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带非凸二次约束的二次比式和问题的全局优化算法(英文).docx

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带非凸二次约束的二次比式和问题的全局优化算法(英文) Introduction Quadraticequationsandproblemsareafundamentalpartofmathematicaloptimization.Theyariseinvariousfields,suchasengineering,finance,andeconomics,andfindingtheirglobalminimumormaximumisachallengingtask.Oneofthechallengesthatariseinquadraticprogrammingisthepresenceofnonconvexquadraticconstraints.Non-convexconstraintscomplicatetheoptimizationproblem,andfindingtheirglobaloptimumrequiresspecializedtechniques. Inthispaper,wewilldiscusstheclassofquadraticratioproblemswithnonconvexquadraticconstraints.Wewillreviewdifferentapproachestosolvetheproblem,evaluatethestrengthsandweaknessesofeachalgorithm,andprovidesuggestionsforfuturework. QuadraticRatioProblemswithNonconvexQuadraticConstraints Quadraticratioproblemswithnonconvexquadraticconstraintsarequadraticoptimizationproblemsthatinvolvearatiooftwoquadraticfunctionssubjecttoconstraintsthatarenon-convex.Atypicalquadraticratioproblemwithnon-convexquadraticconstraintsisformulatedas: Maximize(x^TQ1x)/(x^TQ2x) Subjectto: Cx<=d Gx=h wherexistheoptimizationvector,Q1andQ2arepositivedefinitequadraticmatrices,C,G,andd,harethenon-convexconstraints. Problemswithnon-convexquadraticconstraintsarechallengingtosolvebecausetheyaregenerallynotamenabletoclosed-formsolutions.Furthermore,thenon-convexityoftheconstraintscausestheoptimizationproblemtobenon-convex,whichmeansthattheoptimizationalgorithmscannotguaranteefindingtheglobaloptimum.Thepresenceofthequadraticratiofurtheraddstothecomplexityoftheproblem.Traditionally,optimizationproblemswithratioconstraintsareformulatedasaminimizationproblem,whichinvolvestheintroductionofaslackvariable.Usingthisapproach,theoptimizationproblemwithnon-convexquadraticconstraintscanbeformulatedasanon-convexquadraticallyconstrainedquadraticprogrammingproblem. GlobalOptimizationAlgorithmforQuadraticRatioProblemswithNonconvexQuadraticConstraints Differentmethodshavebeenproposedtosolvethequadraticratioproblemswithnonconvexquadraticconstraints.Thesemethodscanbeclassifiedintotwobroadcateg