Linear convergence of a primal-dual algorithm for distributed interval optimization

Yinghui Wang; Jiuwei Wang; Xiaobo Song; Yanpeng Hu; Yinghui Wang; Jiuwei Wang; Xiaobo Song; Yanpeng Hu

doi:10.3934/era.2024041

Electronic Research Archive

2024, Volume 32, Issue 2: 857-873. doi: 10.3934/era.2024041

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Linear convergence of a primal-dual algorithm for distributed interval optimization

1.
Key Laboratory of Knowledge Automation for Industrial Processes of Ministry of Education, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China
2.
Beijing Engineering Research Center of Industrial Spectrum Imaging, School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, China
3.
University of Chinese Academy of Sciences, Beijing 100049, China
4.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Received: 09 October 2023 Revised: 18 December 2023 Accepted: 27 December 2023 Published: 12 January 2024

In this paper, we investigate a distributed interval optimization problem whose local functions are interval functions rather than scalar functions. Focusing on distributed interval optimization, this paper presents a distributed primal-dual algorithm. A criterion is introduced under which linear convergence to the Pareto solution of distributed interval optimization problems can be achieved without strong convexity. Lastly, a numerical simulation is presented to illustrate the linear convergence of the algorithm that has been proposed.
- distributed interval optimization problem,
- Pareto solution,
- linear convergence rates,
- primal-dual algorithm
Citation: Yinghui Wang, Jiuwei Wang, Xiaobo Song, Yanpeng Hu. Linear convergence of a primal-dual algorithm for distributed interval optimization[J]. Electronic Research Archive, 2024, 32(2): 857-873. doi: 10.3934/era.2024041

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Abstract

In this paper, we investigate a distributed interval optimization problem whose local functions are interval functions rather than scalar functions. Focusing on distributed interval optimization, this paper presents a distributed primal-dual algorithm. A criterion is introduced under which linear convergence to the Pareto solution of distributed interval optimization problems can be achieved without strong convexity. Lastly, a numerical simulation is presented to illustrate the linear convergence of the algorithm that has been proposed.

