The bifurcation of constrained optimization optimal solutions and its applications

Tengmu Li; Zhiyuan Wang; Tengmu Li; Zhiyuan Wang

doi:10.3934/math.2023622

AIMS Mathematics

2023, Volume 8, Issue 5: 12373-12397. doi: 10.3934/math.2023622

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Research article Special Issues

The bifurcation of constrained optimization optimal solutions and its applications

Tengmu Li ^†,
Zhiyuan Wang ^{†
,
,}

School of Electrical and Information Engineering, Tianjin University, No. 92 Weijin Road, Nankai District, Tianjin 300072, China
† These authors contributed equally and should be considered co-first authors.

Received: 07 January 2023 Revised: 26 February 2023 Accepted: 06 March 2023 Published: 24 March 2023
MSC : 37G35, 37D05, 37N30

The appearance and disappearance of the optimal solution for the change of system parameters in optimization theory is a fundamental problem. This paper aims to address this issue by transforming the solutions of a constrained optimization problem into equilibrium points (EPs) of a dynamical system. The bifurcation of EPs is then used to describe the appearance and disappearance of the optimal solution and saddle point through two classes of bifurcation, namely the pseudo bifurcation and saddle-node bifurcation. Moreover, a new class of pseudo-bifurcation phenomena is introduced to describe the transformation of regular and degenerate EPs, which sheds light on the relationship between the optimal solution and a class of infeasible points. This development also promotes the proposal of a tool for predicting optimal solutions based on this phenomenon. The study finds that the bifurcation of the optimal solution is closely related to the bifurcation of the feasible region, as demonstrated by the 5-bus and 9-bus optimal power flow problems.
- bifurcation,
- constrained optimization problem,
- parametric nonlinear programming,
- dynamic systems
Citation: Tengmu Li, Zhiyuan Wang. The bifurcation of constrained optimization optimal solutions and its applications[J]. AIMS Mathematics, 2023, 8(5): 12373-12397. doi: 10.3934/math.2023622

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Abstract

The appearance and disappearance of the optimal solution for the change of system parameters in optimization theory is a fundamental problem. This paper aims to address this issue by transforming the solutions of a constrained optimization problem into equilibrium points (EPs) of a dynamical system. The bifurcation of EPs is then used to describe the appearance and disappearance of the optimal solution and saddle point through two classes of bifurcation, namely the pseudo bifurcation and saddle-node bifurcation. Moreover, a new class of pseudo-bifurcation phenomena is introduced to describe the transformation of regular and degenerate EPs, which sheds light on the relationship between the optimal solution and a class of infeasible points. This development also promotes the proposal of a tool for predicting optimal solutions based on this phenomenon. The study finds that the bifurcation of the optimal solution is closely related to the bifurcation of the feasible region, as demonstrated by the 5-bus and 9-bus optimal power flow problems.

