Parameter-Separable Prox-Lagrangian Method for Convex–Concave Saddle Point Problem

This letter considers the algorithm design and convergence analysis for the non-smooth convex-concave saddle point problem (CCSPP) with a bilinear coupling term. By using a prediction-correction structure and proximal regularizer, we develop a parameter-separable proximal Lagrangian method (Prox-LM) to tackle this problem. Specifically, we obtain a predicted solution from the minimax subproblems that are regularized by a proximal-point term, and then it is further modified in an inertial way to generate a new solution. One distinctive feature of our method is the decoupling of the algorithm parameters from the bilinear term, allowing for independent determination. We demonstrate that our proposed Prox-LM can achieve an
$\mathcal {O}(1/{k})$
convergence rate towards the saddle point in an ergodic sense. Numerical experiments on the minimax games and the constrained multiagent problem have verified the effectiveness and performance of our method.