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求线性比式和问题全局解的新方法.docx

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求线性比式和问题全局解的新方法 Linearprogrammingisamathematicalmethodforsolvingoptimizationproblemswheretheobjectivefunctionandconstraintsarelinear.Thetraditionalapproachtolinearprogramminginvolvestheuseofthesimplexmethod,butthisalgorithmcanbetime-consumingandinefficientforproblemswithlargenumbersofvariablesandconstraints.Inrecentyears,researchershaveproposednewmethodsforsolvinglinearprogrammingproblemsthatarefasterandmoreeffective.Inthispaper,weexploreoneofthesemethods:theinteriorpointmethod. Theinteriorpointmethodisatechniqueforsolvinglinearprogrammingproblemsthatinvolvesfindingapointwithinthefeasibleregionoftheproblemandthenmovingtowardstheoptimalsolutionalongapaththatliesentirelywithinthefeasibleregion.Themethodisbasedontheobservationthattheoptimalsolutiontoalinearprogrammingproblemmustlieattheintersectionofoneoftheconstraintplanesandalevelsetoftheobjectivefunction.Bymovingalongapaththatliesentirelywithinthefeasibleregion,theinteriorpointmethodcanfindthisintersectionpointmoreefficientlythanthesimplexmethod,whichmovesalongtheedgesofthefeasibleregion. Tounderstandhowtheinteriorpointmethodworks,wefirstneedtodefinesometerms.Inalinearprogrammingproblem,theobjectivefunctionisalinearequationthatrepresentsthequantitytobeoptimized.Theconstraintsarelinearinequalitiesorequalitiesthatrepresentrestrictionsonthevariablesintheproblem.Thefeasibleregionisthesetofvaluesofthevariablesthatsatisfyalloftheconstraints.Theoptimalsolutionisthesetofvariablevaluesthatmaximizeorminimizetheobjectivefunctionwhilesatisfyingalloftheconstraints. Toapplytheinteriorpointmethod,webeginbytransformingthelinearprogrammingproblemintoanequivalentformbyintroducinganewvariable,calledaslackvariable,foreachconstraint.Theslackvariableisusedtoconverttheinequalityconstraintintoanequalityconstraint.Thistransformationallowsustowritetheproblemintermsofastandardform: minimizec^Tx subjecttoAx=b x>=0 wherecisavectorofcoefficientsrepresentingtheobjectivefunction,xisavectorofvariables,Aisamatrixrepresentingtheconstraints,andbisavec