A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks

Abstract

In this paper we propose a distributed implementation of the relaxed Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization of a separable convex cost function, whose terms are stored by a set of interacting agents, one for each agent. Specifically the local cost stored by each node is in general a function of both the state of the node and the states of its neighbors, a framework that we refer to as 'partition-based' optimization. This framework presents a great flexibility and can be adapted to a large number of different applications. We show that the partition-based R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford Splitting (R-PRS) operator which, historically, has been introduced in the literature to find the zeros of sum of functions. Interestingly, making use of non expansive operator theory, the proposed algorithm is shown to be provably robust against random packet losses that might occur in the communication between neighboring nodes. Finally, the effectiveness of the proposed algorithm is confirmed by a set of compelling numerical simulations run over random geometric graphs subject to i.i.d. random packet losses.

Distributed Non-convex Optimization of Multi-agent Systems Using Boosting Functions to Escape Local Optima

“…In there, the step size is kept as fixed value at 1 K i where K i = ∑ j∈ Bi K 1 j according to [31] (also see Remark III-C.2). Finally, in order to assess the convergence, we use relative error profile e k [18] where…”

“…In simulations, fixed dimensional parameters N = 25, m = 3 and r = 4 are used. To generate the inter agent connections (i.e., the graph G ) random geometric graph generation is used with taking 0.25 as the communication range parameter [18]. In each realization, the remaining problem parameters A i j , b i ,C i , s i,0 ∀i ∈ V , ∀ j ∈ Bi are generated randomly (keeping the graph G fixed).…”

“…In these applications, gradientbased techniques are typically used due to their simplicity (see the survey paper [16]). However, more computationally complex schemes, e.g., using the Alternating Direction Method of Multipliers (ADMM) [17], [18], are also gaining popularity due to their greater generality.…”

“…In some multi-agent systems, properties of the associated objective function, such as convexity, can be exploited to achieve a global optimum. For example, the Relaxed-ADMM approach in [18] converges to the global optimum for convex objective functions (along with a few minor additional conditions). On the other hand, there are many settings where the objective function takes a non-convex form making it difficult to attain a global optimum (e.g., see [1], [19], [16], [17]).…”

“…The relationship between local and global objective functions is not restricted to any specific form. For example, two common possibilities are the additive form [18]…”