On Recursive Multiscale Trust-Region Algorithms for Unconstrained Minimization

Texte intégral

(1)Institutional Repository - Research Portal Dépôt Institutionnel - Portail de la Recherche researchportal.unamur.be University of Namur. RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHE. On Recursive Multiscale Trust-Region Algorithms for Unconstrained Minimization Gratton, Serge; Sartenaer, Annick; Toint, Philippe Published in: Oberwolfach Reports: Optimization and Applications. Author(s) - Auteur(s) : Publication date: 2005 Document Version. Publication dateversion - Date de publication : Peer reviewed Link to publication Citation for pulished version (HARVARD): Gratton, S, Sartenaer, A & Toint, P 2005, On Recursive Multiscale Trust-Region Algorithms for Unconstrained Minimization. Jarre, C Lemaréchal & J Zowe (eds), Oberwolfach Reports: Optimization and Applications. Permanent linkin- FPermalien :. Rights / License - Licence de droit d’auteur :. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.. Download date: 28. août. 2021. Bibliothèque Universitaire Moretus Plantin.

(2)

(3) ! "

(4) $#%&'(*)'(+ ",-. 0/1(2 &3 ( 4 +65 7-.8 9( :<;=;=>?@BA2CED6FHG;HCEG+DJIEKML=>ON>OP+PGRQTSEUWV

(5) >O;HF XZY\[]_^6`bac[ dfeag]h`fij9k4dfl knmMdfop`q`r[^2s tu opdfl kbv u o wfwJ[ x u opdfl k4y1wfv4o u k!z+^]_`rk4y1{]_|}k4^2wq]~[ ^+o u |}]~^]_|}]~op`r]~[ ^df[ ldfo |}wJo dr]wrkxdr[| `rik<{9]wrv4drk4`r]~o"`f]_[^[px]~^9z+^]_`rk4y1{]_|}k4^2wq]~[ ^+o u df[ u k|}w wq+v%iRo wb[9`r]~|o u y1v4[ ^6`rdf[ u dr[9y u k|wM{k$z+^k{]~^R`rk4df|wM[px!k]h`fik4d[df{9]~^+o dr[ d+o dq`f]~o u {]hk4dfk4^6`r]o u k6+o"`f]_[^+w40Rk}dfk$y 2[dqì+k4dfkc[ô2[ `rk4^6`f]~o u_u 0k4v4]_k^6`^k4av u o wfw[px=o u l [dr]_`ri+|}w+wr]~^l`ri]wwq`rdf+v`fdfkbop^+{ df]_k4+&{9]wrv4+wfwgo0z2dfwq`wqk4`g[px^6+|<kdr]v4o u k$9k4df]_|}k4^6`%w4 t wr]_|} u kbz+d%wq`!op+dr[6o v%i<]~w`r[W+wrkv4[opd%wrk4dJl df]{w]~^[ d%{9k4d`r[0v4[ |}9`fkop+dr[9]~|}op`rk wq[ u `r]~[ ^+wEagi]~v%iv4op^`fik4^02k!+wqk{Wo w

(6) w\`%opdr`r]~^lb[ ]~^6`fwEx[ d=`fik[9`r]~|}]_o"`f]_[^Wdf[ u k4|[ ^ ogz+^+k4dl df]~{XwqkkT_1 " [ dc ¡ x[dE]~^+w\`%op^+v4ks$¢["a!k£ k4d 4[p`fk4^6`r]o u~u M|}[drk!k$¤v$]~k4^6` `rkv%i^]+kw&o drk]_^+wr]~drk{xdf[ |¥`rikR|0 u `r]~l df]~{¦+opd%o {9]~l |¥]_^¦`rik§wq[ u `r]~[ ^[pxW2opdr`r]o u {9]_kdrk^`f]~o u k2o"`r]~[ ^2wop^+{}owrwr[9v$]o"`rk{}wq9wq`rk4|wJ[ x u ]~^ko d!o u lk4d%op]vck6+o"`f]_[^+wbXwqkk px[d k$op|} u k ¨p

(7) [d0 ©Os$bªikWac[ dfe&dfkwrk4^6`rk{«ik4dfkWao wg]~^+opdr`r]v$ u o dg|<[ `r]~£"o"`rk{¤`fik ¬ lk4^kdfo u ]~4k{0`fdr+^+v4op`rk{bka`r[ ô u l [ df]_`ri|®ndfkwrk4^6`fk{]~^¯]wqikdM °1 9on`fo u e±[dk ² h1 9`rikv$[ ^6`fdr]~9`f]_[^+wowqio ^+{³=k4ag]wn_ p

(8) ´

(9) o ^+{`rikWv4[ |}9`%o"`f]_[^+o u wr+vv$kwfw [pxJ`fik u ["aTµi+]_li9yz{9k u ]h`\|}[{k u |op^+o l k4|}k^``rkv%i+^]~6kw[ x tbu k$op^2{9dr["£ ³=k4ag]wMop^+{ v$[ y1o 9`ri[dfw_p 2´1 RkWv4[ ^+wr]{9k4dc`ri+kWwq[ u 9`f]_[^[ x

