{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T18:20:11Z","timestamp":1775326811683,"version":"3.50.1"},"reference-count":80,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T00:00:00Z","timestamp":1699401600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Phasor Measurement Units (PMUs) are the backbone of smart grids that are able to measure power system observability in real-time. The deployment of synchronized sensors in power networks opens up the advantage of real-time monitoring of the network state. An optimal number of PMUs must be installed to ensure system observability. For that reason, an objective function is minimized, reflecting the cost of PMU installation around the power grid. As a result, a minimization model is declared where the objective function is defined over an adequate number of constraints on a binary decision variable domain. To achieve maximum network observability, there is a need to find the best number of PMUs and put them in appropriate locations around the power grid. Hence, maximization models are declared in a decision-making way to obtain optimality satisfying a guaranteed stopping and optimality criteria. The best performance metrics are achieved using binary integer, semi-definite, and binary polynomial models to encounter the optimal number of PMUs with suitable PMU positioning sites. All optimization models are implemented with powerful optimization solvers in MATLAB to obtain the global solution point.<\/jats:p>","DOI":"10.3390\/axioms12111040","type":"journal-article","created":{"date-parts":[[2023,11,8]],"date-time":"2023-11-08T07:02:05Z","timestamp":1699426925000},"page":"1040","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["The Branch-and-Bound Algorithm in Optimizing Mathematical Programming Models to Achieve Power Grid Observability"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0666-9544","authenticated-orcid":false,"given":"Nikolaos P.","family":"Theodorakatos","sequence":"first","affiliation":[{"name":"School of Electrical & Computer Engineering, National Technical University of Athens, 157 80 Athens, Greece"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4128-4428","authenticated-orcid":false,"given":"Rohit","family":"Babu","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronics Engineering, Alliance University, Anekal, Bengaluru 562 106, India"}]},{"given":"Angelos P.","family":"Moschoudis","sequence":"additional","affiliation":[{"name":"Department of Electrical and Electronics Engineering, University of West Attica, 122 44 Athens, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Phadke, A.G., and Thorp, J.S. (2017). Synchronized Phasor Measurements and Their Applications, Springer. [2nd ed.].","DOI":"10.1007\/978-3-319-50584-8"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Mohamed, E. (2020). El-Hawary Advances in Electric Power and Energy: Static State Estimation, John Wiley & Sons.","DOI":"10.1002\/9781119480402"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Cyman, J., and Raczek, J. (2022). Application of Doubly Connected Dominating Sets to Safe Rectangular Smart Grids. Energies, 15.","DOI":"10.3390\/en15092969"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1007\/s10878-019-00380-7","article-title":"Connected power domination in graphs","volume":"38","author":"Brimkov","year":"2019","journal-title":"J. Comb. Optim."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"519","DOI":"10.1137\/S0895480100375831","article-title":"Domination in graphs applied to electric power networks","volume":"15","author":"Haynes","year":"2002","journal-title":"SIAM J. Discret. Math."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Diestel, R. (2017). Graph Theory, Springer.","DOI":"10.1007\/978-3-662-53622-3"},{"key":"ref_7","unstructured":"Narsingh, D. (1974). Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall Inc."},{"key":"ref_8","unstructured":"Christofides, N. (1975). Graph Theory. An Algorithmic Approach\u2019, Academic Press Inc."},{"key":"ref_9","unstructured":"Arora, J.S. (2016). Introduction to Optimum Design, Elsevier Inc.. [4th ed.]."},{"key":"ref_10","unstructured":"Williams, H.P. (2013). Model Building in Mathematical Programming, John Wiley & Sons."},{"key":"ref_11","unstructured":"Karlof, J. (2006). Integer Programming and Practice, CRC Press Taylor & Francis Group, 6000 Broken Sound Parkway NW, Suite 300. [1st ed.]."