boot if ifaf - university of illinoispublish.illinois.edu/ymb/files/2019/01/weekjan21p1.pdfna 6...

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Basiere.su tsouoptuiuizhrs f u IR UCR Take setEU B Csi ball of radius r around na We say that at is a 6 if for some r o first feel see Berta c re local aphunzer if f Cnt e fca for all if sane true for U That of Gateaux did off at at is O s r if it's a critical ph there are pts ad a f al feat fca in any Brest Weierstrgf Theorem if f is continuous on a coup act set k Then f possesses a global duininger Car maximizer Boot If a cunninger doesnt exist we can find Seg of points haeen3 f Can Ifaf But by compactness of k there is a subsequence slate f Cae riff e see sick Then f Cat f Ca for all a Ek D Maru Definition Reminder Coupact setsa Open cover admits finite sub cover Any sequence lies convergent subsequence bounded closed in 42 i coup A function on metric space D is called coercive if f as as 11 es 11 dist se O origin in R some base point sugeireed The If f is contineus coercive on closed mines of Rh Then f has a global minimum Drood LH E c is either empty or bounded closed compact has a Unhium D

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Page 1: Boot If Ifaf - University Of Illinoispublish.illinois.edu/ymb/files/2019/01/WeekJan21p1.pdfna 6 there f slate sick convergent metric Basiere.sutsouoptuiuizhrs f u IR UCR Take setEU

Basiere.sutsouoptuiuizhrs

f u IR UCR Take setEU B Csi ballof radius raround na We say that

at is a 6 if for some r o first feelsee Berta c re local aphunzer if fCnt e fca for all

if sane true for UThat of Gateaux did off at at is O

s r if it's a critical ph there are pts ad a

f al feat fca in any Brest

Weierstrgf Theorem if f is continuous on a coupactsetkThen f possesses a global duiningerCar maximizerBoot If a cunninger doesnt exist we can find Seg

of points haeen3 fCan Ifaf But by compactnessofk there is a subsequence slate f Cae riff e

see sick Then fCat f Ca for all a Ek DMaruDefinitionReminder Coupactsetsa Open cover admits finite subcoverAny sequence liesconvergentsubsequencebounded closed in 42 i coup

A function on metricspace D is called coercive iff as as 1 1 es 1 1 dist seO

origin in R somebasepointsugeireedThe If f is contineus coercive on closed mines of RhThen f has a global minimum

Drood LH E c is either empty or bounded closed

compact has a Unhium D

Page 2: Boot If Ifaf - University Of Illinoispublish.illinois.edu/ymb/files/2019/01/WeekJan21p1.pdfna 6 there f slate sick convergent metric Basiere.sutsouoptuiuizhrs f u IR UCR Take setEU

comp D

AppG hy polynomial PCH Etat t tan has aroot in CI

Baat If l is aerave so it allows a global inkat Z assume it o But if p Aot a.cz't a it t

Flt coefficient 0then one can find ve asVk Tao Aso real

For small u a.kz is made bigger than rest of termslaota.cz't I Kol D

TnnstwdLnmsAeotherofakoeptueohtycouohhuusiThm_Iffa.r

R is Gateaux differentiable on operaand set is a unzer then DfCoE o

Dod Taylor formula DSaeeexaupSternerpoint findpoint on the lanepuinini u sayingdisagreesto vortices of a triangle fCo7i lAolt1BoltlC0l2unoh0E.B

Solution If ItTztT i'zoo rule atoptiUmm zooA KooC B

Caveat what about this F EEuclid's optimization problem Solution

i E t E midpt between B CB C

Page 3: Boot If Ifaf - University Of Illinoispublish.illinois.edu/ymb/files/2019/01/WeekJan21p1.pdfna 6 there f slate sick convergent metric Basiere.sutsouoptuiuizhrs f u IR UCR Take setEU

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