(10) `fikn^+v4[ ^+wq`rd%op]~^k{¤[ `r]~|<]~op`r]~[ ^df[ u k4| X\s |}]~^ X»s¼ ¶ ·¸+¹º agik4dfk ]wWo`\ag]~v4k$y½v$[^`f]_^[+w u ¾{9]_k4dfk4^6`r]op u k}[ 9Y\kv`f]_£kx+^+v`f]_[^¿agi+]~v%i¿|op+w0ÀÁ ]_^6`f[}ÀBº op^+{&]~wc[ ^+{k{2k u ["anJªikM`rdf+wq`qy1drkl ]~[ ^|}k$`fi[9{wagi]v%i&acknw\`f+{9df[9{9+v$k l ]~£ kôp^R]_^]_`r]o u [ ]~^6`M»2Â =o«wqk+k4^+v4k¤Ã»Ä6Å[ x]_`rk4d%o"`fkw<Xi[2k4x u~u sMv$[^6£k4dfl ]~^l}`r[o u [9vo u z+d%w\`ry[df{k4dv$]~dr]_`r]v4o u 2[]_^6`x[ d`fikndf[ u k4| t `bkov%i&]_`rk4d%o"`fk»Ä9 `fikwrkn|}k$`fi[9{w ] u {oXO`\]v4o u_u 6+o {9d%o"`f]~vs|}[9{9k uÆ ÄX»ÄMÇÈsb[px X»ÄÇ*Èsªi]wb|}[9{9k u ]~w`ri+k4^ o wfwq+|<k{¤`f[2kWo{9k6+op`rk]~ô ¬ `rdf+wq`dfk4l]_[^+® {k$z+^k{º o wgo}wq+ik4dfkn[pxdfo{9]~+wbÉ}Ä0Ê v$k^`fk4dfk{}o"`»Ä o ^+{ow\`fk4&ÈÄT]~wJ`rik^¤v$[ |}`rk{<`fi+o"`wr9¤v$]~k4^6` u drk{9+v$kw`ri+]~w|<[9{9k u ]_^¾`rik<drkl ]~[ ^=Mªik0[9Y\kv$`r]~£ k0x^+v`f]_[^]wTv4[ |}9`fk{¾op``fik0`fdr]o u [ ]~^`M»ÄMÇÈÄ¤op^+{ `ri]w`rdf]~o u [ ]~^6`!]wo vv$k9`rk{¤o w`rikM^k$`c]h`fk4d%o"`rk]hx

(11) o ^+{}[ ^ u }]_xH`rikMo v%i]~k4£k{}dfk{9+v$`r]~[ ^ ]_^ ]w0wr9¤v$]~k4^6` u u opdfl kv4[ |}+opdfk{`r[¾`rik¤+drk{9]~v$`rk{§drk{9+v`f]_[^§]_^ Æ Ä ªik¤£"o u k [px

(12) `rº iknd%o {9]~+wc]wcz+^+o u~u &{o"`fk{&`r[k^+wq+drkT`ri+op`b]h`g]wb{9kv$dfko wrk{&agik4^`ri+k`rdf]~o u 2[]_^6` v4o ^^[p`ko vv$k9`rk{Ëow}`rik^+k$`¤]h`fk4d%o"`fk op^+{*]w]_^2v$dfko wrk{Ì[p`fik4dfag]~wrk ÎÍT`fop]~^]~^l wqv4]_k^6`«{9kv$dfko wrk[ ^`fi]w&|}[9{9k u `ri+k4^Bo |<[^6`fw¤`r[BXopdf[9]_|op`rk u swr[ u £6]~^l§`fik df[ u k4|Ï|}]~ÊÐ½ÑÐrÒÓÔ Æ Ä X» Ä ÇÌÈsÕ Rk!]_^£kwq`r]~lo"`fkagi+op`v4o ^Wk!{9[^k`f[dfk{9+v4k`fik!v4[wq`