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ahmed, M.M., Amjad, M., Qureshi, M.A., Imran, K., Haider, Z.M., and Khan, M.O. (2022). A Critical Review of State-of-the-Art Optimal PMU Placement Techniques. Energies, 15.","DOI":"10.3390\/en15062125"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Biswal, C., Sahu, B.K., Mishra, M., and Rout, P.K. (2023). Real-Time Grid Monitoring and Protection: A Comprehensive Survey on the Advantages of Phasor Measurement Units. Energies, 16.","DOI":"10.3390\/en16104054"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Paramo, G., Bretas, A., and Meyn, S. (2022). Research Trends and Applications of PMUs. Energies, 15.","DOI":"10.3390\/en15155329"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/15325008.2023.2240320","article-title":"A Survey on the Application of Phasor Measurement Units to the Protection of Transmission and Smart Distribution Systems","volume":"1","author":"Menezes","year":"2023","journal-title":"Electr. Power Compon. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"107428","DOI":"10.1016\/j.epsr.2021.107428","article-title":"Synchrophasor measurement applications and optimal PMU placement: A review","volume":"199","author":"Joshi","year":"2021","journal-title":"Electr. Power Syst. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1080\/15325008.2015.1117538","article-title":"A Brief Review of Phasor Measurement Units as Sensors for Smart Grid","volume":"44","author":"Mohanta","year":"2016","journal-title":"Electr. Power Compon. Syst."},{"key":"ref_18","first-page":"e12698","article-title":"A critical review of methods for optimal placement of phasor measurement units","volume":"31","author":"Johnson","year":"2020","journal-title":"Int. Trans. Electr. Energy Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"104","DOI":"10.1002\/net.21684","article-title":"The power edge set problem","volume":"68","author":"Poirion","year":"2016","journal-title":"Networks"},{"key":"ref_20","unstructured":"Bei, X., Yoon, Y.J., and Abur, A. (2005). Optimal Placement and Utilization of Phasor Measurements for State Estimation, PSERC Publication."},{"key":"ref_21","unstructured":"Xu, B., and Abur, A. (2004, January 10\u201313). Observability Analysis and Measurement Placement for Systems with PMUs. Proceedings of the IEEE PES Power Systems Conference and Exposition, New York, NY, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1812","DOI":"10.1109\/TPWRD.2008.919046","article-title":"Optimal Multistage Scheduling of PMU Placement: An ILP Approach","volume":"23","author":"Dua","year":"2008","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1099","DOI":"10.1109\/TPWRS.2008.926475","article-title":"Generalized integer linear programming formulation for optimal PMU placement","volume":"23","author":"Gou","year":"2008","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1109\/TPWRD.2008.2008430","article-title":"Placement of synchronized measurements for power system observability","volume":"24","author":"Chakrabarti","year":"2009","journal-title":"IEEE Trans. Power Deliv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105427","DOI":"10.1016\/j.ijepes.2019.105427","article-title":"Groebner bases algorithm for optimal PMU placement","volume":"115","author":"Stefanov","year":"2020","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2427","DOI":"10.1049\/iet-gtd.2015.0662","article-title":"Optimal phasor measurement unit placement for numerical observability in the presence of conventional measurement using semi-definite programming","volume":"9","author":"Korres","year":"2015","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Almunif, A., and Fan, L. (2018, January 9\u201311). DC State Estimation Model-Based Mixed Integer Semidefinite Programming for Optimal PMU Placement. Proceedings of the 2018 North American Power Symposium (NAPS), Fargo, ND, USA.","DOI":"10.1109\/NAPS.2018.8600578"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3403","DOI":"10.1109\/TIM.2019.2932208","article-title":"Optimal Allocation of Phasor Measurement Units Considering Various Contingencies and Measurement Redundancy","volume":"69","author":"Manousakis","year":"2020","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Hyacinth, L.R., and Gomathi, V. (2021). Optimal pmu placement technique to maximize measurement redundancy based on closed neighbourhood search. Energies, 14.","DOI":"10.3390\/en14164782"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.epsr.2014.08.023","article-title":"A graph theory based methodology for optimal PMUs placement and multiarea power system state estimation","volume":"119","author":"Xie","year":"2015","journal-title":"Electr. Power Syst. Res."