A Decentralized Multi-objective Optimization Algorithm
During the past few decades, multi-agent optimization problems have drawn increased attention from the research community. When multiple objective functions are present among agents, many works optimize the sum of these objective functions. However, this formulation implies a decision regarding the...
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| Vydané v: | Journal of optimization theory and applications Ročník 189; číslo 2; s. 458 - 485 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
Springer US
01.05.2021
Springer Nature B.V |
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| ISSN: | 0022-3239, 1573-2878 |
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| Abstract | During the past few decades, multi-agent optimization problems have drawn increased attention from the research community. When multiple objective functions are present among agents, many works optimize the sum of these objective functions. However, this formulation implies a decision regarding the relative importance of each objective: optimizing the sum is a special case of a multi-objective problem in which all objectives are prioritized equally. To enable more general prioritizations, we present a distributed optimization algorithm that explores Pareto optimal solutions for non-homogeneously weighted sums of objective functions. This exploration is performed through a new rule based on agents’ priorities that generates edge weights in agents’ communication graph. These weights determine how agents update their decision variables with information received from other agents in the network. Agents initially disagree on the priorities of objective functions, though they are driven to agree upon them as they optimize. As a result, agents still reach a common solution. The network-level weight matrix is (non-doubly) stochastic, contrasting with many works on the subject in which the network-level weight matrix is doubly-stochastic. New theoretical analyses are therefore developed to ensure convergence of the proposed algorithm. This paper provides a gradient-based optimization algorithm, proof of convergence to solutions, and convergence rates of the proposed algorithm. It is shown that agents’ initial priorities influence the convergence rate of the proposed algorithm and that these initial choices affect its long-run behavior. Numerical results performed with different numbers of agents illustrate the performance and effectiveness of the proposed algorithm. |
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| AbstractList | During the past few decades, multi-agent optimization problems have drawn increased attention from the research community. When multiple objective functions are present among agents, many works optimize the sum of these objective functions. However, this formulation implies a decision regarding the relative importance of each objective: optimizing the sum is a special case of a multi-objective problem in which all objectives are prioritized equally. To enable more general prioritizations, we present a distributed optimization algorithm that explores Pareto optimal solutions for non-homogeneously weighted sums of objective functions. This exploration is performed through a new rule based on agents’ priorities that generates edge weights in agents’ communication graph. These weights determine how agents update their decision variables with information received from other agents in the network. Agents initially disagree on the priorities of objective functions, though they are driven to agree upon them as they optimize. As a result, agents still reach a common solution. The network-level weight matrix is (non-doubly) stochastic, contrasting with many works on the subject in which the network-level weight matrix is doubly-stochastic. New theoretical analyses are therefore developed to ensure convergence of the proposed algorithm. This paper provides a gradient-based optimization algorithm, proof of convergence to solutions, and convergence rates of the proposed algorithm. It is shown that agents’ initial priorities influence the convergence rate of the proposed algorithm and that these initial choices affect its long-run behavior. Numerical results performed with different numbers of agents illustrate the performance and effectiveness of the proposed algorithm. |
| Author | Blondin, Maude J. Hale, Matthew |
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| Cites_doi | 10.1016/j.automatica.2014.10.022 10.1109/TIE.2016.2636810 10.1137/110837462 10.1109/TAC.2015.2416927 10.1007/s10957-010-9737-7 10.1109/EnergyTech.2013.6645336 10.1007/978-3-662-08883-8 10.1016/j.jpdc.2006.08.010 10.1137/S1052623499362111 10.1109/Allerton.2012.6483403 10.1016/j.automatica.2012.05.025 10.23919/ECC.2018.8550343 10.1109/ALLERTON.2010.5706956 10.23919/ACC45564.2020.9148017 10.1016/j.pisc.2015.11.022 10.1109/TAC.2008.2009515 10.1109/JAS.2014.7004621 10.1109/JPROC.2006.887293 10.1109/TAC.2010.2041686 10.1007/s10107-011-0467-x 10.1109/TAC.2011.2161027 |
