Implicit Tracking-based Distributed Constraint-coupled Optimization

Abstract

In this work, a kind of distributed constraintcoupled optimization problem is studied. We first propose an augmented primal-dual dynamics and study its convergence, but the need for knowing the violation of the coupled constraint prevents it from being implemented distributedly. By comprehending a classical continuous-time distributed optimization algorithm from a new perspective, the novel implicit tracking approach is proposed to track the violation distributedly, which leads to the implicit tracking-based distributed augmented primaldual dynamics (IDEA). To deal with the cases where local constrained sets exist, we further develop a projected variant of IDEA, i.e., Proj-IDEA. Under the undirected and connected communication topology, the convergence of IDEA can be guaranteed when local objective functions are only convex, and if local objective functions are strongly convex and smooth, IDEA can converge exponentially with a quite weak condition about the constraint matrix. In line with IDEA, Proj-IDEA can also guarantee convergence when local objective functions are only convex. Furthermore, we also explore the convergences of IDEA and Proj-IDEA under the directed, strongly connected, and weight-balanced communication topology. Finally, numerical experiments are taken to corroborate our theoretical results.

Gradient Tracking: A Unified Approach to Smooth Distributed Optimization

“…Though they do not possess the explicit gradient tracking structure, we found that they do have an implicit one, hence they can also be seen as gradient tracking algorithms, but with the implicit tracking approach. The implicit tracking mechanism (approach) is first proposed in [18], which is employed to develop efficient algorithms for distributed constraint-coupled optimization problems. Specifically speaking, [18] offers a new perspective to understand a classical continuous-time distributed optimization algorithm [20], which is given as…”

“…The implicit tracking mechanism (approach) is first proposed in [18], which is employed to develop efficient algorithms for distributed constraint-coupled optimization problems. Specifically speaking, [18] offers a new perspective to understand a classical continuous-time distributed optimization algorithm [20], which is given as…”

“…Different from [20], ( 4) is derived from a brand-new way in [18]. Concretely speaking, it is easy to verify that…”

“…As mentioned before, the implicit tracking approach is used in the algorithm design for distributed constraint-coupled optimization problems in [18]. Different from [18], we focus on the classical set-up (P1) and develop a unified algorithmic framework for it, with the help of a brand-new understanding of existing distributed optimization algorithms that the implicit tracking mechanism offers.…”

“…Though some algorithms do not have the explicit gradient tracking structure [3]- [6], [17], they still possess certain implicit structure that can track the global gradient, which we call the implicit tracking mechanism. The idea originates from our earlier work [18], where the implicit tracking mechanism is proposed and further used to design efficient algorithms for distributed constraintcoupled optimization problems. Therefore, our framework can be seen as a generalized gradient tracking method, which is the reason we call it unified gradient tracking (UGT).…”