Continuous-time proportional-integral distributed optimisation for networked systems

Abstract

In this paper we explore the relationship between dual decomposition and the consensus-based method for distributed optimization. The relationship is developed by examining the similarities between the two approaches and their relationship to gradient-based constrained optimization. By formulating each algorithm in continuous-time, it is seen that both approaches use a gradient method for optimization with one using a proportional control term and the other using an integral control term to drive the system to the constraint set. Therefore, a significant contribution of this paper is to combine these methods to develop a continuous-time proportional-integral distributed optimization method. Furthermore, we establish convergence using Lyapunov stability techniques and utilizing properties from the network structure of the multi-agent system. is solved by these methods. Specifically, we formulate both the mentioned dual-decomposition method and the consensus-based method in [8] in control theoretic terms to draw parallels and gain intuition behind why they can naturally be joined together. In fact, it will become apparent that dual-decomposition is very closely related to integral (I) control and the consensus method is closely related to proportional (P) control. Therefore, a significant contribution of this paper is to combine these two methods to form a new, proportional-integral (PI) distributed optimization method. This formulation will be similar to the PI distributed optimization method introduced in [9] and extended in [11,19]. However, due to the fact that we create the PI optimization method from the perspective of the dual-decomposition method, which involves a set of constraints associated with the interconnections of agents, it enables us to form a different type of integral control term and reduce the required communication.While much of the work on distributed optimization has been developed in discrete-time formulations, which are amenable for implementation, e.g. [7,10,12,20], a great deal of work recently has been made in continuous-time [8,9,11,13,14,21,19]. Continuous-time analysis has proven useful as it allows Lyapunov stability conditions to be directly applied to the update-equations for convergence analysis. It also allows for an intuitive connection between the optimization algorithm proposed in this paper and proportional-integral control. Moreover, a discretization of the framework proposed in this paper does not pose a significant contribution.The proportional element has been evaluated in discrete time in [10] and the integral element has been evaluated in discrete time in [12,13,14]. Furthermore, a closely related PI distributed optimization algorithm developed in [9,11,19] (discussed further in Section 5.1) was discretized in [9].The remainder of this paper will proceed as follows. We first introduce the necessary background material for the analysis of the distributed optimization algorithms, including the problem formulation, the graph-based multi-agent model of the network, and a ...

Distributed gradient algorithm for constrained optimization with application to load sharing in power systems

“…Note that (6) contains an additional second-order consensus dynamics to ensure all the agents to reach the same optimal point, differing from that given in [25]. Moreover, if there were no constraints in our problem formulation, our algorithm would be consistent with those algorithms for the unconstrained optimization in [11,[15][16][17].…”

“…Different from the existing results such as those given in [2][3][4][5][6][7][8] and [25,[11][12][13][14][15][16][17], our distributed algorithm enables the agents to find the optimal point with respect to the sum of the local objective functions while satisfying all the local constraints. Note that the optimal solution must be within the intersection set of each agent's local private feasible set specified by the local constraints, while neither local objective function nor local constraint functions of each agent can be known or shared by other agents.…”

“…The quadratic penalty term in the objective function utilizes the augmented Lagrangian method (see section 4.7 of [26]) for the consensus, and it plays a damping role in the algorithm (as in [15]). …”

“…A second-order distributed dynamics was proposed to solve an unconstrained optimization in [11], while a similar algorithm was also constructed with non-smooth objective functions in [16]. Moreover, a distributed optimization algorithm based on proportional-integral control was given in [15], and later internal model principle was employed to achieve exact optimization with capability of rejecting external disturbance in [13]. However, to our knowledge, very few continuous-time distributed algorithms for constrained optimization have ever been documented.…”

“…The continuous-time dynamics for distributed optimization attract more and more attention by taking advantage of the welldeveloped continuous-time control techniques and by concerning the implementation of physical systems (see [11][12][13][14][15][16][17]). The continuous-time optimization algorithm was studied in the seminal work [18] and then investigated with various backgrounds (referring to [19,20]).…”