| ContentType | Journal Article |
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| DOI | 10.1007/s10957-021-01840-z |
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| References | Nedić, A., Ozdaglar, A., Parrilo, P.: Constrained consensus. arXiv preprint arXiv:0802.3922 (2008) Bianchi, P., Fort, G., Hachem, P., Jakubowicz, J.: Performance analysis of a distributed Robbins-Monro algorithm for sensor networks. In: European Signal Processing Conference, pp. 1030–1034 (2011) Blondel, V.D., Hendrickx, J.M., Olshevsky, A., Tsitsiklis, J.N.: Convergence in multiagent coordination, consensus, and flocking. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2996–3000 (2005) Khim, S.: The Frobenius–Perron theorem. Doctoral Dissertation, PhD thesis, The University of Chicago (2007) Blondin, M. J., Hale, M.: An algorithm for multi-objective multi-agent optimization. In: American control conference (ACC), pp. 1489–1494. Denver, CO (2020). https://doi.org/10.23919/ACC45564.2020.9148017 ZhangYLouYHongYAn approximate gradient algorithm for constrained distributed convex optimizationIEEE/CAA J. Autom. Sinica20141616710.1109/JAS.2014.7004621 NedicABertsekasDPIncremental subgradient methods for nondifferentiable optimizationSIAM J. Optim.2001121109138187058810.1137/S1052623499362111 Byungchul, K., Lavrova, O.: Optimal power flow and energy-sharing among multi-agent smart buildings in the smart grid. In: IEEE Energytech, pp. 1–5 (2013) Filotheou, A., Nikou, A., Dimarogonas, D.V.: Decentralized control of uncertain multi-agent systems with connectivity maintenance and collision avoidance. In: European Control Conference, pp. 8–13 (2018) ColletteYSiarryPMultiobjective Optimization: Principles and Case Studies2004BerlinSpringer10.1007/978-3-662-08883-8 DuchiJCAgarwalAWainwrightMJDual averaging for distributed optimization: convergence analysis and network scalingIEEE Trans. Autom. Control2011573592606293281810.1109/TAC.2011.2161027 OhKKParkMCAhnHSA survey of multi-agent formation controlAutomatica20155342440331861810.1016/j.automatica.2014.10.022 Olfati-SaberRFaxJAMurrayRMConsensus and cooperation in networked multi-agent systemsProc. IEEE.200795121523310.1109/JPROC.2006.887293 NedićAOzdaglarAParriloPConstrained consensus and optimization in multi-agent networksIEEE Trans. Autom. Control2010554922938265443210.1109/TAC.2010.2041686 LiuQWangJA second-order multi-agent network for bound-constrained distributed optimizationIEEE Trans. Autom. Control2015601233103325343270010.1109/TAC.2015.2416927 NedićAOzdaglarADistributed subgradient methods for multi-agent optimizationIEEE Trans. Autom. Control20095414861247807010.1109/TAC.2008.2009515 LobelIOzdaglarAFeijerDDistributed multi-agent optimization with state-dependent communicationMath. Program.20111292255284283788210.1007/s10107-011-0467-x OlshevskyATsitsiklisJNConvergence speed in distributed consensus and averagingSIAM Rev.2011534747772286126510.1137/110837462 TouriBNedicAOn backward product of stochastic matricesAutomatica201848814771488295039710.1016/j.automatica.2012.05.025 WangXSuHWangXChenGAn overview of coordinated control for multi-agent systems subject to input saturationPerspect. Sci.201671333910.1016/j.pisc.2015.11.022 XiaoLBoydSKimSJDistributed average consensus with least-mean-square deviationJ. Parallel Distrib. Comput.2007671334610.1016/j.jpdc.2006.08.010 QinJMaQShiYWangLRecent advances in consensus of multi-agent systems: a brief surveyIEEE Trans. Ind. Electron.20166464972498310.1109/TIE.2016.2636810 Agarwal, A., Duchi, J.C.: Distributed delayed stochastic optimization. In: Advances in Neural Information Processing Systems, pp. 873–881 (2011) NedićAOzdaglarAPalomarDEldarYCooperative distributed multi-agentConvex Optimization in Signal Processing and Communications2010CambridgeCambridge University Press3403861241.90100 Tsianos, K.I., Lawlor, S., Rabbat, M.G.: Consensus-based distributed optimization: practical issues and applications in large-scale machine learning. In: Annual Allerton IEEE Conference on Communication, Control, and Computing, pp. 1543–1550 (2012) RamSSNedićAVeeravalliVVDistributed stochastic subgradient projection algorithms for convex optimizationJ. Optim. Theory Appl.20101473516545273399210.1007/s10957-010-9737-7 MiettinenKMNonlinear Multiobjective Optimiation1999New YorkKluwer Academic Publishers Wang, J., Elia, N.: Control approach to distributed optimization. In: Annual Allerton Conference on Communication, Control, and Computing, pp. 557–561 (2010) A Nedić (1840_CR14) 2009; 54 1840_CR2 1840_CR1 Y Zhang (1840_CR28) 2014; 1 1840_CR5 1840_CR4 1840_CR3 1840_CR9 1840_CR8 I Lobel (1840_CR11) 2011; 129 1840_CR16 JC Duchi (1840_CR7) 2011; 57 Y Collette (1840_CR6) 2004 J Qin (1840_CR21) 2016; 64 KM Miettinen (1840_CR12) 1999 A Nedic (1840_CR13) 2001; 12 A Olshevsky (1840_CR20) 2011; 53 L Xiao (1840_CR27) 2007; 67 X Wang (1840_CR26) 2016; 7 R Olfati-Saber (1840_CR19) 2007; 95 KK Oh (1840_CR18) 2015; 53 1840_CR25 1840_CR24 A Nedić (1840_CR17) 2010; 55 SS Ram (1840_CR22) 2010; 147 A Nedić (1840_CR15) 2010 B Touri (1840_CR23) 2018; 48 Q Liu (1840_CR10) 2015; 60 |
