lecture04_simplex method

# lecture04_simplex method - Yinyu Ye MS&E Stanford...

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

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