Network-decentralised optimisation and control: An explicit saturated solution

This paper proposes a decentralised explicit (closed-form) iterative formula that solves convex programming problems with linear equality constraints and interval bounds on the decision variables. In particular, we consider a team of decision agents, each setting the value of a subset of the variables, and a team of information agents, in charge of ensuring that the equality constraints are fulfilled. The structure of the constraint matrix imposes a communication pattern between decision and information agents, which can be represented as a bipartite graph. We associate each information agent with an integral variable and each decision agent with a saturated function, which takes the interval bounds into account, and we design a decentralised dynamic mechanism that globally converges to the optimal solution. Under mild conditions, the convergence is shown to be exponential. We also provide a discrete-time algorithm, based on the Euler system, and we give an upper bound for the step parameter to ensure convergence. Although the considered optimisation problem is static, we show that the proposed scheme can be successfully applied to find the optimal solution of network-decentralised dynamic control problems.