problems and mantains feasibility at each iteration. Convergence to a st ationary solution of the original nonconvex optimization is establi shed. Our framework is very general and flexible; it unifies several existing Successive Convex Approximation (SCA)-based alg orithms such as (proximal) gradient or Newton type methods, block coordinate (parallel) descent schemes, difference o f convex functions methods, and improves on their convergence prope rties. More importantly, and differently from current SCA approac hes, it naturally leads to distributed and parallelizable implementations for a large class of nonconvex problems. This Part I is devoted to the description of the framework in its generality. In Part II we customize our general methods to se veral multi-agent optimization problems, mainly in communications and networking; the result is a new class of (distributed) al gorithms that compare favorably to existing ad-hoc (centralized) schemes (when they exist).