Lectures on convex sets - Niels Lauritzen

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1、LECTURESONCONVEXSETSNielsLauritzenThereisnotrivialmathematics,thereareonlytrivialmathematicians!Amathematicianistrivialifheorshebelievesthatthereexiststrivialmathematics.D.ZeilbergerNIELSLAURITZENDEPARTMENTOFMATHEMATICALSCIENCESUNIVERSITYOFAARHUSDENMARKEMAIL:niels@imf.au.

2、dkURL:http://home.imf.au.dk/niels/March2009March2010ContentsPrefacevNotationvii1Introduction11.1Linearinequalities.........................21.1.1Twovariables........................21.2Polyhedra..............................51.3Exercises...............................82Ba

3、sics112.1ConvexsubsetsofRn........................112.2Theconvexhull...........................132.3Extremalpoints...........................172.4Thecharacteristicconeforaconvexset.............182.5Convexcones............................182.6Affineindependence.............

4、...........212.7Carath´eodory’stheorem......................222.8Thepolarcone............................262.9Exercises...............................293Separation333.1Separationofapointfromaclosedconvexset.........353.2Supportinghyperplanes......................383.3Sepa

5、rationofdisjointconvexsets.................393.4Anapplication............................403.5Farkas’lemma............................413.6Exercises...............................444Polyhedra474.1Thedoubledescriptionmethod..................48iiiivContents4.1.1Thekeylemma.

6、......................494.2Extremalandadjacentrays....................524.3Farkas:fromgeneratorstohalfspaces..............554.4Polyhedra:generallinearinequalities..............574.5Thedecompositiontheoremforpolyhedra...........584.6Extremalpointsinpolyhedra...............

7、....594.7Exercises...............................63ALinear(in)dependence65A.1Lineardependenceandlinearequations.............65A.2Therankofamatrix........................67A.3Exercises...............................69BAnalysis71B.1Measuringdistances........................

8、71B.2Sequences..............................73B.2.1Supremumandinfimum..................76B.3Bound