Linear Convergence of First- and Zeroth-Order Primal–Dual Algorithms for Distributed Nonconvex Optimization

This article considers the distributed nonconvex optimization problem of minimizing a global cost function formed by sum local functions using information exchange. We first consider first-order primal–dual algorithm. show that it converges sublinearly to stationary point if each is smooth and linearly optimum under an additional condition satisfies Polyak–Łojasiewicz condition. weaker than strong convexity, which standard for proving linear convergence algorithms, minimizer not necessarily unique. Motivated situations where gradients are unavailable, we then propose zeroth-order algorithm, derived from considered algorithm deterministic gradient estimator, has same properties as conditions. The theoretical results illustrated numerical simulations.