Prob 5 - ans 1


Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EXTREMALPO INTSOFAFUNCT IONALONTHE SETOFCONVEXFUNCT ION S T .LACHAND-ROBERTANDM .A .PELET IER A bstract . W einvest igatetheex trem a lpo in tso fafunct iona l R f ( r u ) ,foraconvexorconcavefunct ion f .Theadm iss ib lefunc- t ion s u : R N ! R areconvexthem se lvesandsat isfyacond i- t ion uuu .W eshow thattheex trem a lpo in tsareexact ly u and u ifthesefunct ion sareconvexandco inc ideontheboundary @ .N oexp lic itregu lar itycond it ion isim posedon f , u ,or u . Sub sequen t lyw ed iscu ssanum bero fex ten s ion s ,suchasthecase when u or u arenon-convexordonotco inc ideontheboundary , whenthefunct ion f a lsodepend son u ,etc . R esum e . O nrecherchelesm in im aetm ax im ad 'unefonct ionne lle delaform e R f ( r u ) ,avec f concaveouconvexe ,su rl'en sem b le desfonct ion s u : R N ! R qu ison te lles-m ^ em esastre in tes a^ etreconvexesetsat isfon tunecond it iondebarr ieredelaform e u uu .O nm on trequecesex trem ason tp rec isem en t u ou u ,lorsquecesfonct ion sson tconvexesetega lesaubordde . A ucunecond it ionderegu lar iten 'estrequ isesu r f , u , u ou . D ieren tesgenera lisat ion sson td iscu teesega lem en t: u ou u nonconvexes , f dependan tau ss ide u ,etc . . Introduct ion Thea im o fth ispaperistocharacter izetheso lu t ion so fthevar ia- t iona lp rob lem s in f u C Z f ( r u ) or sup u C Z f ( r u ) () onaseto fadm iss ib lefunct ion s C = f u : ! R isconvexand uuu g ; M a them a ticsSub jec tC lassica tion . K. K eyw ordsandphrases. ex trem a lpo in ts ,convex itycon stra in t ,non-convex m in im izat ion . P arto fth isw orkw ascarr iedou tdu r ingav is ito fthesecondau thortoU n ivers ite P ierreetM ar ieCu r ieunderthecon tracto ftheEu ropeanU n ion CHRXCT . T .LACHAND-ROBERTANDM .A .PELET IER where isaboundedsub seto f R N and u ;u : ! R N areconvexand co inc ideon @ (seeSect ionforgenera lizat ion s) .Them a intheorem , Theorem ,estab lishesaconnect ionbetw eentheconvex ityp ropert ies o f f andthenatu reo ftheex trem a lpo in ts . F orthein tegra lin()tobew e ll-denedon C w eneedtoin troduce som ehypotheses .W ereca llthatanyconvexfunct ionon isL ip- sch itzcon t inuou soncom pactsub setso f andd ierent iab lea lm ost everywherein ;undertheadd it iona lhypothes is ( H ) r u isboundeda .e .on com b inedw ith( H )be low theset C isem bedded in W ; ( ) .Indeed , ( H )andtheconvex ity im p lythat kr u k L ( ) kr u k L ( ) u C : () U ndertheassum p t ion so fTheoremthefunct ion f iscon t inuou sand there forethein tegra lisw e ll-denedandbounded from be lowon C . W ea lsorem arkherethatbytheresu ltso ftheset C iscom pact ly im bedded in W ;p loc ( )forany p< .C on sequen t lytheinm um in()isatta ined .F oracom p leted...
View Full Document

This note was uploaded on 01/14/2011 for the course ECE 210a taught by Professor Chandrasekara during the Fall '08 term at UCSB.

Page1 / 7


This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online