Distributed Dual Gradient Tracking for Resource Allocation in Unbalanced Networks

Abstract

This paper proposes a distributed conjugate gradient tracking algorithm (DCGT) to solve resource allocation problems in a possibly unbalanced network, where each node of the network computes its optimal resource via interacting only with its neighboring nodes. Our key idea is the novel use of the celebrated AB algorithm to the dual of the resource allocation problem. To study the convergence of DCGT, we first establish the sublinear convergence of AB for non-convex objective functions, which advances the existing results on AB as they require the strong-convexity of objective functions. Then we show that DCGT converges linearly for strongly convex and Lipschitz smooth objective functions, and sublinearly without the Lipschitz smoothness. Finally, simulation results validate that DCGT outperforms state-of-the-art algorithms in distributed resource allocation problems.

Implicit Tracking-based Distributed Constraint-coupled Optimization

“…As a result, DC-ADMM can be seen as a dual variant of C-ADMM. Similarly, the algorithms proposed in [13], [14], [15], and [16] correspond to those proposed in [6], [8], [11], and [5] respectively. However, the convergences of all algorithms mentioned above rely on the strict or strong convexity of f i , which limits their application scenarios.…”

“…Nevertheless, one can easily verify that the dual problem of (P1) has the same form with (1), which implies that we can employ many existing distributed algorithms mentioned above to solve the dual problem of (P1). Therefore, if the strong duality holds, (P1) can be solved conveniently via this approach, which has been adopted by many works [12]- [16]. Concretely speaking, (P1) and (P2) are both considered in [12], C-ADMM, a distributed version of Alternating Direction Method of Multipliers (ADMM), is first proposed to solve (P2), then by applying C-ADMM to the dual problem of (P1), DC-ADMM is further derived to solve (P1).…”

“…By contrast, X i can be unbounded in (P1) and there is no subproblem that needs to be solved exactly in Proj-IDEA. (ii) When local constrained sets are absent, there exists several exponentially convergent algorithms for (P1), all of whom require that f i is strongly convex and smooth, and further impose some different conditions on A i , such as [23] (A i = 1), [13] (A i = I), [14] (A i = I), [16] (A i = I), [24] (A i = I), [12] (A i has full row rank), and [15] (A i has full row rank). With the same condition about f i , IDEA can also converge exponentially over undigraphs, but distinguished from these algorithms, IDEA only requires that A has full row rank, where…”

“…(iii) For digraphs, the convergence of Proj-IDEA can be guaranteed when f i is strongly convex. Though the algorithms proposed in [14], [23] and [16] can also converge over digraphs for strongly convex f i , the problems they considered are all special cases of (P1). In addition, IDEA can converge exponentially over digraphs if we further assume that A i = I.…”

Distributed Dual Gradient Tracking for Economic Dispatch in Power Systems with Noisy Information

“…Based on the consensus optimization algorithm, D-DLM was proposed in [11] by integrating the gradient tracking technique with two types of momentum terms. In [12], [13], researchers applied the Push-Pull algorithm [14], [15] to EDP, providing an algorithm for directed networks which converges sublinearly for strongly convex cost functions. Interesting readers may refer to [16] for more discussions on EDP in power systems.…”