American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2020061

Distributed convex optimization with coupling constraints over time-varying directed graphs†

Bingru Zhang 1, , Chuanye Gu 2, and  Jueyou Li 3,,

1. 

School of Finance and Economics, Yangtze Normal University, Chongqing, 408100, China
2. 

Department of Mathematics and Statistics, Curtin University, Bentley, WA, 6102, Australia
3. 

School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, China

* Corresponding author: Jueyou Li

Received  March 2019 Revised  November 2019 Published  March 2020

Fund Project: † This research was partially supported by the NSFC 11971083 and 11871128, by the Natural Science Foundation Projection of Chongqing cstc2017jcyjAX0253, by the Fund Program of Chongqing Social Science Planning of China 2017YBGL137 and by the Science and Technology Research Program of Chongqing Municipal Education Commission KJQN201800520.

This paper considers a distributed convex optimization problem over a time-varying multi-agent network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling constraints. Over directed graphs, we propose a distributed algorithm that incorporates the push-sum protocol into dual sub-gradient methods. Under the convexity assumption, the optimality of primal and dual variables, and the constraint violation are first established. Then the explicit convergence rates of the proposed algorithm are obtained. Finally, numerical experiments on the economic dispatch problem are provided to demonstrate the efficacy of the proposed algorithm.

Keywords: Distributed optimization, Multi-agent network, Dual decomposition, Push-sum protocol, Directed graph.
Mathematics Subject Classification: Primary: 47N10; Secondary: 68W15.
Citation: Bingru Zhang, Chuanye Gu, Jueyou Li. Distributed convex optimization with coupling constraints over time-varying directed graphs†. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020061
References:
[1]

N. S. Aybat and E. Y. Hamedani, Distributed primal-dual method for multi-agent sharing problem with conic constraints, 2016 50th Asilomar Conference on Signals, Systems and Computers, (2016), 777–782. doi: 10.1109/ACSSC.2016.7869152.  Google Scholar

[2]

N. S. Aybat and E. Y. Hamedani, A distributed ADMM-like method for resource sharing over time-varying networks, SIAM J. Optim., 29 (2019), 3036-3068.  doi: 10.1137/17M1151973.  Google Scholar

[3]

B. Baingana, G. Mateos and G. B. Giannakis, Proximal-gradient algorithms for tracking cascades over social networks, IEEE Journal of Selected Topics in Signal Processing, 8 (2014), 563-575.  doi: 10.1109/JSTSP.2014.2317284.  Google Scholar

[4]

A. Beck, A. Nedić, A. Ozdaglar and M. Teboulle, An $O(1/k)$ gradient method for network resource allocation problems, IEEE Trans. Control Netw. Syst., 1 (2014), 64-73.  doi: 10.1109/TCNS.2014.2309751.  Google Scholar

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[6]

T.-H. Chang, A. Nedić and A. Scaglione, Distributed constrained optimization by consensus-based primal-dual perturbation method, IEEE Trans. Automat. Control, 59 (2014), 1524-1538.  doi: 10.1109/TAC.2014.2308612.  Google Scholar

[7]

T.-H. Chang, M. Hong and X. Wang, Multi-agent distributed optimization via inexact consensus ADMM, IEEE Trans. Signal Process., 63 (2015), 482-497.  doi: 10.1109/TSP.2014.2367458.  Google Scholar

[8]

T.-H. Chang, A proximal dual consensus ADMM method for multi-agent constrained optimization, IEEE Trans. Signal Process., 64 (2016), 3719-3734.  doi: 10.1109/TSP.2016.2544743.  Google Scholar

[9]

J. C. Duchi, A. Agarwal and M. J. Wainwright, Dual averaging for distributed optimization: Convergence analysis and network scaling, IEEE Trans. Automat. Control, 57 (2012) 592–606. doi: 10.1109/TAC.2011.2161027.  Google Scholar

[10]

A. Falsone, K. Margellos, S. Garatti and M. Prandini, Dual decomposition for multi-agent distributed optimization with coupling constraints, Automatica J. IFAC, 84 (2017), 149-158.  doi: 10.1016/j.automatica.2017.07.003.  Google Scholar

[11]

