Regularized dual gradient distributed method for constrained convex optimization over unbalanced directed graphs

Abstract

This paper investigates a distributed optimization problem over a cooperative multi-agent time–varying network, where each agent has its own decision variables that should be set so as to minimize its individual objective subjected to global coupled constraints. Based on push-sum protocol and dual decomposition, we design a regularized dual gradient distributed algorithm to solve this problem, in which the algorithm is implemented in unbalanced time–varying directed graphs only requiring the column stochasticity of communication matrices. By augmenting the corresponding Lagrangian function with a quadratic regularization term, we first obtain the bound of the Lagrangian multipliers which does not require constructing a compact set containing the dual optimal set when compared with most of primal-dual based methods. Then, we obtain that the convergence rate of the proposed method can achieve the order of \(\mathcal {O}(\ln T/T)\) for strongly convex objective functions, where T is the number of iterations. Moreover, the explicit bound of constraint violations is also given. Finally, numerical results on the network utility maximum problem are used to demonstrate the efficiency of the proposed algorithm.

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Funding

This research was partially supported by the NSFC 11501070, 11671362 and 11871128, by the Natural Science Foundation Projection of Chongqing cstc2017jcyjA0788 and cstc2018jcyjAX0172, and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN201800520).

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Correspondence to Jueyou Li.

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Gu, C., Wu, Z. & Li, J. Regularized dual gradient distributed method for constrained convex optimization over unbalanced directed graphs. Numer Algor 84, 91–115 (2020). https://doi.org/10.1007/s11075-019-00746-2

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Keywords

  • Convex optimization
  • Distributed algorithm
  • Dual decomposition
  • Regularization
  • Multi-agent network