  We consider solving multi-objective optimization problems in a distributed
manner by a network of cooperating and learning agents. The problem is
equivalent to optimizing a global cost that is the sum of individual
components. The optimizers of the individual components do not necessarily
coincide and the network therefore needs to seek Pareto optimal solutions. We
develop a distributed solution that relies on a general class of adaptive
diffusion strategies. We show how the diffusion process can be represented as
the cascade composition of three operators: two combination operators and a
gradient descent operator. Using the Banach fixed-point theorem, we establish
the existence of a unique fixed point for the composite cascade. We then study
how close each agent converges towards this fixed point, and also examine how
close the Pareto solution is to the fixed point. We perform a detailed
mean-square error analysis and establish that all agents are able to converge
to the same Pareto optimal solution within a sufficiently small
mean-square-error (MSE) bound even for constant step-sizes. We illustrate one
application of the theory to collaborative decision making in finance by a
network of agents.