X. S. Han, H. B. Gooi and D. S. Kirschen, Dynamic economic dispatch: Feasible and optimal solutions, 2001 Power Engineering Society Summer Meeting. Conference Proceedings, 16 (2001), 22-28.  doi: 10.1109/PESS.2001.970332.  Google Scholar

[12]

J. Li, C. Wu, Z. Wu and Q. Long, Gradient-free method for nonsmooth distributed optimization, J. Global Optim., 61 (2015), 325-340.  doi: 10.1007/s10898-014-0174-2.  Google Scholar

[13]

J. Li, G. Chen, Z. Dong and Z. Wu, A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints, Comput. Optim. Appl., 64 (2016), 671-697.  doi: 10.1007/s10589-016-9826-0.  Google Scholar

[14]

J. Li, C. Gu, Z. Wu and C. Wu, Distributed optimization methods for nonconvex problems with inequality constraints over time-varying networks, Complexity, 2017 (2017), Article ID 3610283, 10 pp. doi: 10.1155/2017/3610283.  Google Scholar

[15]

P. Di Lorenzo and G. Scutari, NEXT: In-network nonconvex optimization, IEEE Trans. Signal Inform. Process. Netw., 2 (2016), 120-136.  doi: 10.1109/TSIPN.2016.2524588.  Google Scholar

[16]

G. Mateos and G. B. Giannakis, Distributed recursive least-squares: Stability and performance analysis, IEEE Trans. Signal Process., 60 (2012), 3740-3754.  doi: 10.1109/TSP.2012.2194290.  Google Scholar

[17]

D. K. Molzahn, F. Dörfler, H. Sandberg, S. H. Low, S. Chakrabarti, R. Baldick and J. Lavaei, A survey of distributed optimization and control algorithms for electric power systems, IEEE Transactions on Smart Grid, 8 (2017), 2941-2962.   Google Scholar

[18]

A. Nedić and A. Ozdaglar, Distributed subgradient methods for multi-agent optimization, IEEE Trans. Automat. Control, 54 (2009), 48-61.  doi: 10.1109/TAC.2008.2009515.  Google Scholar

[19]

A. Nedić and A. Olshevsky, Distributed optimization over time-varying directed graphs, IEEE Trans. Automat. Control, 60 (2015), 601-615.  doi: 10.1109/TAC.2014.2364096.  Google Scholar

[20]

A. Nedić, A. Olshevsky and W. Shi, Achieving geometric convergence for distributed optimization over time-varying graphs, SIAM J. Optim., 27 (2017), 2597-2633.  doi: 10.1137/16M1084316.  Google Scholar

[21]

B. T. Polyak, Introduction to Optimization, Optimization Software, Inc., Publications Division, New York, 1987.  Google Scholar

[22]

S. S. Ram, A. Nedić and V. V. Veeravalli, Distributed stochastic subgradient projection algorithms for convex optimization, J. Optim. Theory Appl., 147 (2010), 516-545.  doi: 10.1007/s10957-010-9737-7.  Google Scholar

[23]

W. Shi, Q. Ling, K. Yuan, G. Wu and W. Yin, On the linear convergence of the ADMM in decentralized consensus optimization, IEEE Trans. Signal Process., 62 (2014), 1750-1761.  doi: 10.1109/TSP.2014.2304432.  Google Scholar

[24]

W. Shi, Q. Ling, G. Wu and W. Yin, EXTRA: An exact first-order algorithm for decentralized consensus optimization, SIAM J. Optim., 25 (2015), 944-966.  doi: 10.1137/14096668X.  Google Scholar

[25]

K. I. Tsianos, S. Lawlor and M. G. Rabbat, Push-sum distributed dual averaging for convex optimization, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), (2012), 5453–5458. doi: 10.1109/CDC.2012.6426375.  Google Scholar

[26]

S. Wang and C. Li, Distributed robust optimization in networked system, IEEE Transactions on Cybernetics, 47 (2017), 2321-2333.  doi: 10.1109/TCYB.2016.2613129.  Google Scholar

[27]

X. Xia and A. M. Elaiw, Optimal dynamic economic dispatch of generation: A review, Electric Power Systems Research, 80 (2010), 975-986.  doi: 10.1016/j.epsr.2009.12.012.  Google Scholar

