This theorem gave us a method to prove that a given 64258ow is optimal simply exhibit a cut with the same value This theorem for 64258ows and cuts in a graph is a speci64257c instance of the LP Duality Theorem which relates the optimal values of LP ID: 23918
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LECTURE5.LPDUALITY2this,lettingy1;y2;y3bethecoecientsofourlinearcombination.Thenwemusthave4y1+2y2+3y228y1+y2+2y33y1;y2;y30andweseekmin(12y1+3y2+4y3)ThistooisanLP!WerefertothisLPasthedualandtheoriginalLPastheprimal.Theactualchoiceofwhichproblemistheprimalandwhichisthedualisnotimportantsincethedualofthedualisequaltotheprimal.Wedesignedthedualtoserveasamethodofconstructinganupperboundontheoptimalvalueoftheprimal,soifyisafeasiblesolutionforthedualandxisafeasiblesolutionfortheprimal,then2x1+3x212y1+3y2+4y3.Ifwecanndtwofeasiblesolutionsthatmaketheseequal,thenweknowwehavefoundtheoptimalvaluesoftheseLP.Inthiscasethefeasiblesolutionsx1=1 2;x2=5 4andy1=5 16;y2=0;y3=1 4givethesamevalue4.75,whichthereforemustbetheoptimalvalue.5.1.1GeneralizationIngeneral,theprimalLPP=max(c�xjAxb;x0;x2Rn)correspondstothedualLP,D=min(b�yjA�yc;y0;y2Rm)whereAisanmnmatrix.Whenthereareequalityconstraintsorvariablesthatmaybenegative,theprimalLPP=max(c�x)s.t.aixbifori2I1aix=bifori2I2xj0forj2J1xj2Rforj2J2correspondstothedualLPD=min(b�y)s.t.yi0fori2I1yi2Rfori2I2Ajycjforj2J1Ajy=cjforj2J2 LECTURE5.LPDUALITY4 Figure5.1:TheobjectivevectorliesintheconespannedbytheconstraintvectorsSupposeforcontradictionthatcdoesnotlieinthiscone.ThentheremustexistaseparatinghyperplanebetweencandK:i.e.,thereexistsavectord2Rnsuchthata�id0foralli2I,butc�d�0.Nowconsiderthepointz=x+dforsometiny�0.Notethefollowing:Forsmallenough,thepointzsatisifestheconstraintsAzb.Considera�jzbforj62I:sincethisconstraintwasnottightforx,wewon'tviolateitifissmallenough.Andfora�jzbwithj2Iwehavea�jz=a�jx+a�jd=b+a�jdbsince�0anda�jd0.Theobjectivefunctionvalueincreasessincec�z=c�x+c�d�c�x.Thiscontradictsthefactthatxwasoptimal.Thereforethevectorclieswithintheconemadeofthenormalstotheconstraints,socisapositivelinearcombinationofthesenormals.Chooseifori2Isothatc=Pi2Iiai;0andsetj=0forj62I.Weknow0.A�=Pi2[m]iai=Pi2Iiai=c.b�=Pi2Ibii=Pi2I(aix)i=Pi2Iiaix=c�x.Thereforeisasolutiontothedualwithc�x=b�,sobyTheWeakDualityTheorem,OPT(P)=OPT(D). AsomewhatmorerigorousproofnotrelyingonourgeometricintuitionthatthereshouldbeaseparatinghyperplanebetweenaconeandavectornotspannedbytheconereliesonalemmabyFarkasthatoftencomesinseveralforms.TheformsweshalluseareasfollowsTheorem5.3(Farkas'Lemma(1894)-Form1).GivenA2Rmnandb2Rm,exactlyoneofthefollowingstatementsistrue.