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1049\/iet-gtd.2015.1005","article-title":"Genetic algorithm-based phasor measurement unit placement method considering observability and security criteria","volume":"10","author":"Castro","year":"2016","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1080\/15325008.2019.1605635","article-title":"Optimal Phasor Measurement Unit Placement for Numerical Observability Using Branch-and-Bound and a Binary-Coded Genetic Algorithm","volume":"47","author":"Theodorakatos","year":"2019","journal-title":"Electr. Power Compon. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2817","DOI":"10.1049\/iet-gtd.2016.0287","article-title":"Optimal PMU placement for full observability of the power network with maximum redundancy using modified binary cuckoo optimization algorithm","volume":"10","author":"Dalali","year":"2016","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1832","DOI":"10.1080\/15325008.2017.1378775","article-title":"A Multi-objective PMU Placement Method in Power System via Binary Gravitational Search Algorithm","volume":"45","author":"Singh","year":"2017","journal-title":"Electr. Power Compon. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"625","DOI":"10.1016\/j.jare.2016.06.003","article-title":"Optimal PMU placement using topology transformation method in power systems","volume":"7","author":"Rahman","year":"2016","journal-title":"J. Adv. Res."},{"key":"ref_36","unstructured":"Chakrabarti, S., Venayagamoorthy, G.K., and Kyriakides, E. (2008, January 14\u201317). PMU placement for power system observability using binary particle swarm optimization. Proceedings of the 2008 Australasian Universities Power Engineering Conference, Sydney, NSW, Australia."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"2550","DOI":"10.1109\/TIA.2017.2666091","article-title":"Multiple Solutions of Optimal PMU Placement Using Exponential Binary PSO Algorithm for Smart Grid Applications","volume":"53","author":"Maji","year":"2017","journal-title":"IEEE Trans. Ind. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"109472","DOI":"10.1016\/j.asoc.2022.109472","article-title":"Security-constrained optimal placement of PMUs using Crow Search Algorithm","volume":"128","author":"Johnson","year":"2022","journal-title":"Appl. Soft Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/15325008.2021.1977428","article-title":"Realistic method for placement of phasor measurement units through optimization problem formulation with conflicting objectives","volume":"49","author":"Ramasamy","year":"2021","journal-title":"Electr. Power Compon. Syst."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Logeshwari, V., Abirami, M., Subramanian, S., and Manoharan, H. (2023). Multi-objective precise phasor measurement locations to assess small-signal stability using dingo optimizer. Optim. Control. Appl. Methods.","DOI":"10.1002\/oca.3053"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.segan.2015.03.002","article-title":"Redundancy based PMU placement in state estimation","volume":"2","author":"Xia","year":"2015","journal-title":"Sustain. Energy Grids Netw."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Yang, X.S. (2010). Engineering Optimization: An Introduction with Metaheuristic Applications, Wiley.","DOI":"10.1002\/9780470640425"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"3499","DOI":"10.1109\/TPWRS.2013.2242698","article-title":"A Weighted Least Squares Algorithm for optimal PMU placement","volume":"28","author":"Manousakis","year":"2013","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_44","unstructured":"Chinneck, J.W. (2008). Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods, Springer."