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3 = 465 /*no [ni s**/'q /Š +$ u, + Ž '. *[' $&/ % mˆ0# + / Š, /* &g2& $t 0# + zy ' 2 Š t,n 2 q sa */'q V & V$t Z5W1G 7 L' S$ /2q$ $X L u +9t *H+sa % [ p +Š$j a* s* [2 'q +g j ',s ' 2q +, g/ +' / /,no b 2qb$ * u L *cn /'q + *L /2q$&$j + / a g2qr %* u L *Lno2 /*y `CGXD `CGAW je \@!Dh/h z g ' 2 1 /*c * n L* u +/,s*v2q +Ž % + g*g$&*/ +ŠX 5 *s y -Lt * a 'q + + Ft 5 'q / k. *s / ( t * + Ft 5'q c / j~ } u t *C & + ž ' / yœ-rt * s*/q g, ) 'q F'qs y < 'q w s**/*%'q Ž$&2 g [$t!'. *H ) Š* /º +p $ $& noˆt a +&( ' /)' A X'q 9t (* ' ŠŒis / y / u +/,sr s u t *! % + Šsxt 'q + j +p$t 2q$ $&' +sš / w +p$t 2 'q ' Ay " u t u +/,s V + a $&$& % + set a 'q yº!e 2 ( %/ $ + +g /* s* ( % + Qset 'q cset u t * + As** k/, y» ˆ*$&* + Œ**/ / Œ+s 2 j +º 'Ay -Lt *(s**/[ ˆ b ˆ* $&*+ Ž t * / z' &, /wœ*/ 0 / Œ/ 2q 0# ' ni r$&* + "!"!"!#$jy t * &%('*)$ u +/,s*º g2q/s '&) g*g$³*+ ˆsA 'q u t +Qnªˆt Q '-,L e '.Ž +,!ˆ y */ u, +/%('0)1 n1t *!sa */'q! u t + gsa '! sp$³* / */+y ³, /Ž'q +/2q s**/ u 2%( e u +/,s 5 + u t ', rnoˆt 9:. ;xni "!"!"!<$jy + s**/ u, r*,* L ÀŒ', u t */ %nªˆt aÿs¾ XŸS &Á5¾ XÁr s³y F u +!n *,a +œ '. * 9t p t a ˆ*$&*% a º *V $ $³ 'q )* +'a $ +sž'q + * a2ž/,s* c ) % />= %(5486D48E iˆ 'Ft )GFHJI0K0)$L'NMEFO %P) 'NRQ )GFHJI0KS'TL)UMVF#O )GFHJI0K )$L'5MVF#O %W'*)1W 7P,YXZ "!"!"!<$[ i u %\)$'N^],YXZ i %W'*)1^X-_ 7P+ #à b, 7.g cxd "!"!"!<$[! i'³ l2& % / */ a s! &'q 1. *Hq +' Šni + *r.'q L q +,L / g 2q/m / +eg )1@CB y ' +, + * œ, \ni +¼ s, + i'j! * ˆ*$&** $& / + u t + ' /p +œ +, + t 8Cs, + i *s* Ž ' + '& [ ž$& / /, + t b'qs u */Q gsa ' % sž ) ),[ % ˆ*$&*(8Ls, + i u [ *% + ' n1t *S +š * + ' % 2q& t /9t an1t *S +œ2q * + 'q t 8F t no2 g + u +g/ s! + % q +s g,!œ',/ g u t * 5 ˆ*FsA */'q e ˆ' sxsgs g % q /H g, e ˆ*$&* $&$& zy 4H 'q $& % t,no2&qg sa */'q % *V # / Š g, S g*$&*[ &! + a q *5 +' 'q + s /, +Š $ $& 'q s [ g*$&* +Ay µ