| References_xml | – reference: Byungchul, K., Lavrova, O.: Optimal power flow and energy-sharing among multi-agent smart buildings in the smart grid. In: IEEE Energytech, pp. 1–5 (2013) – reference: OlshevskyATsitsiklisJNConvergence speed in distributed consensus and averagingSIAM Rev.2011534747772286126510.1137/110837462 – reference: NedićAOzdaglarAParriloPConstrained consensus and optimization in multi-agent networksIEEE Trans. Autom. Control2010554922938265443210.1109/TAC.2010.2041686 – reference: LobelIOzdaglarAFeijerDDistributed multi-agent optimization with state-dependent communicationMath. Program.20111292255284283788210.1007/s10107-011-0467-x – reference: Agarwal, A., Duchi, J.C.: Distributed delayed stochastic optimization. In: Advances in Neural Information Processing Systems, pp. 873–881 (2011) – reference: XiaoLBoydSKimSJDistributed average consensus with least-mean-square deviationJ. Parallel Distrib. Comput.2007671334610.1016/j.jpdc.2006.08.010 – reference: Nedić, A., Ozdaglar, A., Parrilo, P.: Constrained consensus. arXiv preprint arXiv:0802.3922 (2008) – reference: DuchiJCAgarwalAWainwrightMJDual averaging for distributed optimization: convergence analysis and network scalingIEEE Trans. Autom. Control2011573592606293281810.1109/TAC.2011.2161027 – reference: NedićAOzdaglarAPalomarDEldarYCooperative distributed multi-agentConvex Optimization in Signal Processing and Communications2010CambridgeCambridge University Press3403861241.90100 – reference: Khim, S.: The Frobenius–Perron theorem. Doctoral Dissertation, PhD thesis, The University of Chicago (2007) – reference: Wang, J., Elia, N.: Control approach to distributed optimization. In: Annual Allerton Conference on Communication, Control, and Computing, pp. 557–561 (2010) – reference: WangXSuHWangXChenGAn overview of coordinated control for multi-agent systems subject to input saturationPerspect. Sci.201671333910.1016/j.pisc.2015.11.022 – reference: Blondin, M. J., Hale, M.: An algorithm for multi-objective multi-agent optimization. In: American control conference (ACC), pp. 1489–1494. Denver, CO (2020). https://doi.org/10.23919/ACC45564.2020.9148017 – reference: NedicABertsekasDPIncremental subgradient methods for nondifferentiable optimizationSIAM J. Optim.2001121109138187058810.1137/S1052623499362111 – reference: Olfati-SaberRFaxJAMurrayRMConsensus and cooperation in networked multi-agent systemsProc. IEEE.200795121523310.1109/JPROC.2006.887293 – reference: LiuQWangJA second-order multi-agent network for bound-constrained distributed optimizationIEEE Trans. Autom. Control2015601233103325343270010.1109/TAC.2015.2416927 – reference: RamSSNedićAVeeravalliVVDistributed stochastic subgradient projection algorithms for convex optimizationJ. Optim. Theory Appl.20101473516545273399210.1007/s10957-010-9737-7 – reference: Blondel, V.D., Hendrickx, J.M., Olshevsky, A., Tsitsiklis, J.N.: Convergence in multiagent coordination, consensus, and flocking. In: Proceedings of the 44th IEEE Conference on Decision and Control, pp. 2996–3000 (2005) – reference: QinJMaQShiYWangLRecent advances in consensus of multi-agent systems: a brief surveyIEEE Trans. Ind. Electron.20166464972498310.1109/TIE.2016.2636810 – reference: Filotheou, A., Nikou, A., Dimarogonas, D.V.: Decentralized control of uncertain multi-agent systems with connectivity maintenance and collision avoidance. In: European Control Conference, pp. 8–13 (2018) – reference: Tsianos, K.I., Lawlor, S., Rabbat, M.G.: Consensus-based distributed optimization: practical issues and applications in large-scale machine learning. In: Annual Allerton IEEE Conference on Communication, Control, and Computing, pp. 1543–1550 (2012) – reference: TouriBNedicAOn backward product of stochastic matricesAutomatica201848814771488295039710.1016/j.automatica.2012.05.025 – reference: MiettinenKMNonlinear Multiobjective Optimiation1999New YorkKluwer Academic Publishers – reference: Bianchi, P., Fort, G., Hachem, P., Jakubowicz, J.: Performance analysis of a distributed Robbins-Monro algorithm for sensor networks. In: European Signal Processing Conference, pp. 1030–1034 (2011) – reference: ZhangYLouYHongYAn approximate gradient algorithm for constrained distributed convex optimizationIEEE/CAA J. Autom. 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| SubjectTerms | Algorithms Applications of Mathematics Calculus of Variations and Optimal Control; Optimization Convergence Engineering Mathematics Mathematics and Statistics Multiagent systems Multiple objective analysis Operations Research/Decision Theory Optimization Optimization algorithms Pareto optimization Pareto optimum Priorities Theory of Computation Weight |
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| Title | A Decentralized Multi-objective Optimization Algorithm |
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