Distributed zero‐gradient‐sum algorithm for convex optimization with time‐varying communication delays and switching networks

Abstract

Summary The distributed convex optimization problem subject to time‐varying communication delays and switching network topologies is addressed in this paper. Based on continuous‐time Zero‐Gradient‐Sum scheme, the novel distributed algorithms are proposed to minimize the global objective function which is composed of a sum of strictly convex local cost functions. In the fixed network topology case, by constructing a new Lyapunov‐Krasovskii function, two explicit sufficient conditions for the maximum admissible time delay are derived to guarantee that all agents' states converge to the optimal solution. In the switching network topology case, the stability condition is derived by the common Lyapunov function theory. In addition, two sufficient conditions about the maximum admissible time delays are also derived for the fixed and switching weight‐balanced network topologies, respectively. Several simulation tests are used to illustrate the effectiveness of our obtained theoretical results.

Distributed continuous‐time algorithm for a general nonsmooth monotropic optimization problem

“…where l ii is the ith diagonal element of the Laplacian matrix L that equals ∑ N =1 a i , and hence, one has 0 < k (L) ≤ 2 for all k ≠ 1. As a consequence, 8 k (L) − 2 k (L) > 0 for all k ≠ 1, which implies that (A23) is equivalent to (23). In consideration of (A19), in what follows, it will be shown that N (L)(5 N (L) + 2) + 2 N (L)…”

“…Consider the case when i , g i ∈ 1 for all i ∈ ℐ N . Under Assumption 1, if condition (23) holds, then the statements in Theorem 2 are still correct.…”

“…To date, although a wide spectrum of results have been reported for discrete-time networks with various scenarios in the literature, ranging from distributed optimization problems in the absence of constraints to those subject to constraints, [1][2][3][4][5] continuous-time algorithms have attracted an increasing interest in recent years mostly due to the fact that a lot of physical systems operate in a continuum domain, such as the current flow in smart grid. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] For instance, distributed convex optimization problems have been studied in the work of Yang et al 17 subject to local feasible constraints, local inequality and equality constraints, where a proportional-integral continuous-time algorithm has been designed with output information exchange.…”

“…To show (A30), define M 34 = M 3 − M 4 , and then, by appealing to Schur complement, it can be easily verified that M 34 ≥ 0 for small enough 1 , 2 > 0 under condition (23). Thus, combining (A27), (10), (11), (13), (A28) with (A30), one has…”

“…which is negative semidefinite under condition (23). Therefore, one can obtain that V(t) ≤ V(0), indicating that x, , , s are all bounded according to the definition of V, where V(t), V(0) mean the function at time instant t and 0.…”