Distributed Nonsmooth Optimization With Coupled Inequality Constraints via Modified Lagrangian Function

Abstract

This technical note considers a distributed convex optimization problem with nonsmooth cost functions and coupled nonlinear inequality constraints. To solve the problem, we first propose a modified Lagrangian function containing local multipliers and a nonsmooth penalty function. Then we construct a distributed continuous-time algorithm by virtue of a projected primal-dual subgradient dynamics.Based on the nonsmooth analysis and Lyapunov function, we obtain the existence of the solution to the nonsmooth algorithm and its convergence.

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

“…In comparison, the algorithm (6)-(9) proposed in this paper has been rigorously proved to be effective to cope with problem (5) even with inequality constraints, and the key points are to employ i instead of i in algorithm (6) and to introduce constant gains r, c in algorithm (6)- (9). Furthermore, we also note that a relatively more general scenario with coupled inequality constraints has been investigated in the literature, such as the work of Chang et al 38 for discrete-time algorithms and the work of Liang et al 39 for continuous-time algorithms, but the aforementioned works usually depends on some stringent assumptions, such as compactness of X i 's, differentiability of f i , g i 's, Lipschitz continuous gradients of g i 's, or strict convexity of f i 's, which make the problem relatively well behaved, and the techniques in the aforementioned works 38,39 are not directly applicable if these assumptions do not hold.…”

Distributed sub-optimal resource allocation over weight-balanced graph via singular perturbation

“…Here, (4) contains coupled inequality constraints, which can be linear or nonlinear. This problem with continuous-time design has also been considered in (Liang et al 2018a) for undirected graphs, while we allow for weight-balanced graphs.…”

Multi-agent network flows that solve linear complementarity problems

“…• Theorems 3.2 -3.6 provide a complete procedure to prove that the algorithm (13) or (31) solves LCP(q, M ) in a distributed manner.…”

“…Recently, multi-agent networks have received much attention in various research fields such as distributed optimization and game [12,13,14,29,32], distributed machine learning [25] and distributed computation of equations [22,31]. In contrast to centralized computations, network based distributed algorithms do not require overall information to accomplish a task and introduce an inherent robustness to communication or sensor failures, and environmental uncertainties.…”