[28]

C. Yang, Y. Xu, L. Zhou and Y. Sun, Model-free composite control of flexible manipulators based on adaptive dynamic programming, Complexity, 2018 (2018), Article ID 9720309, 9 pp. doi: 10.1155/2018/9720309.  Google Scholar

[29]

D. Yuan, S. Xu and H. Zhao, Distributed primal-dual subgradient method for multiagent optimization via consensus algorithms, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 41 (2011), 1715-1724.   Google Scholar

[30]

D. Yuan, D. W. Ho and Y. Hong, On convergence rate of distributed stochastic gradient algorithm for convex optimization with inequality constraints, SIAM J. Control Optim., 54 (2016), 2872-2892.  doi: 10.1137/15M1048896.  Google Scholar

[31]

D. Yuan, D. W. C. Ho and S. Xu, Regularized primal-dual subgradient method for distributed constrained optimization, IEEE Transactions on Cybernetics, 46 (2016), 2109-2118.  doi: 10.1109/TCYB.2015.2464255.  Google Scholar

[32]

Y. Zhang and G. B. Giannakis, Distributed stochastic market clearing with high-penetration wind power, IEEE Transactions on Power Systems, 31 (2016), 895-906.   Google Scholar

[33]

Z. Zhang and M. Y. Chow, Convergence analysis of the incremental cost consensus algorithm under different communication network topologies in a smart grid, IEEE Transactions on Power Systems, 27 (2012), 1761-1768.  doi: 10.1109/TPWRS.2012.2188912.  Google Scholar

[34]

M. Zhu and S. Martínez, On distributed convex optimization under inequality and equality constraints, IEEE Trans. Automat. Control, 57 (2012), 151-164.  doi: 10.1109/TAC.2011.2167817.  Google Scholar

show all references

References:
[1]

N. S. Aybat and E. Y. Hamedani, Distributed primal-dual method for multi-agent sharing problem with conic constraints, 2016 50th Asilomar Conference on Signals, Systems and Computers, (2016), 777–782. doi: 10.1109/ACSSC.2016.7869152.  Google Scholar

[2]

N. S. Aybat and E. Y. Hamedani, A distributed ADMM-like method for resource sharing over time-varying networks, SIAM J. Optim., 29 (2019), 3036-3068.  doi: 10.1137/17M1151973.  Google Scholar

[3]

B. Baingana, G. Mateos and G. B. Giannakis, Proximal-gradient algorithms for tracking cascades over social networks, IEEE Journal of Selected Topics in Signal Processing, 8 (2014), 563-575.  doi: 10.1109/JSTSP.2014.2317284.  Google Scholar

[4]

A. Beck, A. Nedić, A. Ozdaglar and M. Teboulle, An $O(1/k)$ gradient method for network resource allocation problems, IEEE Trans. Control Netw. Syst., 1 (2014), 64-73.  doi: 10.1109/TCNS.2014.2309751.  Google Scholar

[5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar
[6]

T.-H. Chang, A. Nedić and A. Scaglione, Distributed constrained optimization by consensus-based primal-dual perturbation method, IEEE Trans. Automat. Control, 59 (2014), 1524-1538.  doi: 10.1109/TAC.2014.2308612.  Google Scholar

[7]

T.-H. Chang, M. Hong and X. Wang, Multi-agent distributed optimization via inexact consensus ADMM, IEEE Trans. Signal Process., 63 (2015), 482-497.  doi: 10.1109/TSP.2014.2367458.  Google Scholar

[8]

T.-H. Chang, A proximal dual consensus ADMM method for multi-agent constrained optimization, IEEE Trans. Signal Process., 64 (2016), 3719-3734.  doi: 10.1109/TSP.2016.2544743.  Google Scholar

[9]

J. C. Duchi, A. Agarwal and M. J. Wainwright, Dual averaging for distributed optimization: Convergence analysis and network scaling, IEEE Trans. Automat. Control, 57 (2012) 592–606. doi: 10.1109/TAC.2011.2161027.  Google Scholar

[10]

A. Falsone, K. Margellos, S. Garatti and M. Prandini, Dual decomposition for multi-agent distributed optimization with coupling constraints, Automatica J. IFAC, 84 (2017), 149-158.  doi: 10.1016/j.automatica.2017.07.003.  Google Scholar

