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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(5),3060-3068
PublishedOnlineMay2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.115326
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OptimizationLogarithmicBarrierMethod
ChenxuanZheng
SchoolofStatisticsandMathematics,YunnanUniversityofFinanceandEconomics,Kunming
Yunnan
Received:Apr.27
th
,2022;accepted:May21
st
,2022;published:May31
st
,2022
Abstract
Themethodofsolvingthelogarithmicbarrierfunctionmethodisverypopularin
solvingthenon-equalityconstraintoptimizationproblem,itiswellknownthatthe
©ÙÚ^:x•Z.•`zéêæN{[J].A^êÆ?Ð,2022,11(5):3060-3068.
DOI:10.12677/aam.2022.115326
x•Z
logarithmicbarrierfunctionplaysanimportantroleinlinearplanningandlinear
semi-planning.Thispapermainlyintroducesthelogarithmicbarriermethodandits
algorithm,andtheeffectivenessofthismethodisillustratedbycalculatingexamples.
Keywords
LogarithmicBarrier,Validity
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.1153263065A^êÆ?Ð
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ë•©z
[1]æ‰,š©Õ.•`zn؆•{[M].®:‰Æч,1997.
[2]Polak,E.,Higgins,J.E.andMayne,D.Q.(1992)ABarrierFunctionMethodforMinimax
Problems.MathematicalProgramming,54,155-176.https://doi.org/10.1007/BF01586049
[3]Pourmohamad, T. and Lee, H.K.H. (2022) BayesianOptimization via Barrier Functions. Jour-
nalofComputationalandGraphicalStatistics,31,74-83.
https://doi.org/10.1080/10618600.2021.1935270
[4]Zhang,R.,Mei,J.,Dai,B.,Schuurmans,D.andLi,N.(2022)OntheEffectofLog-Barrier
Regularization inDecentralized Softmax Gradient Play inMultiagent Systems. arXivpreprint
arXiv:2202.00872
[5]•.•`zn؆Ž{[M].®:˜uŒÆчk•úi,2005.
[6]Frisch,K.R.(1955)TheLogarithmicPotentialMethodofConvexProgramming.Memoran-
dum,UniversityInstituteofEconomics,Oslo,5.
[7]Wright,M.H.(1992)InteriorMethodsforConstrainedOptimization.ActaNumerica,1,341-
407.https://doi.org/10.1017/S0962492900002300
[8]Fiacco, A.V.andMcCormick, G.P.(1990)NonlinearProgramming:Sequential Unconstrained
MinimizationTechniques.SocietyforIndustrialandAppliedMathematics,Philadelphia,PA.
https://doi.org/10.1137/1.9781611971316
[9]Fang,S.-C.andPuthenpura,S.(1993)LinearOptimizationandExtensions:Theoryand
Algorithms.Prentice-Hall,Inc.,Hoboken.
DOI:10.12677/aam.2022.1153263068A^êÆ?Ð

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