A fixed step distributed proximal gradient push‐pull algorithm based on integral quadratic constraint

In order to solve the distributed optimization problem with smooth + nonsmooth structure of the objective function on unbalanced directed networks, this article uses the proximal operator to deal with the nonsmooth part of the objective function, and designs and analyzes the fixed step proximal gradient Push-Pull (PG-Push-Pull) algorithm. Firstly, the Integral Quadratic Constraint (IQC) suitable for proximal gradient Push-Pull algorithm is given. When the smooth part of the objective function is strongly convex and the gradient satisfies the Lipchitz condition, the convergence of the algorithm is proved, and the convergence analysis is transformed into solving a linear matrix inequality by using this IQC framework. Its feasibility can ensure that the proposed algorithm has linear convergence rate, which is the same as that of Push-Pull gradient algorithm. Then, the upper bound of convergence rate can be found by solving a Non-Linear Programming problem. Finally, an example is given to analyze the upper bound of the convergence rate and verify the effectiveness of the proposed algorithm.