[11]

X. S. Han, H. B. Gooi and D. S. Kirschen, Dynamic economic dispatch: Feasible and optimal solutions, 2001 Power Engineering Society Summer Meeting. Conference Proceedings, 16 (2001), 22-28.  doi: 10.1109/PESS.2001.970332.  Google Scholar

[12]

J. Li, C. Wu, Z. Wu and Q. Long, Gradient-free method for nonsmooth distributed optimization, J. Global Optim., 61 (2015), 325-340.  doi: 10.1007/s10898-014-0174-2.  Google Scholar

[13]

J. Li, G. Chen, Z. Dong and Z. Wu, A fast dual proximal-gradient method for separable convex optimization with linear coupled constraints, Comput. Optim. Appl., 64 (2016), 671-697.  doi: 10.1007/s10589-016-9826-0.  Google Scholar

[14]

J. Li, C. Gu, Z. Wu and C. Wu, Distributed optimization methods for nonconvex problems with inequality constraints over time-varying networks, Complexity, 2017 (2017), Article ID 3610283, 10 pp. doi: 10.1155/2017/3610283.  Google Scholar

[15]

P. Di Lorenzo and G. Scutari, NEXT: In-network nonconvex optimization, IEEE Trans. Signal Inform. Process. Netw., 2 (2016), 120-136.  doi: 10.1109/TSIPN.2016.2524588.  Google Scholar

[16]

G. Mateos and G. B. Giannakis, Distributed recursive least-squares: Stability and performance analysis, IEEE Trans. Signal Process., 60 (2012), 3740-3754.  doi: 10.1109/TSP.2012.2194290.  Google Scholar

[17]

D. K. Molzahn, F. Dörfler, H. Sandberg, S. H. Low, S. Chakrabarti, R. Baldick and J. Lavaei, A survey of distributed optimization and control algorithms for electric power systems, IEEE Transactions on Smart Grid, 8 (2017), 2941-2962.   Google Scholar

[18]

A. Nedić and A. Ozdaglar, Distributed subgradient methods for multi-agent optimization, IEEE Trans. Automat. Control, 54 (2009), 48-61.  doi: 10.1109/TAC.2008.2009515.  Google Scholar

[19]

A. Nedić and A. Olshevsky, Distributed optimization over time-varying directed graphs, IEEE Trans. Automat. Control, 60 (2015), 601-615.  doi: 10.1109/TAC.2014.2364096.  Google Scholar

[20]

A. Nedić, A. Olshevsky and W. Shi, Achieving geometric convergence for distributed optimization over time-varying graphs, SIAM J. Optim., 27 (2017), 2597-2633.  doi: 10.1137/16M1084316.  Google Scholar

[21]

B. T. Polyak, Introduction to Optimization, Optimization Software, Inc., Publications Division, New York, 1987.  Google Scholar

[22]

S. S. Ram, A. Nedić and V. V. Veeravalli, Distributed stochastic subgradient projection algorithms for convex optimization, J. Optim. Theory Appl., 147 (2010), 516-545.  doi: 10.1007/s10957-010-9737-7.  Google Scholar

[23]

W. Shi, Q. Ling, K. Yuan, G. Wu and W. Yin, On the linear convergence of the ADMM in decentralized consensus optimization, IEEE Trans. Signal Process., 62 (2014), 1750-1761.  doi: 10.1109/TSP.2014.2304432.  Google Scholar

[24]

W. Shi, Q. Ling, G. Wu and W. Yin, EXTRA: An exact first-order algorithm for decentralized consensus optimization, SIAM J. Optim., 25 (2015), 944-966.  doi: 10.1137/14096668X.  Google Scholar

[25]

K. I. Tsianos, S. Lawlor and M. G. Rabbat, Push-sum distributed dual averaging for convex optimization, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), (2012), 5453–5458. doi: 10.1109/CDC.2012.6426375.  Google Scholar

[26]

S. Wang and C. Li, Distributed robust optimization in networked system, IEEE Transactions on Cybernetics, 47 (2017), 2321-2333.  doi: 10.1109/TCYB.2016.2613129.  Google Scholar

[27]

