Distributed Optimization for Systems with Time-varying Quadratic Objective Functions

This paper considers a distributed optimization problem under undirected graph. Different from most of the existing distributed optimization works that consider the optimal solutions to be constants, the optimal solution and the objective functions at the optimal solution are both assumed to be time-varying. A gradient based searching method is proposed to track the unknown optimal solution. Uncoupled problems are firstly considered followed by neighboring coupled distributed optimization problems. At last, generally coupled problems are solved by using a penalty function based method. Convergence analysis is conducted by using Lyapunov analysis. It is shown that the proposed method enables the agents' strategies to converge asymptotically to the optimal solution for systems with decoupled or neighboring coupled objective functions. For generally coupled systems, the proposed method enables the agents to approximate the optimal solution. A numerical example is presented to verify the effectiveness of the proposed method.