References

[1]	I. Necoara, Y. Nesterov, F. Glineur, Linear convergence of first order methods for non-strongly convex optimization, Math. Program., 175 (2019), 69–107. https://doi.org/10.1007/s10107-018-1232-1 doi: 10.1007/s10107-018-1232-1
[2]	A. Makhdoumi, A. Ozdaglar, Convergence rate of distributed admm over networks, IEEE Trans. Autom. Control, 62 (2017), 5082–5095. https://doi.org/10.1109/TAC.2017.2677879 doi: 10.1109/TAC.2017.2677879
[3]	X. He, T. Huang, J. Yu, C. Li, Y. Zhang, A continuous-time algorithm for distributed optimization based on multiagent networks, IEEE Trans. Syst. Man Cybern.: Syst., 49 (2017), 2700–2709. https://doi.org/10.1109/TSMC.2017.2780194 doi: 10.1109/TSMC.2017.2780194
[4]	H. Li, Q. Lü, X. Liao, T. Huang, Accelerated convergence algorithm for distributed constrained optimization under time-varying general directed graphs, IEEE Trans. Syst. Man Cybern.: Syst., 50 (2018), 2612–2622. https://doi.org/10.1109/TSMC.2018.2823901 doi: 10.1109/TSMC.2018.2823901
[5]	Q. Wang, J. Chen, B. Xin, X. Zeng, Distributed optimal consensus for euler-lagrange systems based on event-triggered control, IEEE Trans. Syst. Man Cybern.: Syst., 51 (2021), 4588–4598. https://doi.org/10.1109/TSMC.2019.2944857 doi: 10.1109/TSMC.2019.2944857
[6]	J. Guo, R. Jia, R. Su, Y. Zhao, Identification of fir systems with binary-valued observations against data tampering attacks, IEEE Trans. Syst. Man Cybern.: Syst., 53 (2023), 5861–5873. https://doi.org/10.1109/TSMC.2023.3276352 doi: 10.1109/TSMC.2023.3276352
[7]	J. Guo, X. Wang, W. Xue, Y. Zhao, System identification with binary-valued observations under data tampering attacks, IEEE Trans. Autom. Control, 66 (2020), 3825–3832. https://doi.org/10.1109/TAC.2020.3029325 doi: 10.1109/TAC.2020.3029325
[8]	X. Zeng, Y. Peng, Y. Hong, Distributed algorithm for robust resource allocation with polyhedral uncertain allocation parameters, J. Syst. Sci. Complexity, 31 (2018), 103–119. https://doi.org/10.1007/s11424-018-7145-5 doi: 10.1007/s11424-018-7145-5
[9]	V. Kekatos, G. B. Giannakis, Distributed robust power system state estimation, IEEE Trans. Power Syst., 28 (2013), 1617–1626. https://doi.org/10.1109/TPWRS.2012.2219629 doi: 10.1109/TPWRS.2012.2219629
[10]	S. Sra, S. Nowozin, S. J. Wright, Optimization for Machine Learning, Mit Press, 2012.
[11]	B. Q. Hu, S. Wang, A novel approach in uncertain programming part Ⅰ: new arithmetic and order relation for interval numbers, J. Ind. Manage. Optim., 2 (2006), 351–371. https://doi.org/10.3934/jimo.2006.2.351 doi: 10.3934/jimo.2006.2.351
[12]	L. Wu, M. Shahidehpour, Z. Li, Comparison of scenario-based and interval optimization approaches to stochastic SCUC, IEEE Trans. Power Syst., 27 (2012), 913–921. https://doi.org/10.1109/TPWRS.2011.2164947 doi: 10.1109/TPWRS.2011.2164947
[13]	A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, 1990.
[14]	J. Rohn, Positive definiteness and stability of interval matrices, SIAM J. Matrix Anal. Appl., 15 (1994), 175–184. https://doi.org/10.1137/S0895479891219216 doi: 10.1137/S0895479891219216
[15]	V. I. Levin, Nonlinear optimization under interval uncertainty, Cybern. Syst. Anal., 35 (1999), 297–306. https://doi.org/10.1007/BF02733477 doi: 10.1007/BF02733477
[16]	T. Saeed, S. Treană, New classes of interval-valued variational problems and inequalities, Results Control Optim., 13 (2023), 100324. https://doi.org/10.1016/j.rico.2023.100324 doi: 10.1016/j.rico.2023.100324
[17]	M. Ciontescu, S. Treană, On some connections between interval-valued variational control problems and the associated inequalities, Results Control Optim., 12 (2023), 100300. https://doi.org/10.1016/j.rico.2023.100300 doi: 10.1016/j.rico.2023.100300
[18]	Y. Guo, G. Ye, W. Liu, D. Zhao, S. Treanţǎ, Solving nonsmooth interval optimization problems based on interval-valued symmetric invexity, Chaos, Solitons Fractals, 174 (2023), 113834. https://doi.org/10.1016/j.chaos.2023.113834 doi: 10.1016/j.chaos.2023.113834
[19]	S. Treană, T. Saeed, On weak variational control inequalities via interval analysis, Mathematics, 11 (2023), 2177. https://doi.org/10.3390/math11092177 doi: 10.3390/math11092177
[20]	I. Hisao, T. Hideo, Multiobjective programming in optimization of the interval objective function, Eur. J. Oper. Res., 48 (1990), 219–225. https://doi.org/10.1016/0377-2217(90)90375-L doi: 10.1016/0377-2217(90)90375-L