X. Xia and A. M. Elaiw, Optimal dynamic economic dispatch of generation: A review, Electric Power Systems Research, 80 (2010), 975-986.  doi: 10.1016/j.epsr.2009.12.012.  Google Scholar

[28]

C. Yang, Y. Xu, L. Zhou and Y. Sun, Model-free composite control of flexible manipulators based on adaptive dynamic programming, Complexity, 2018 (2018), Article ID 9720309, 9 pp. doi: 10.1155/2018/9720309.  Google Scholar

[29]

D. Yuan, S. Xu and H. Zhao, Distributed primal-dual subgradient method for multiagent optimization via consensus algorithms, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 41 (2011), 1715-1724.   Google Scholar

[30]

D. Yuan, D. W. Ho and Y. Hong, On convergence rate of distributed stochastic gradient algorithm for convex optimization with inequality constraints, SIAM J. Control Optim., 54 (2016), 2872-2892.  doi: 10.1137/15M1048896.  Google Scholar

[31]

D. Yuan, D. W. C. Ho and S. Xu, Regularized primal-dual subgradient method for distributed constrained optimization, IEEE Transactions on Cybernetics, 46 (2016), 2109-2118.  doi: 10.1109/TCYB.2015.2464255.  Google Scholar

[32]

Y. Zhang and G. B. Giannakis, Distributed stochastic market clearing with high-penetration wind power, IEEE Transactions on Power Systems, 31 (2016), 895-906.   Google Scholar

[33]

Z. Zhang and M. Y. Chow, Convergence analysis of the incremental cost consensus algorithm under different communication network topologies in a smart grid, IEEE Transactions on Power Systems, 27 (2012), 1761-1768.  doi: 10.1109/TPWRS.2012.2188912.  Google Scholar

[34]

M. Zhu and S. Martínez, On distributed convex optimization under inequality and equality constraints, IEEE Trans. Automat. Control, 57 (2012), 151-164.  doi: 10.1109/TAC.2011.2167817.  Google Scholar

Figure 1.  The directed connected graph at time $ t $ with $ \mathscr{G}[t] = (\mathscr{V}, \mathscr{E}[t]) $
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Figure 2.  Dual variables (fanxiexian_myfh/MWh) v.s. iterations
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Figure 3.  Generation outputs (MW) v.s. iterations
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Figure 4.  Total generation v.s. demand
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Figure 5.  Value of total cost function v.s. iterations
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Table 1.  Parameters for 7 generators in IEEE 57-bus system
Gen. $ a_{i} $ $ b_{i} $ $ c_{i} $ [$ p_{i}^{\mathrm{min}},p_{i}^{\mathrm{max}} $]
1 0.0775795 20 0 [0,575.88]
2 0.01 40 0 [0,100]
3 0.25 20 0 [0,140]
6 0.01 40 0 [0,100]
8 0.0222222 20 0 [0,550]
9 0.01 40 0 [0,100]
12 0.0322581 20 0 [0,410]
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Download as excel

Gen. $ a_{i} $ $ b_{i} $ $ c_{i} $ [$ p_{i}^{\mathrm{min}},p_{i}^{\mathrm{max}} $]
1 0.0775795 20 0 [0,575.88]
2 0.01 40 0 [0,100]
3 0.25 20 0 [0,140]
6 0.01 40 0 [0,100]
8 0.0222222 20 0 [0,550]
9 0.01 40 0 [0,100]
12 0.0322581 20 0 [0,410]
Table 2.  Number of iterations with different nodes for both Algorithms DDSG-PS and DDP
Number of node $ m $ Alg. DDSG-PS Alg. DDSG
10 38 34
50 97 99
100 203 201
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Download as excel

Number of node $ m $ Alg. DDSG-PS Alg. DDSG
10 38 34
50 97 99
100 203 201
Table 3.  Number of iterations with different dimensions for both Algorithms DDSG-PS and DDP
Dimension of $ \widehat{\mathbf{x}}_{i} $ Alg. DDSG-PS Alg. DDSG
3 50 84
5 77 98
9 108 136
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Dimension of $ \widehat{\mathbf{x}}_{i} $ Alg. DDSG-PS Alg. DDSG
3 50 84
5 77 98
9 108 136
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