lecture02

lecture02 - Yinyu Ye, MS&E, Stanford...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 1 LinearProgramming YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 2 SparsestDataFittingI Wewanttofindasparsestsolutiontofitexactdatameasurements,thatis,to minimizethenumberofnon-zeroentriesin x suchthat A x = b : minimize | support ( x ) | subjectto A x = b ; wherethesetfunction support ( x )= { j : x j 6 =0 } . Sometimesthisobjectivecanbeaccomplishedby minimize k x k 1 = n j =1 | x j | subjectto A x = b . Thisisa linearprogram ? Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 3 SparsestDataFittingII Yes,sinceitcanbeequivalentlyrepresentedby minimize n j =1 x j subjectto A x = b ,- x x x ; or minimize n j =1 ( x j + x 00 j ) subjectto A ( x- x 00 )= b . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #02 4 LPExample5:CombinatorialAuctionI Given m potential states thataremutuallyexclusiveandexactlyoneofthemwill berealizedatthematurity. An order isabetononeora combination ofstates,witha pricelimit (the maximumpricetheparticipantiswillingtopayforoneunitoftheorder)anda quantitylimit (themaximumnumberofunitsorsharestheparticipantiswillingto...
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lecture02 - Yinyu Ye, MS&E, Stanford...

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