Distributed Interval Optimization Over Time-Varying Networks: A Numerical Programming Perspective

In this paper, we investigate a distributed interval optimization problem which is modelled with optimizing a sum of convex interval objective functions subject to global convex constraints, corresponding to agents over the time-varying network. We first reformulate the distributed interval optimization problem as a distributed constrained optimization problem by scalarization of the distributed interval optimization problem. Then, we show that the Pareto optimal solutions of the reformulated problem. The optimal solutions of the distributed constrained optimization problem is equivalent to Pareto optimal solutions to the distributed interval optimization problem, and we design a distributed subgradient-free algorithm for the distributed constrained optimization problems through constructing random differences of reformulated optimal objective functions. Moreover, we prove that for the proposed algorithm, a Pareto optimal solution can be achieved almost surely over the time-varying network. Finally, we give a numerical example to illustrate the effectiveness of the proposed algorithm.