Abstract
In this paper, we consider a distributed resource allocation problem of minimizing a global convex function formed by a sum of local convex functions with coupling constraints. Based on neighbor communication and stochastic gradient, a distributed stochastic mirror descent algorithm is designed for the distributed resource allocation problem. Sublinear convergence to an optimal solution of the proposed algorithm is given when the second moments of the gradient noises are summable. A numerical example is also given to illustrate the effectiveness of the proposed algorithm.
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Acknowledgements
This work was supported by the National Key Research and Development Program of China (No. 2016YFB0901900), the National Natural Science Foundation of China (No. 61733018) and the China Special Postdoctoral Science Foundation Funded Project (No. Y990075G21).
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Appendix
Appendix
Proof of Lemma 4
According to Lemma 2, we have
and
Applying (A1), (A2) and (A3) to (12), we get
where
Summing \(\theta _{k}\) over \(k=1,2,\ldots , T\), we obtain
where (b) follows from \({\varvec{y}}-\varvec{y^{k}}={\varvec{y}}-\varvec{y^{k-1}}+\varvec{y^{k-1}}-\varvec{y^{k}}\), (c) follows from \({\varvec{x}}^{-1}={\varvec{x}}^{0}\) and \({\varvec{z}}^{-1}={\varvec{z}}^{0}\), (d) and (f) follow from \(b\langle u,v \rangle \leqslant \dfrac{a}{2}\Vert v\Vert ^{2}+\dfrac{b^{2}\Vert u\Vert ^{2}}{2a}\) for all \(a>0\) and (14), and (e) follows from \(\Vert {\varvec{y}}-{\varvec{y}}^{0}\Vert ^{2}-\Vert {\varvec{y}}-{\varvec{y}}^{T}\Vert ^{2}=\Vert {\varvec{y}}^{0}\Vert ^{2}-\Vert {\varvec{y}}^{N}\Vert ^{2}-2\langle {\varvec{y}}, {\varvec{y}}^{0}-{\varvec{y}}^{T}\rangle\). (15) follows from (A4) and (A6).
From (A6) (d), Assumption 1 and the fact that \(\textstyle \sum \limits _{k=1}^{T}\mathrm {E}Q(\varvec{w^{k}},\varvec{w^{*}})\geqslant 0\), if we fix \({\varvec{w}}={\varvec{w}}^{*}\), then
and
from which (16) follows.\(\square\)
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Wang, Y., Tu, Z. & Qin, H. Distributed stochastic mirror descent algorithm for resource allocation problem. Control Theory Technol. 18, 339–347 (2020). https://doi.org/10.1007/s11768-020-00018-8
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DOI: https://doi.org/10.1007/s11768-020-00018-8