Control of distributed convex optimization

Abstract

This paper addresses the problem of solving unconstrained, separable, convex optimization problems over networks and introduces a new approach to the problem: control of distributed convex optimization. We first develop Hopwise Equalizing (HE), a non-gradient-based, distributed asynchronous iterative algorithm that is asymptotically convergent and that is capable of solving the problem. Based on the framework provided by HE, we then develop Controlled Hopwise Equalizing (CHE), showing that a common Lyapunov function, constructed based on the first-order convexity condition, can be used to incorporate the notion of greedy, decentralized, feedback iteration control, whereby individual nodes use potential drops in the value of the Lyapunov function to control, on their own, when to initiate an iteration. Finally, via extensive simulation on wirelessly connected random geometric graphs, we show that CHE is significantly more bandwidth/energy efficient than several existing subgradient algorithms, requiring far less communications to solve a convex optimization problem.

Zero-Gradient-Sum Algorithms for Distributed Convex Optimization: The Continuous-Time Case

“…As it turns out and will be shown in Section 4, each ϕ i and χ i in (9) and (10) do not have to depend on f N i , so that the nodes do not have to exchange their f i 's. We note that the algorithm PB in [9] also exhibits this feature, but the algorithms PE in [9] and CHE in [10] do not.…”

Push-Pull Finite-Time Convergence Distributed Optimization Algorithm

“…At present, a series of algorithms on problem (1-1) have been extensively studied. In general, these algorithms can be divided into two categories: discrete-time algorithms [1] [2] [7] [8] [9] and continuous-time algorithms [10]- [16]. Most of the former adopt iterative method, and based on the consistency of the dynamic system to achieve the goal.…”

“…Most of the former adopt iterative method, and based on the consistency of the dynamic system to achieve the goal. For example, in reference [1], the authors propose a non-gradient distributed random iterative algorithm, which can achieve asymptotic convergence with less information transmission, which is better than some existing gradient-based algorithms. In [2], the authors propose a new event-driven zero-gradient and algorithm that can be widely applied to most network models.…”

Reaching optimal consensus for multi-agent systems based on approximate projection

“…Minimizing a sum of convex functions, where each component is known only to a particular node, has attracted much attention recently, due to its simple formulation and wide applications [22], [20], [21], [26], [27], [23], [31], [30], [28], [29], [32], [25], [24]. The key idea is that properly designed distributed control protocols or computation algorithms can lead to a collective optimization, based on simple exchanged information and individual optimum observation.…”

“…Non-subgradientbased methods also showed up in the literature. For instance, a non-gradient-based algorithm was proposed, where each node starts at its own optimal solution and updates using a pairwise equalizing protocol [28], [29], and later an augmented Lagrangian method was introduced in [32].…”