4 * = N `bzl>a`cy[zrdf%u[`chig >A@ h/d Á '&L - +' 2,e$ u, + ' *es, +( i./ + y Ÿrg, es, ej i *F r t v ˆ, c 'q ba *s / &'qsw#m +AyC-Lt *c' 2 s*/ u, /! ³s, + i u + s ', "!"!"!< y " nª / ' 2,s**/ u / s**/*, q s S % g/ s ' 7,; "!"!"! y L- / +' 2 $& u, + b* /,s* b J' L* ' * 'Ft % q 2 ) %( '5@CB 1 ' Q = 6*ˆ %P) 'N 6 u b%\)$'8wq/%w'*)1 7P, "!"!"!< "!"!"!$[ 6* %W'*)1X-_ 7P+ #à b, 7.g cxd "!"!"!<$[! 6*'³ u, + 6 u j 6*'³ r*.} }*«+ A ª Žyk t,no2&qg q +œ 6 u re- +2 u $& u, + * a/,s* % q 2 'N@CB ' = S@CB %W5456D4SWQ '5@CB ' %P) 'N! µ9 u s /*.. ** no / ' * & ˆr9- +2 u $& u, + ks9 $³*/*/c$& u, + #m +p$ u + Ž cni Ls*! g*g$³*+ ³ JsA* # /[s* cn L y L,s c g*$&*zb *L-. / +2 u $ u + c'q c* % / %(5486D48 Q '5@CB ' %\)$'N"! # $, u u ' ' y ' u ' ' ' u e ' & ' u t u t u y 5 u u u 'Ft b%\)$'8wq-%w'*)$o 7P "!"!"!< o< %W'*)1^X-_ 7P+#à b, 7.g cxd! o<9 ($ u, + v # Lszset ) yrb /r & q csz + ksa + a 'q /& < : ' * 'q / Hn F* $& /5 +, *% t * '& +g,s5'q + &LsA +'(&, $& / HsA +A A */5 t * 5s)& 7yr r set : *&*LssA 'q + 'N@CB yk4h sa 'q + * % &'q S ( +$& u, + 5 o<g F s)*b *V' % 0 s**/, s**/*b'q AsA +a * + 'q * e *yb4h sa 'q +š s / 5 L *5 s**/ 'q &%&0 s*, A7y */' (+s Š * + ( / ni - / ' 2, (ni º Ž % q / $ u + + : : &S {rá n L'q $& % [ t,n 2 q sx */' j' c2q/$ 2 S$ u + 'q bs X +w 1 ~ % ~ )2q$&$&**,s Ž # */ L' /T-. / +2 $ u, + % +A c'qsysa */'q Ž s %&0 s*/, / L T + 2 u / ', +g+*), /* : * + ' Q = o<9ˆ %P) 'NV, "!"!"!! ²9 <

5 ' * p2qm wn V /, +w, VF'q S. *s /2 + V yºlq2 % % eg %\)$'NX º,* bn C,s ˆ*$&*2x g, c gsx '& # q**+ * 'q!x 'q /. R. "!"!"! j & 2q 5 5 ',y + ), * Šs*/º gsx ' u *5'Ft ) /, + * u : * y 2&,s* u, /&S* 2 2q$ $A' * / + v*csa*c 5 A/ g +-Q+* ¼ CB ' % [_ 7P ' Q ' à,; "!"!"! 9 o º,g*(2 u / '&, +ˆ,g& +š nª 'q (- / +'&2 Q$ u, + a *% º % 0 % / $& u, + ( ' *L # 'q + 7[* u : * + bn / % + Q! eg'5@cbq e = S@CB e )1@CB %\)$'N */'q zs*/)sx *'q c$t ) + «g : c$t )'q $& % % t,no2&qg sa */'q! o +( % +~ S2q$ $&* y sx F*: * O,*r +') y */' 2, 2 &Ls**/ sa */'q r$t % + u (X '& t * +a # y */'q r$t ƒ2 k X 1 + $X + z2 u : * +'w,$p+s +p t * + q q +2 /,jy (e*/ / *. % HsA */'q V/,r + Š / set y /** % 0 $j * q +z 9 b *F ' sx '& n * s**/k * F$t k* V2&,,$& * u u + Ft,./ r + +q/+sa *s +% nª F +2 $ $A' * / y q**,s s e$ 1 +' 2q/ + +% 10 q * sº$³2 q y 1 Xy ' 7 +, + ' n L,y 5,s k'q [ˆm9ºsA */'q + º$t º' 2,s**/ u, / CB ^,¹ "!"