Global Social Cost Minimization With Possibly Nonconvex Objective Functions: An Extremum Seeking-Based Approach

A social cost minimization problem is addressed in this article. In the considered problem, a network of agents work collaboratively to minimize the social cost function, which is defined as the sum of the agents' local objective functions. The engaged agents are supposed to be equipped with an undirected and connected communication graph. Different from most of the existing works, the social cost function in the considered problem is allowed to be nonconvex and possibly admits local extrema. To avoid local extrema and achieve the global minimization of the social cost function, an extremum-seeking-based approach is proposed by introducing a dynamic average consensus protocol to the sinusoidal-dither-signal-based extremum seeking scheme. The dynamic average consensus protocol is leveraged in the proposed extremum-seeker for information sharing and the sinusoidal probing signal is utilized for information extraction. For the avoidance of local extrema, the amplitude of the dither signal is designed to be adaptive. Through Lyapunov stability analysis, it is shown that the proposed method enables the decision variable to converge to a neighborhood of the global minimum point if the conditions on the network connectivity, the existence of unique global minimum and achievability of the global minimum are satisfied. The theoretical result is verified via simulating a numerical example.