Extended Zero-Gradient-Sum Approach for Constrained Distributed Optimization With Free Initialization

This article proposes an extended zero-gradient-sum (EZGS) approach for solving constrained distributed optimization with free initialization and desired convergence properties. A Newton-based continuous-time algorithm is first designed for general constrained optimization, which is adapted to handle inequality constraints by using log-barrier penalty functions. Then, a general class of EZGS dynamics is developed to address equation-constrained distributed optimization, where an auxiliary dynamics is introduced to ensure the final ZGS property from any initialization. It is demonstrated that for typical consensus protocols and auxiliary dynamics, the proposed EZGS dynamics can achieve the performance with exponential/finite/fixed/prescribed-time (PT) convergence. Particularly, the nonlinear consensus protocols for finite-time EZGS algorithms allow for heterogeneous power coefficients. Significantly, the proposed PT EZGS dynamics is continuous, uniformly bounded, and capable of reaching the optimal solution in a single stage. Furthermore, the barrier method is employed to handle the inequality constraints effectively. Finally, the efficiency and performance of the proposed algorithms are validated through numerical examples, highlighting their superiority over existing methods. In particular, by selecting appropriate protocols, the proposed EZGS dynamics can achieve desired convergence performance.