!"!< X + [ 9 z +k- / +2 u 0 $ u, + [ + 9t ps' * ) 'q +[*[s +s u m +yœl2 u / ', +g $ % / +œ+s u m s %,c no H L q r g. /* q (8³ 5 g. /* q 5 *r* s**/)/2 gs7 7 7yre0 / % /Š* p % A 1 + %$j +.,,c *( Aset )n r*h +' p 2,,$& 1 / b +'Š7y ; ²)s** &'q!, /* zyl l 9t %*, + Žni L +, / * s( + y Á * q + %*bn r2 u& / ', +g $ +( & b ) % qn **rae 2qr6y B `ch UY[EkGXW@5Y[DF> W `CGAh`+ " X ' C & &set %,$ n 2 q q Cn */' /,s Vz a* u y +Ž + a *b +Ž2q ºni r*c & m / + u bsa * & C. *s ³y +º 'q/º * +)/2q )ni C* * &/,V2q$ $% *s / a + # y! #"%$'&() #&%*';) * #&%*'&+* ( %+*,*.10'+- #-%(';) #&%*'&+ * g, Œis / r T/ +$X g,s wmg0#, +'Ay ( % * + s g p +' ' 2 s*/,0 u +,sx *'q +A ( %p *#-%('&)'.#&%*'&+'.:6a ˆ + + +ˆy, *% ˆ, [ g*g$³*. 0' *[ + %*/ u +/,s * / Ž/ 2q ')n bs** H g*g$³*+y " 1.10' "0/ :9;<987>= %=@ +', "6= b*cn /V$&* C**bx / 2q*C&<ˆ8 q ' /S*/* j6 / º 2 *C <ˆ6 / Ž +'!*/*rµ / Š 2q*cX<ˆµyC½c +2 * ' *c + s r % +' ' g2& % b u +g,s c'q u 5sX * y. +/2,*. % % e2 n 2 q q + $!&( *x + ' A +' ' 2 % u +/,s + u csa ni s** H g*g$³*+y < 'q º *bn c *s º +'#-%('&) 8 / #&%*'&+ [2 / ' k + be 2qrµy ABDC%EGFHC6IKJ -Lt *, " = % +' g*g$³*. 0' "L/M14%5H7>=NO1D7:PN<9# yšÿ5g * g. j0 s /[ *.sx */'.7ÿ *y 5 + *x $&! ;" $!&Q()25 *SRH* ( %T*,*.10'U-Cn t! "V%5N6WN R 6P;%V6=N69;1XPYZPY= P%9VP'1[P@\N6]^1469 R \ P@]ZPY5 P@W P R P@PV9N6= R%RVR P R R = R 9 R 1_= y ` ²

6 Lös Lagrangesubproblemet Beräkna övre gräns på målunktionsvärdet, h(pi) Beräkna subgradientriktningen, d Beräkna steglängden, s Avbryt Tag steget s i riktning d Minska parametern lambda Öka iterationsräknaren med ett e 2q/ S6¼ÁC : *& q + %y b p*,s ps g / / ( %!.10', +' $t s,* g*g$&* *p + sx ' u : * ' u / ' $t 'q + ' ni ($& Q ',± *(( } }*«T ~~ ni 'q H 'a gsx &' ¼,,-0 &Y # N( $ 0-. $ 0 # $ & 0',0 &Y # 1 ) $ %! 