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Theodorakatos, N.P., Lytras, M., and Babu, R. (2020). Towards Smart Energy Grids: A Box-Constrained Nonlinear Underdetermined Model for Power System Observability Using Recursive Quadratic Programming. Energies, 13.","DOI":"10.3390\/en13071724"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"012125","DOI":"10.1088\/1742-6596\/2090\/1\/012125","article-title":"A Generalized Pattern Search Algorithm Methodology for solving an Under-Determined System of Equality Constraints to achieve Power System Observability using Synchrophasors","volume":"2090","author":"Theodorakatos","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"e12977","DOI":"10.1002\/2050-7038.12977","article-title":"An approach for economic design of wide area monitoring system by co-optimizing phasor measurement unit placement and associated communication infrastructure","volume":"31","author":"Patel","year":"2021","journal-title":"Int. Trans. Electr. Energy Syst."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Baba, M., Nor, N.B., Aman Sheikh, M., Irfan, M., and Tahir, M. (2020). A Strategic and Significant Method for the Optimal Placement of Phasor Measurement Unit for Power System Network. Symmetry, 12.","DOI":"10.3390\/sym12071174"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Baba, M., Nor, N.B.M., Sheikh, M.A., Baba, A.M., Irfan, M., Glowacz, A., Kozik, J., and Kumar, A. (2021). Optimization of Phasor Measurement Unit Placement Using Several Proposed Case Factors for Power Network Monitoring. Energies, 14.","DOI":"10.3390\/en14185596"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Bonavolont\u00e0, F., Caragallo, V., Fatica, A., Liccardo, A., Masone, A., and Sterle, C. (2021). Optimization of IEDs Position in MV Smart Grids through Integer Linear Programming. Energies, 14.","DOI":"10.3390\/en14113346"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Almunif, A., and Fan, L. (2017, January 17\u201319). Mixed integer linear programming and nonlinear programming for optimal PMU placement. Proceedings of the 2017 North American Power Symposium (NAPS), Morgantown, WV, USA.","DOI":"10.1109\/NAPS.2017.8107398"},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Fortuna, L., and Buscarino, A. (2022). Nonlinear Technologies in Advanced Power Systems: Analysis and Control. Energies, 15.","DOI":"10.3390\/en15145167"},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Livanos, N.-A.I., Hammal, S., Giamarelos, N., Alifragkis, V., Psomopoulos, C.S., and Zois, E.N. (2023). OpenEdgePMU: An Open PMU Architecture with Edge Processing for Future Resilient Smart Grids. Energies, 16.","DOI":"10.3390\/en16062756"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"529724","DOI":"10.1155\/2015\/529724","article-title":"Optimized Extreme Learning Machine for Power System Transient Stability Prediction Using Synchrophasors","volume":"2015","author":"Zhang","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Arefin, A.A., Baba, M., Singh, N.S.S., Nor, N.B.M., Sheikh, M.A., Kannan, R., Abro, G.E.M., and Mathur, N. (2022). Review of the Techniques of the Data Analytics and Islanding Detection of Distribution Systems Using Phasor Measurement Unit Data. Electronics, 11.","DOI":"10.3390\/electronics11182967"},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Salda\u00f1a-Gonz\u00e1lez, A.E., Sumper, A., Arag\u00fc\u00e9s-Pe\u00f1alba, M., and Smolnikar, M. (2020). Advanced Distribution Measurement Technologies and Data Applications for Smart Grids: A Review. Energies, 13.","DOI":"10.3390\/en13143730"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Dusabimana, E., and Yoon, S.-G. (2020). A Survey on the Micro-Phasor Measurement Unit in Distribution Networks. Electronics, 9.","DOI":"10.3390\/electronics9020305"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Shahriar, M.S., Habiballah, I.O., and Hussein, H. (2018). Optimization of Phasor Measurement Unit (PMU) Placement in Supervisory Control and Data Acquisition (SCADA)-Based Power System for Better State-Estimation Performance. Energies, 11.","DOI":"10.3390\/en11030570"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"e2580","DOI":"10.1002\/etep.2580","article-title":"A hybrid power system state estimator using synchronized and unsynchronized sensors","volume":"28","author":"Manousakis","year":"2018","journal-title":"Int. Trans. Electr. Energy Syst."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"3205","DOI":"10.1109\/TPWRS.2016.2628344","article-title":"A Robust Iterated Extended Kalman Filter for Power System Dynamic State Estimation","volume":"32","author":"Zhao","year":"2017","journal-title":"IEEE Trans. Power Syst."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1109\/JSTSP.2018.2827261","article-title":"A Robust Generalized-Maximum Likelihood Unscented Kalman Filter for Power System Dynamic State Estimation","volume":"12","author":"Zhao","year":"2018","journal-title":"IEEE J. Sel. Top. Signal Process."