Multi-agent systems reaching optimal consensus based on simple Bernoulli decisions

In this paper, we formulate and solve a randomized optimal consensus problem for multi-agent systems with stochastically time-varying interconnection topology. The considered multi-agent system with a simple randomized iterating rule achieves an almost sure consensus meanwhile solving the optimization problem min(zϵR^d) Σ_i=1ⁿ f_i(z), in which the optimal solution set of objective function f_i corresponding to agent i can only be observed by agent i itself. At each time step, each agent independently and randomly chooses either taking an average among its neighbor set, or projecting onto the optimal solution set of its own optimization component. Both directed and bidirectional communication graphs are studied. Connectivity conditions are proposed to guarantee an optimal consensus almost surely with proper convexity and intersection assumptions. The convergence analysis is carried out using convex analysis. The results illustrate that a group of autonomous agents can reach an optimal opinion with probability one by each node simply making a randomized trade-off between following its neighbors or sticking to its own opinion at each time step.