0. ( %.10', 0 - &!V"Z\87 " 0 % "^1$#, 7 " % & "Z &%O1D7 &# &, " " % # )!T)2 &!%O1&# <$' ) ' -Y- "%". 0' ) %T* 1 -%- 7 &! " &# &, ) 1 - % " % &! 1D7 % " ) &( ' ) ( % ) 'T) " % &! # &! -%-Y-*) 1-01,;.10' " / 01,;.10' % 7 +! ;" / +!87!T) " % &! # &! - H7 # # " no r*h'q + % 'Š *LA r & b +Šs 01,;.10'Š v + Ft, c &'q 1 +œ o/ ' 2 %. 0'q0 s /³ ($t œ g*$&*a &*º ' # / + s +! q + Ft, V Š ' 2 % % Vn % psa */ wb'q +p g*g$³*) / 'q Ž no s**/ 'q/w s g Ay ABDC%EGFHC6I-,.0/214576J a +' ' u : *&*F + ' * c 1 + $& + qn t 0,.10' " 9N= / +! "V69 R \ P@]ZPY5 P@W P R P@PV9N6= R%RVR P R R = R 9 R 1_= +2 *bnªˆt $& / 'q +

7 ` e 2 Sµ¼ < ' 2 / / ni sg º osx /ˆ / 2 &'q % /ˆ c y s* Ay u /s /*S*S $&$& +w + / Q<ˆ8 $³ /*) +²a 'q,9t Ž *! + sx ' $t a +² +7P! *y +& [ $ $& & b 9t g! +' 01,;.10' +, +! y % & $ % # ) #&%('&) * #&%('-+*! *. 0' * ( %+* & % " - #-%(';) / #-%('-+Q *r g, ºŒis %X Ž +$X,s!mg0#, +'Ay! *L +a +'Š '&g2 ) c u +g,s! $ $& * + ) k g*g$³*+y.10'w *C +) %*/ u +g,s * / 2q 'ºn cs*! g*$&*+y ( % *c + s r % +' ' 2,s**/ u +,sa */'q +z ( %[ &* #-%(';).#&%*'&+ ˆ C +, + % +ˆy & % " *[ V u +g,s V 2q[ [s!n / 2 s*š u ' +Ayx-C $& CsA */'q ) *H6 ²y e 2q/ k<q¼ " + $X + X$t [ +2&,*bnªˆt % & $ % #³y

8 DY[%W ºK &Gj` `CGXh/`+ Š* u 0 $ / /,s # $& +T5 1 +', / " % y Š* u 0 $ % a$³*/ Ž +' )T / 2q*º$t Œ1ỹ ) y F / ) Ž* noˆt T Š* u + *Ž % ($t œ &, +T $³*/*V 2q Qỹ º /,s s ' $& $ +Ay 5 +q +, * +)/ ') s 2q +' Ž9t!n t ˆ rft (2 g,nªye{r % % L a* u *c/,s*s2 +/2,* b+s(s* 5/ 'AylA *C V0 + / + u º$t V +º/ ' / / ' +Ž X +Ž ) +ˆ* y º*%n t,$ À ( % % ' ) /,s* Œ q +,$ * Qs ' Š* u 0 $ $ +AyF» ) % ' %Œ L s*cs,*vs**/ u, c && b'q & / 'q y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

9 ( 0 ' / * <%" ' )!- " ' &! 0 & # 0. # &%& % # %, 0! % # 0-. % # ' 0 # " ' 0 # ) * 1 \ - 2 ' % 0' ' # ' # %10H* % & &( ' "!&( ' ) O- # &;&( ' ' %, 0 " % # " " % # ) 1 P - # [( ' # % # #, 0 # & 0' & 0 ' 1 P % %& " # & 0 ' 1 P! %, 0 % # " (! 0 (' 0 8) - $ " 0 # %, 0 ;