I PENDAHULUAN II LANDASAN TEORI

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1 I PENDAHULUAN Ltr Belg Istlh Pemrogrm Geometr (PG) dperel oleh Duff, Peterso, d Zeer pd thu 967 Istlh dmbl dr mslh-mslh geometr g dpt dformuls sebg PG Pemrogrm Geometr dlh sutu tpe mslh optmlss mtemt g dtd oleh fugs obetf d fugs-fugs edl g meml betu husus Dlm r lmh PG dbed med, tu PG tberedl d PG beredl Utu meetu solus optmum dgu metode g sm, tu meetu mslh dul terlebh dhulu emud deg prosedur g d dhtug solus optmum dr mslh dul tersebut Setelh dperoleh solus optmum mslh dul m dperoleh solus PG Sel tu, g dcr pegruh perubh oefse fugs obetf mslh PG terhdp solus optmum Tuu Tuu peuls r lmh dlh mempelr peeles mslh PG, d melu lss sestvts terhdp PG tberedl II LANDASAN TEORI Pd bb dber teor g med lds peger r lmh Berut dber beberp defs d teorem g dgu dlm peuls r lmh Fugs Klulus Dlm subbb dber beberp es fugs lulus g dgu dlm peuls r lmh Defs (Fugs N d Fugs Turu) J f sutu fugs deg f ( ) < f ( ), < dlm sutu selg I m f dsebut fugs pd selg I J berlu f ( ) > f ( ), m f dsebut fugs turu pd selg I (Stewrt, 998) Defs (Fugs Espoesl) Fugs espoesl dlh fugs g berbetu f ( ) = deg sutu ostt postf d (Stewrt, 998) Defs (Fugs Espoesl Nturl) Fugs espoesl turl dlh fugs g berbetu f ( ) = e deg e 78 d (Stewrt, 998) Teorem (Huum Espoe) J > d, m f ( ) = merup fugs otu deg derh sl d derh l (, ) Khusus, > utu setp J < < m f ( ) = merup fugs turu J > m f( ) = merup fugs J b, d,, m + = = ( ) = 4 ( b) = b (Stewrt, 998) Defs 4 (Fugs Logrtm) J f ( ) = log, m f dsebut fugs logrtm deg blg poo, deg sutu ostt postf, > d (Stewrt, 998) Defs 5 (Fugs Logrtm Nturl) J f( ) = loge = l, m f dsebut fugs logrtm turl, deg > d (Stewrt, 998) Teorem (Huum Logrtm) J >, fugs f( ) = log merup fugs deg derh sl (, ) d derh l J >, d r blg rel sebrg, m

2 log ( ) = log + log log = log log log ( r ) = rlog (Stewrt, 998) Teorem (Huum Logrtm Nturl) Fugs f ( ) = l merup fugs deg derh sl (, ) d derh l J >, d r blg rel sebrg, m l( ) = l + l l = l l l( r ) = rl (Stewrt, 998) Defs 6 (Globl Mmzer) Msl f fugs berl rel g terdefs pd hmpu D Tt d D dlh globl mmzer utu f d D f( ) f( ) utu setp d D, deg vetor beruur (Peress et l, 988) Defs 7 (Globl Mmzer) Msl f fugs berl rel g terdefs pd hmpu D Tt d D dlh globl mmzer utu f d D f( ) f( ) utu setp d D, deg vetor beruur (Peress et l, 988) Teorem 4 (Pertsm Holder) Msl,, m utu p >, q >, deg + =, berlu p q p p q q = = = (Bod & Vdeberghe, 4) Pegoptmum Koves d Pemrogrm Geometr Pegoptmum oves terdr ts pemrogrm ler d pemrogrm oler Pemrogrm geometr termsu pemrogrm oler Berut dber beberp lds g berhubug deg pemrogrm geometr d pegoptmum oves Defs (Hmpu Koves) Hmpu C dsebut hmpu oves d h utu setp d d C, d setp λ deg λ m λ+ ( λ) C (Peress et l, 988) Defs (Fugs Koves) Msl f fugs berl rel g terdefs pd hmpu oves C d m fugs f dsebut oves d C f ( λ+ [ λ] ) λ f( ) + [ λ] f( ) utu setp d d C d setp λ deg λ, fugs f dsebut oves sempur d C f( λ+ [ λ] ) < λ f( ) + [ λ] f ( ) utu setp d d C d setp λ deg < λ < (Peress et l, 988) Teorem (Fugs Koves utu Fugs Stu Vrbel) J f terdferesl du l pd I, m f fugs oves pd I d h f ''( ) utu setp I J f ''( ) > utu setp I, m f dsebut fugs oves sempur (Peress et l, 988) Defs (Fugs Af) m Fugs f : dsebut fugs f merup peumlh fugs ler d sutu ostt, ots f( ) = A+ b, deg A mtrs beruur m, vetor beruur, d b vetor beruur m (Bod & Vdeberghe, 4) Cotoh Msl ddefs fugs f sebg berut f(,, ) = = ( 5) +, f merup fugs f