References

[1]	C. Arancibia-Ibarra, P. Aguirre, J. Flores, P. van Heijster, Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response, Appl. Math. Comput., 402 (2021), 126152. https://doi.org/10.1016/j.amc.2021.126152 doi: 10.1016/j.amc.2021.126152
[2]	A. M. Dehrouyeh-Semnani, On bifurcation behavior of hard magnetic soft cantilevers, Int. J. Non-Linear Mech., 134 (2021), 103746. https://doi.org/10.1016/j.ijnonlinmec.2021.103746 doi: 10.1016/j.ijnonlinmec.2021.103746
[3]	R. Zhou, Y. Gu, J. Cui, G. Ren, S. Yu, Nonlinear dynamic analysis of supercritical and subcritical Hopf bifurcations in gas foil bearing-rotor systems, Nonlinear Dyn., 103 (2021), 2241–2256. https://doi.org/10.1007/s11071-021-06234-4 doi: 10.1007/s11071-021-06234-4
[4]	J. Nocedal, S. J. Wright, Numerical optimization, New York: Springer, 1999.
[5]	A. B. Poore, Bifurcations in parametric nonlinear programming, Ann. Oper. Res., 27 (1990), 343–369. https://doi.org/10.1007/BF02055201 doi: 10.1007/BF02055201
[6]	A. B. Poore, C. A. Tiahrt, Bifurcation problems in nonlinear parametric programming, Math. Program., 39 (1987), 189–205. https://doi.org/10.1007/bf02592952 doi: 10.1007/bf02592952
[7]	M. Kojima, Strongly stable stationary solutions in nonlinear programs, In: Analysis and computation of fixed points, Elsevier, 1980, 93–138.
[8]	M. Kojima, R. Hirabayashi, Continuous deformation of nonlinear programs, In: A. V. Fiacco, Sensitivity, stability and parametric analysis, Mathematical Programming Studies, Springer Berlin Heidelberg, Berlin, Heidelberg, 21 (1984), 150–198. https://doi.org/10.1007/BFb0121217
[9]	H. D. Chiang, C. Y. Jiang, Feasible region of optimal power flow: characterization and applications, IEEE Trans. Power Syst., 33 (2018), 236–244. https://doi.org/10.1109/TPWRS.2017.2692268 doi: 10.1109/TPWRS.2017.2692268
[10]	W. A. Bukhsh, A. Grothey, K. I. M. McKinnon, P. A. Trodden, Local solutions of the optimal power flow problem, IEEE Trans. Power Syst., 28 (2013), 4780–4788. https://doi.org/10.1109/TPWRS.2013.2274577 doi: 10.1109/TPWRS.2013.2274577
[11]	C. Y. Jiang, H. D. Chiang, Pseudo-pitchfork bifurcation of feasible regions in power systems, Int. J. Bifurcation Chaos, 28 (2018), 1830002. https://doi.org/10.1142/S0218127418300021 doi: 10.1142/S0218127418300021
[12]	D. P. Bertsekas, Nonlinear programming, 2 Eds., Athena Scientific, Belmont, Massachusetts, 2003.
[13]	M. Ye, J. Shen, G. Lin, T. Xiang, L. Shao, S. C. H. Hoi, Deep learning for person re-identification: a survey and outlook, IEEE Trans. Pattern Anal. Mach. Intell., 44 (2022), 2872–2893. https://doi.org/10.1109/TPAMI.2021.3054775 doi: 10.1109/TPAMI.2021.3054775
[14]	A. M. Shaheen, R. A. El-Sehiemy, H. M. Hasanien, A. R. Ginidi, An improved heap optimization algorithm for efficient energy management based optimal power flow model, Energy, 250 (2022), 123795. https://doi.org/10.1016/j.energy.2022.123795 doi: 10.1016/j.energy.2022.123795
[15]	Z. Y. Wu, F. S. Bai, X. Q. Yang, L. S. Zhang, An exact lower order penalty function and its smoothing in nonlinear programming, Optimization, 53 (2004), 51–68. https://doi.org/10.1080/02331930410001662199 doi: 10.1080/02331930410001662199
[16]	H. D. Chiang, L. F. C. Alberto, Stability regions of nonlinear dynamic systems: theory, estimation, and applications, Cambridge, U.K.: Cambridge University Press, 2015. https://doi.org/10.1017/CBO9781139548861
[17]	F. Capitanescu, Critical review of recent advances and further developments needed in AC optimal power flow, Electr. Power Syst. Res., 136 (2016), 57–68. https://doi.org/10.1016/j.epsr.2016.02.008 doi: 10.1016/j.epsr.2016.02.008
[18]	M. B. Cain, R. P. O'Neill, A. Castillo, History of optimal power flow and formulations, Federal Energy Regul. Comm., 1 (2012), 1–36.
[19]	N. Yang, Z. Dong, L. Wu, L. Zhang, X. Shen, D. Chen, et al., A comprehensive review of security-constrained unit commitment, J. Mod. Power Syst. Clean Energy, 10 (2022), 562–576. https://doi.org/10.35833/MPCE.2021.000255 doi: 10.35833/MPCE.2021.000255
[20]	M. Zhang, Q. Wu, J. Wen, Z. Lin, F. Fang, Q. Chen, Optimal operation of integrated electricity and heat system: a review of modeling and solution methods, Renew. Sustain. Energy Rev., 135 (2021), 110098. https://doi.org/10.1016/j.rser.2020.110098 doi: 10.1016/j.rser.2020.110098
[21]	J. Mulvaney-Kemp, S. Fattahi, J. Lavaei, Smoothing property of load variation promotes finding global solutions of time-varying optimal power flow, IEEE Trans. Control Netw Syst., 8 (2021), 1552–1564. https://doi.org/10.1109/TCNS.2021.3084039 doi: 10.1109/TCNS.2021.3084039
[22]	W. A. Bukhsh, A. Grothey, K. McKinnon, P. A. Trodden, Test case archive of optimal power flow (OPF) problems with local optima, 2013. Available from: https://www.maths.ed.ac.uk/optenergy/LocalOpt/.

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