lecture04_simplex method

lecture04_simplex method - Yinyu Ye, MS&E,...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 1 TheSimplexMethodI YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 2 Geometryoflinearprogramming Consider maximize x 1 +2 x 2 subjectto x 1 1 x 2 1 x 1 + x 2 1 . 5 x 1 , x 2 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 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 #04 4 solution (decision,point):anyspecificationofvaluesforalldecisionvariables, regardlessofwhetheritisadesirableorevenallowablechoice feasiblesolution :asolutionforwhichalltheconstraintsaresatisfied. feasibleregion (constraintset,feasibleset):thecollectionofallfeasible solution interior,boundary,face extremeorcornerorvertexpoint objectivefunctioncontour (iso-profit,iso-costline) optimalsolution :afeasiblesolutionthathasthemostfavorablevalueofthe objectivefunction optimalobjectivevalue :thevalueoftheobjectivefunctionevaluatedatan optimalsolution activeconstraint (bindingconstraint) Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 5 TheoryofLinearProgramming AllLPproblemsfallintooneofthreeclasses: Problemis infeasible :Feasibleregionisempty. Problemis unbounded :Feasibleregionisunboundedtowardstheoptimizing direction. Problemis feasibleandbounded .Inthiscase: thereexistsan optimalsolutionoroptimizer . Theremaybea unique optimizeror multiple optimizers. Alloptimizersareona face ofthefeasibleregion. Thereisalwaysatleastone corner(extreme) optimizerifthefacehasa corner. Ifacornerpointisnot worse thanallits adjacentorneighboring corners, thenitisoptimal. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 6 History GeorgeB.Dantzig s SimplexMethod forlinearprogrammingstandsasoneofthe mostsignificantalgorithmicachievementsofthe20thcentury.Itisnowover50 yearsoldandstillgoingstrong. Thebasicideaofthesimplexmethodtoconfinethesearchto cornerpoints ofthe feasibleregion(ofwhichthereareonly finitely many)inamostintelligentway.In contrast, interior-pointmethods willmoveintheinteriorofthefeasibleregion, hopingtoby-passmany cornerpoints ontheboundaryoftheregion. Thekeyforthesimplexmethodistomakecomputers see cornerpoints;andthe keyforinterior-pointmethodsisto stay intheinteriorofthefeasibleregion. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #04 7 Fromgeometrytoalgebra Howtomakecomputerrecognizea cornerpoint ? Howtomakecomputertellthattwocornersare neighboring ? Howtomakecomputer terminate anddeclareoptimality? Howtomakecomputeridentifyabetter neighboringcorner ?...
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This note was uploaded on 06/16/2010 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.

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lecture04_simplex method - Yinyu Ye, MS&E,...

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