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# lecture06 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 1 LinearProgrammingDualityanditsApplications YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 2 LPexampleagain maximize x 1 +2 x 2 subjectto x 1 ≤ 1 x 2 ≤ 1 x 1 + x 2 ≤ 1 . 5 x 1 , x 2 ≥ . minimize- x 1- 2 x 2 subjectto x 1 + x 3 =1 x 2 + x 4 =1 x 1 + x 2 + x 5 =1 . 5 x 1 , x 2 , x 3 , x 4 , x 5 ≥ . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 3 a1 a2 a3 a2 a3 a4 a4 a5 a norm direction cone contained by the norm LP Geometry depicted in two variable space If the direction of c is Objective contour Each corner point has the point is optimal. cone of a corner point, then c Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 4 WhenaBSFisoptimal? GivenaBFSintheLPstandardform A B x B = b , x B ≥ , x N = , anditscompanionshadowpriceandreducedcostvectors: A T B y = c B , ( and r = c- A T y ) . Ifthereducedcostvector r ≥ ,thentheBFSwithbasicvariableset B ,thenthe BFS optimal .(–Whataboutthe maximization ?) Atoptimalitywealwayshave c T x = b T y . Whatdotheshadowpricesandreducedcostsmean? Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 5 IntheLPExample,let B = { 1 , 2 , 3 } sothat A B = 101 010 110 , x B =(0 . 5 , 1 , . 5) T and y T =(0 ,- 1 ,- 1) and r = c- A T y =(0 , , , 1 , 1) T Notethat c T x = b T y =- 2 . 5 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 6 Whatdotheshadowprices y mean? • All inactive constrainthave zeroshadowprice . • Ingeneral,theshadowpriceonagivenactiveconstraintisthe rateofchange intheOVasthe RHS oftheconstraint increases ,ceterisparibus. • IftheOSis degenerate ,theshadowpricemaybevalidfor one-sided changes intheRHS. • TheconstraintRHS ranges givetherangesoftheconstraintRHSoverwhich nochangeinthe optimalbasis willoccur. • Oneofthe allowableincreaseanddecrease foran inactive constraintis infiniteandtheotherequalstotheslackorsurplus. • Ingeneral,whentheRHSofan active constraintchanges,boththeOVand OSwillchange. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 7 Whatdothereducedcosts r mean? • All basic variableshave zeroreducedcost . • Ingeneral,thereducedcostcoefficientofany non-basic variableisthe amounttheobjectivecoefficientofthatvariablewouldhavetochange,withall otherdataheldfixed,inorderforittobecomea basic variableatoptimality. • IftheOSis degenerate ,theobjectivecoefficientofa non-basic variablewould havetochangebyatleast,andpossiblymorethan,thereducedcostinorder tobecomea basic variableintheOS. • Theobjectivecoefficient ranges givetherangesoftheobjectivefunctionover whichnochangeinthe optimalbasis willoccur. • Oneofthe allowableincreaseanddecrease fora non-basic variableisinfinite andtheotheristhereducedcost. • Ifa non-basic variablehas zero reducedcost,thentheremayexistan alternative optimalsolution....
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lecture06 - Yinyu Ye MS&E Stanford MS&E211 Lecture...

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