[21]	S. T. Liu, R. T. Wang, A numerical solution method to interval quadratic programming, Appl. Math. Comput., 189 (2007), 1274–1281. https://doi.org/10.1016/j.amc.2006.12.007 doi: 10.1016/j.amc.2006.12.007
[22]	C. Jiang, X. Han, G. Liu, G. Liu, A nonlinear interval number programming method for uncertain optimization problems, Eur. J. Oper. Res., 188 (2008), 1–13. https://doi.org/10.1016/j.ejor.2007.03.031 doi: 10.1016/j.ejor.2007.03.031
[23]	A. Jayswal, I. Stancu-Minasian, I. Ahmad, On sufficiency and duality for a class of interval-valued programming problems, Appl. Math. Comput., 218 (2011), 4119–4127. https://doi.org/10.1016/j.amc.2011.09.041 doi: 10.1016/j.amc.2011.09.041
[24]	M. Hladık, Interval linear programming: a survey, in Linear Programming-New Frontiers in Theory and Applications, (2012), 85–120.
[25]	A. Bellet, Y. Liang, A. B. Garakani, M. F. Balcan, F. Sha, A distributed Frank-Wolfe algorithm for communication-efficient sparse learning, in Proceedings of the 2015 SIAM International Conference on Data Mining (SDM), (2015), 478–486. https://doi.org/10.1137/1.9781611974010.54
[26]	G. Qu, N. Li, Accelerated distributed Nesterov gradient descent, IEEE Trans. Autom. Control, 65 (2020), 2566–2581. https://doi.org/10.1109/TAC.2019.2937496 doi: 10.1109/TAC.2019.2937496
[27]	A. Nedic, A. Olshevsky, W. Shi, Achieving geometric convergence for distributed optimization over time-varying graphs, SIAM J. Optim., 27 (2017), 2597–2633. https://doi.org/10.1137/16M1084316 doi: 10.1137/16M1084316
[28]	S. Liang, L. Y. Wang, G. Yin, Exponential convergence of distributed primal–dual convex optimization algorithm without strong convexity, Automatica, 105 (2019), 298–306. https://doi.org/10.1016/j.automatica.2019.04.004 doi: 10.1016/j.automatica.2019.04.004
[29]	X. Yi, S. Zhang, T. Yang, K. H. Johansson, T. Chai, Exponential convergence for distributed optimization under the restricted secant inequality condition, IFAC-PapersOnLine, 53 (2020), 2672–2677. https://doi.org/10.1016/j.ifacol.2020.12.383 doi: 10.1016/j.ifacol.2020.12.383
[30]	S. Treană, Lu-optimality conditions in optimization problems with mechanical work objective functionals, IEEE Trans. Neural Networks Learn. Syst., 33 (2021), 4971–4978. https://doi.org/10.1109/TNNLS.2021.3066196 doi: 10.1109/TNNLS.2021.3066196
[31]	H. C. Wu, On interval-valued nonlinear programming problems, J. Math. Anal. Appl., 338 (2008), 299–316. https://doi.org/10.1016/j.jmaa.2007.05.023 doi: 10.1016/j.jmaa.2007.05.023
[32]	B. T. Polyak, Introduction to Optimization, Chapman and Hall, 1987.
[33]	R. Durrett, Probability: Theory and Examples, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511779398
[34]	T. Maeda, On optimization problems with set-valued objective maps: existence and optimality, J. Optim. Theory Appl., 153 (2012), 263–279. https://doi.org/10.1007/s10957-011-9952-x doi: 10.1007/s10957-011-9952-x
[35]	A. K. Bhurjee, G. Panda, Efficient solution of interval optimization problem, Math. Methods Oper. Res., 76 (2012), 273–288. https://doi.org/10.1007/s00186-012-0399-0 doi: 10.1007/s00186-012-0399-0
[36]	S. S. Ram, A. Nedić, V. V. Veeravalli, Distributed stochastic subgradient projection algorithms for convex optimization, J. Optim. Theory Appl., 147 (2010), 516–545. https://doi.org/10.1007/s10957-010-9737-7 doi: 10.1007/s10957-010-9737-7
[37]	A. Nedic, A. Ozdaglar, Distributed subgradient methods for multi-agent optimization, IEEE Trans. Autom. Control, 54 (2009), 48–61. https://doi.org/10.1109/TAC.2008.2009515 doi: 10.1109/TAC.2008.2009515
[38]	A. P. Ruszczyński, A. Ruszczynski, Nonlinear Optimization, Princeton University Press, 2006. https://doi.org/10.1515/9781400841059
[39]	A. Nedic, A. Ozdaglar, P. A. Parrilo, Constrained consensus and optimization in multi-agent networks, IEEE Trans. Autom. Control, 55 (2010), 922–938. https://doi.org/10.1109/TAC.2010.2041686 doi: 10.1109/TAC.2010.2041686
[40]	A. Nedić, A. Olshevsky, Stochastic gradient-push for strongly convex functions on time-varying directed graphs, IEEE Trans. Autom. Control, 61 (2016), 3936–3947. https://doi.org/10.1109/TAC.2016.2529285 doi: 10.1109/TAC.2016.2529285

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