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3HowtousetheFarkas'certicateNotonlycantheFarkas'certicatebeusedtocertifythataproblemisinfeasible,itcanalsobeusedtopinpointthecauseinfeasibility.Usually,ifaproblemisinfeasiblewewouldliketorepairit,oratleastknowwhichpartoftheproblemiscausingtheinfeasibility.Taketheexample(1):Wemaythinkthatchangingtheproblemtominimizex1subjecttox11;x11:5;(10)willremovetheinfeasibility(wehavechangedtheright-handsideforthesecondconstraintfrom2to1.5).However,therevisedFarkas'conditionsarey1+1:5y1�0;y10;y20(11)anditisseenthattheoldcerticatey1=�1andy2=1isstillavalidcerticate.Notethatifwehadchangedtheright-handsideofthesecondconstraintto1thenFarkas'certicatenolongerwouldbevalid.Therefore,ingeneralwhenrepairinganinfeasibleproblemitshouldbechangedatleastsomuchthatthecerticateofinfeasibilityisnolongervalidbecauseotherwisetheproblemstaysinfeasible.Anotherimportantobservationisthatifyi=0thentheithconstraintisnotinvolvedinthepinpointedinfeasibilitysinceiftheithconstraintisremovedfromtheproblemandyiisremovedfromthevectory,thenthecompressedyisstillaninfeasibilitycerticate.Indeed,assumeyisaninfeasibilitycerticateto(2)anddeneI:=fi:yi6=0g:(12)Next,considertherelaxedproblemAI;:x=bIx0whereallconstraintsforwhichyiiszerohavebeenremoved.ItiseasytoverifythatbTIyI�0ATI;:yI0;(13)andhencethatyIisaFarkas'certicatefortherelaxedproblem.Ingeneral,itcanbehopedthatthesetIcontainsonlyfewelements,whichwouldimplythatwehavelocatedasmallsetofconstraintscausingtheinfeasibility,andhencemakingarepaireasy.Wehaveimplicitlymadetheassumptionthattheconstraintx0isnotcausingtheinfeasibility.However,itcanbeveriedthatJ:=fj:ATy0gthenAI;:x=bIxJ0isstillinfeasible.Hence,potentiallymanyofthesimpleinequalityconstraintsisirrelevantfortheinfeasibility.4AninfeasibilityreportOnewaytoexploittheinfeasibilitycerticatesistogeneratetogenerateaninfeasibilityreportlookingsomethinglikethis:Equality5isimportantfortheinfeasibility.y[5]is23.67.Equality15isimportantfortheinfeasibility.y[15]is1.0.basedonthesetsIandJ.HerewehaveassumedthatI=f5;15g.Now,thedenitionofIin(12)workswellintheory,butinpracticesomethinglikeI:=(i:jyij&#x]TJ/;ø 9;&#x.962; Tf;&#x 18.;ԥ ;� Td;&#x [00;" biATi;: 1�1):(14)isbetter,where"isasmallconstant,typically"2[1:0e�8;1:0e�5].ThedenitionofthesetIin(14)isindependentofthescalingoftheconstraintswhichiaanattractiveproperty. www.mosek.comPage3of4 MosekApSFruebjergvej32100CopenhagenDenmarkwww.mosek.cominfo@mosek.com \thefastpathtooptimum" MOSEKApSprovidesoptimizationsoftwarewhichhelpourclientstomakebetterdecisions.Ourcustomerbaseconsistsofnancialin-stitutionsandcompanies,engineeringandsoftwarevendors,amongothers.Thecompanywasestablishedin1997byErlingD.AndersenandKnudD.Andersenanditspecializesincreatingadvancedsoftwareforsolutionofmathematicaloptimizationproblems.Inparticular,thecompanyfocusesonsolutionoflarge-scalelinear,quadratic,andconicoptimizationproblems.