Designing the parameters of high dimensional consensus: Multi-objective optimization and pareto-optimality

In this paper, we study the synthesis problem in linear high dimensional consensus (HDC) algorithms for large-scale networks. In HDC, we partition the network nodes into leaders and followers. Each follower updates its state as a linear combination of its neighboring states, whereas, the state of the leaders remains fixed. Hence, linear HDC can be thought of as a linear time-invariant (LTI) system. The synthesis problem for this LTI system is to design its parameters such that the system converges to a desired pre-specified state. We cast this synthesis problem as a multi-objective optimization problem (MOP) to which we apply Pareto-optimality. We show that the optimal solution of the synthesis problem is a Pareto-optimal (P.O.) solution of the MOP. We then provide a graphical method to extract the optimal MOP solution from the set of all P.O. solutions. Casting the synthesis problem as an MOP naturally lends itself to interesting performance vs speed trade-offs in HDC.