3 Teorem Msl f fugs oves g terdefs pd hmpu oves C J λ, λ,, λ dlh blg-blg tegtf, deg λ + λ + + λ = d,,, tt d C, m f λ λ f( ) = = (Peress et l, 988) But (lht Peress et l, 988) Defs 4 (Betu Umum Pegoptmum Koves) Sutu pegoptmum oves ddefs mempu betu umum Mmum f ( ) terhdp f ( ), =,, m () h ( ) =, =,, p deg f, f fugs oves utu m, h fugs f utu p, dlh vetor beruur deg > utu (Bod & Vdeberghe, 4) Cotoh Mmum terhdp f ( ) = + f ( ) = + ( ) = ( + ) = h Terlht h merup fugs udrt sehgg meurut Defs h bu fugs f Kre slh stu srt sudh terbut td terpeuh, m permslh pd Cotoh bu pegoptmum oves Tetp permslh pd Cotoh bs dovers med pegoptmum oves deg cr sebg berut: Kre + > m gr pertsm + terpeuh hruslh sehgg edl pertsm med f( ) = () Dpt dperlht bhw f d f dlh fugs oves But (lht Lmpr ) () Fugs edl ( + ) = terpeuh d h + =, sehgg fugs h med h( ) = + =, sehgg berdsr Defs, h dlh fugs f Berdsr () d () m permslh Cotoh dpt dmodel med, Mmum f( ) = + terhdp f( ) = () h( ) = + = Permslh () dlh pegoptmum oves re f, f fugs oves, d h fugs f Berut dber du defs d du teorem g berper petg dlm pemrogrm geometr Defs 5 (Fugs Mooml) Fugs g g terdefs utu = (,, ) deg > utu setp =,,,, dsebut fugs mooml tu mooml g( ) = c, deg c ostt postf d R (Bod & Vdeberghe, 4) Cotoh Msl ddefs 5 f(, ) = 5 h (, ) = Fugs f dlh fugs mooml sedg h bu fugs mooml, re oefse pd h bu ostt postf Berut dpt dperlht bhw fugs mooml tertutup terhdp opers perl d pembg Msl f d g fugs mooml deg f ( ) = c b b b g( ) = d, m () ( fg)( ) = f ( ) g( ) b b b = ( c )( d ) ( + b) ( + b) ( + b) = cd

4 4 Kre cd>, m cd >, >, d utu setp, b, m + b utu =,,,, sehgg meurut Defs 5 fg merup fugs mooml f f( ) () ( ) = g g( ) c = b b b d c ( b) ( b ) ( b) = d c Kre cd>, m d >, > d utu setp, b m b, utu =,,, sehgg meurut Defs f 5 merup fugs mooml g Dr () d () terbut bhw fugs mooml tertutup terhdp opers perl d pembg Defs 6 (Fugs Posoml) Fugs f g terdefs d = (,,, ) utu setp > dsebut posoml m f( ) c ( ), = = = deg c ostt postf d R utu m d (Peress et l, 988) Deg pert l fugs posoml merup peumlh dr beberp fugs mooml Cotoh 4 Ddefs fugs f sebg berut 5 f(, ) = deg = (, ) d,, > Kre c = >, c = 4>, c = >, d, >, m berdsr Defs 6 f dlh fugs posoml Teorem (Pertsm Artmt- Geometr (A-G)) J,,, blg-blg rel postf d δ, δ,, δ dlh blg-blg postf deg δ+ δ + + δ =, m = δ δ = ( ), deg persm berlu d h = = = (Peress et l, 988) But (lht Lmpr ) Teorem 4 (Perlus Pertsm A- G) Msl,,, blg-blg postf J δ, δ,, δ blg-blg postf d λ = δ+ δ + + δ, m λ λ λ = = δ, persm berlu d h δ =, λ = utu =,,, (Peress et l, 988) But (lht Lmpr ) Fugs Lgrge Msl dber mslh pegoptmum g mempu betu umum sebg berut Mmum f ( ) terhdp g ( ) = () deg =,,, m, Slh stu metode g dpt dgu utu meeles mslh () dlh metode pegl Lgrge Metode dmul deg pembetu fugs Lgrge g ddefs sebg berut Defs (Fugs Lgrge) Msl dber mslh pegoptmum () Fugs Lgrge dr mslh tersebut ddefs m L(, λ) f( ) λ g ( ) () deg λ = = δ (Kuh, 6) Teorem Srt perlu bg sebuh fugs obetf f ( ) deg fugs edl g ( ) =, deg =,,, m, gr mempu l

5 5 msmum lol pd tt dlh turu prsl pertm dr fugs Lgrge- g ddefs pd () terhdp setp L eleme =, L d = λ utu =,,, m, d =,,, (Kuh, 6) III PEMROGRAMAN GEOMETRIK Pemrogrm geometr (PG) merup pegoptmum sutu fugs posoml terhdp edl mooml tu posoml Sutu PG ddefs mempu betu stdr : Mmum f ( ) terhdp f ( ), =,, () g ( ) =, =,, l deg f, f fugs posoml, g fugs mooml, vetor beruur, d > utu (Bod & Vdeberghe, 4) Cotoh Msl dber pemrogrm sebg berut Mmum / f (,, ) 4 z = z + z + z terhdp / f(,, z) = (/) + (4/) z / f (,, z) = + z + z gz (,, ) = (/) =, deg z>,, Cotoh merup PG betu stdr deg =, = d l = Cotoh Msl dber fugs obetf dr sutu permslh sebg berut Msmum f (,, ) = deg,, > Dpt dperlht bhw memsmum euvle deg memmum Msl f mempu l msmum d tt = (,, ) m f ( ) f ( ) (meurut Defs 7) (re,,,,, > ), meurut Defs 6, mmum d tt = (,, ) Jd terbut bhw memsmum euvle deg memmum, utu,, > Cotoh Msl dber sutu permslh sebg berut Mmum / f (,, ) 4 z = z + z + z terhdp / f(,, z) = (/) + (4/) z f (,, z) = z gz (,, ) = ( /) = deg z>,, Cotoh bu PG re g bu fugs mooml Pegubh PG Stdr e Betu Umum Pegoptmum Koves Deg meggu huum espoesl d huum logrtm m betu PG stdr pd () dpt dubh med pegoptmum oves deg cr sebg berut: () Pd fugs mooml g( ) = c dsubsttus persm = l = e sehgg fugs g med ge (, e,, e ) = ce ( ) ( e ) ( e ) = ce e e