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Aljabrine, A.A., Smadi, A.A., Chakhchoukh, Y., Johnson, B.K., and Lei, H. (2021). Resiliency Improvement of an AC\/DC Power Grid with Embedded LCC-HVDC Using Robust Power System State Estimation. Energies, 14.","DOI":"10.3390\/en14237847"},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Smadi, A.A., Johnson, B.K., Lei, H., and Aljabrine, A.A. (2023, January 16\u201320). Improving Hybrid Ac\/dc Power System Resilience Using Enhanced Hybrid Power State Estimator. Proceedings of the 2023 IEEE Power & Energy Society General Meeting (PESGM), Orlando, FL, USA.","DOI":"10.1109\/PESGM52003.2023.10252644"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/s10898-020-00984-y","article-title":"A general branch-and-bound framework for continuous global multiobjective optimization","volume":"80","author":"Eichfelder","year":"2021","journal-title":"J. Glob. Optim."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.disopt.2016.01.005","article-title":"Branch-and-bound algorithms: A survey of recent advances in searching, branching, and pruning","volume":"19","author":"Morrison","year":"2016","journal-title":"Discret. Optim."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"De Ita, G., Bello, P., and Tovar, M. (2023). A Branch and Bound Algorithm for Counting Independent Sets on Grid Graphs. Comput. Sci. Math. Forum, 7.","DOI":"10.3390\/IOCMA2023-14434"},{"key":"ref_67","doi-asserted-by":"crossref","unstructured":"Su, C.-H., and Wang, J.-Y. (2022). A Branch-and-Bound Algorithm for Minimizing the Total Tardiness of Multiple Developers. Mathematics, 10.","DOI":"10.3390\/math10071200"},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1002\/net.21751","article-title":"Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis","volume":"70","author":"Lewis","year":"2017","journal-title":"Networks"},{"key":"ref_69","unstructured":"L\u00f6fberg, J. (2004, January 2\u20134). YALMIP: A toolbox for modeling optimization in MATLAB. Proceedings of the 2004 IEEE International Conference on Robotics and Automation, New Orleans, LA, USA."},{"key":"ref_70","unstructured":"The MathWorks Inc (2023, September 15). Documentation: Intlinprog. Available online: https:\/\/es.mathworks.com\/help\/optim\/ug\/intlinprog.html."},{"key":"ref_71","unstructured":"MathWorks (2023, September 15). Fmincon. Available online: https:\/\/www.mathworks.com\/help\/optim\/ug\/fmincon.html."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12532-008-0001-1","article-title":"SCIP: Solving constraint integer programs","volume":"1","author":"Achterberg","year":"2009","journal-title":"Math. Progr. Comput."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3585516","article-title":"Enabling research through the SCIP optimization suite 8.0","volume":"49","author":"Bestuzheva","year":"2023","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_74","unstructured":"(2023, July 15). Gurobi. Available online: http:\/\/www.gurobi.com."},{"key":"ref_75","doi-asserted-by":"crossref","unstructured":"Shalileh, S. (2023). An Effective Partitional Crisp Clustering Method Using Gradient Descent Approach. Mathematics, 11.","DOI":"10.3390\/math11122617"},{"key":"ref_76","unstructured":"(2023, August 08). Available online: https:\/\/yalmip.github.io\/_posts\/tutorials\/2016-09-17-globaloptimization\/."},{"key":"ref_77","unstructured":"(2023, September 07). Available online: https:\/\/YALMIP.github.io\/solver\/bmibnb\/."},{"key":"ref_78","unstructured":"(2023, September 15). Available online: https:\/\/yalmip.github.io\/solver\/cutsdp\/."},{"key":"ref_79","unstructured":"(2023, October 10). Parametric Fusion (MOSEK 10.1). Available online: https:\/\/www.mosek.com."},{"key":"ref_80","unstructured":"(2023, September 30). Available online: https:\/\/icseg.iti.illinois.edu\/power-cases\/."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1040\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:19:21Z","timestamp":1760131161000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/11\/1040"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,8]]},"references-count":80,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["axioms12111040"],"URL":"https:\/\/doi.org\/10.3390\/axioms12111040","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,11,8]]}}}