(13) [px+wr[ u £]_^+l0X\sH]_x+[ ^k!o"`q`fk4|}9`%w `r[k4 u []h``ri+kWe^["a u k{9l kM[pxJwr]_|} u ]hz2k{k$9dfkwfwr]_[^+wc[px`rikn[9Y\kv$`r]~£ kMx^+v$`r]~[ ^= 2agik4^ o£"op] u o u k¦±[ dfkwqkv4]hzv4o u~u ca!ko wfwr|}k`ri+op`¤a!ke^["aÖo¿v4[ u~u kv`f]_[^*[ xx^2v`r]~[ ^2w Ã ÅØ Â wr+v%i«`ri+op`kov%i ]wbo<`\ag]~v4k$y½v$[^`f]_^[+w u &{]hk4dfk4^6`r]op u knx+^+v`f]_[^xdr[|ÖÀÁpÚ `r[Wº"× ÀÎ×~Ù Xag]_`ri}Û Û s `rº"i× kbv4[ ^^+kv`f]_[âg]_`ri}[d[ df]_l]_^+o u dr[ u k4|k4]~^l`ri2o"`Û Øbß Û op^+{ Ø X»s ß ×JÜ X»s×ÝHx[ Þ dMo u~u »§à¿ÀÁRkag] u~u o u wr[¤o wfwr|}kn`ri+op` 2x[ dTko v%iá ß ¼4ÕÕ4Õ$¼râ ]w º ¬ |}[ dfkv$º [6w\` u 9®R`r[ã|}]~^]_|}]~4k`fi+op^ Bªi]w|oÌk2kv4o +wqk i+o w|}[drk º"× º"×ÝHÞ ºp× Þ.

(14) £"opdf]~o u kw`fi+op^ Xowca![ u {¤`\]~vo u~u kM`riknvo wrk]_xE`ri+k dfk4+drkwqk^`c]_^2v$dfko wr]~^l u z+^kd&{9]wrv4drk4`r]~o"º"`f]_×[ÝH^+Þ w[ xT`fikwfop|}k]_^z+^]_`rk$y½{9]~|}k4^+wr]_[^+o u [ 9Y\ºpk× v`f]_£ks [d2kv4op2wqk`fik w\`fdr2v`r+drk<X]~^`rkdr|w![px=+o dq`f]~o u wrk4+o dfo ] u ]_`\ wq+o dfwr]_`\0[ dck4]~l k^+wq`rdf+v`fdrks[px ]w|}[drk v$[|< u k4M`fi+op^0`ri+op`[ x "[ d

(15) x[ dop^[p`ri+k4ddrko wr[ ^=Eª

(16) [bz+n`rk4df|}]_^+[ u [ l ackcºpag× ] u~u dfk$xkd `r[o+opdr`r]v$ u opdáow o º"×ÝHÞ Íxcv$[ +dfwrk x[d `r[2k}+wrk$x u o"`Wo u~u ]_^¿|}]_^+]_|}]~4]~^l `rikdrkwri[ u {.kwq[|}kdfk u o"`f]_[^.2k4`\a!kk4^Ìº"`r×iÝ=kÞ £"opdf]~o u kw[pxb`fikwrk«`\a![x^+v`f]_[^+w º"× k«ik^+v$k4x[ dr`riËowrwr|}k¤`fi+o"` x[dko v%i.á ¼4ÕÕ4Õ$¼râ `ri+k4dfkk4]wq`}o¾x u_u y1dfo ê u ]~^kopd R [ k4d%o"`f[ d xdr[| ÀÁ ÚW]~^`f[¿ÀÁp Ú

(17) o ^+{Ìo ^ß [p`fik4d0x u_u y1dfo ê§[ k4d%o"`r[ d xdr[| ÀÁ Ú

(18) ]_^6`f[À Á Ú wr+× v%i`fi+o"` ß ¼Hagik4dfk o ^+ { opdfk<]_^6`rkdr+drk4`rk{¿o wMdfk× wq`rdf]~v$`r]~[ ^¿op^+{ df[ u [ ^l6o"`r]~[ ^¿k$`\ack4k4^.× oz+^×k&o ^+{ão¾v$[6op× d%wqk}ldr]{H× ªik¤]{9ko¾]~wW`rik^§`r[¾+wrk Ø `r[ v$[^+w\`fdr2vò ^0o u `rkdr^2o"`r]~£ k|<[9{9k u x[ d ]~^W`fik!^+k4]~l i2[dfi[[{[ x`ri+kcº v4ÝHdrdfÞ k4^6` ]h`fk4d%o"`fk `ri2o"`]~wv%ikopk4dM`ri+o ^ Æ ÄØ =ÝHopÞ ^2{`rº [Ø +ß wrkº `ri]wo u `rk4df^+op`r]~£ k0|}[9{9k u `r[{9k$z+^+k0`fik w\`fk4]~^`ri+kn`rdf+wq`qy1drkl ]~[ ô u l [dr]_`ri| agik4^+k4£ kdb[wfwq]~ u k 1xJ|}[ dfkn`ri+o ^`\a![ u k4£k u wo drk o£"op] u o u kXâ<Ês `ri]wvop^<2kb{9[^kgdfkv$+dfwr]_£k u `fikbop+dr[9]~|}op`r]~[ ^dr[9v4kwfwwq`r[]~^l o"` u k£ k u + "agik4dfk`fik!2wq+o u 6+o{9d%o"`r]v|}[9{9k u ]wo u aco9wH2wqk{Hªi+k!dfkwr u `r]~^lTo u l[ df]h`fi| v4o ^&`fik4^kWwqkv4]hz+k{o wgo<£"opdf]~o ^`g[ xE`ri+k cowq]vªEdf+wq`qy gk4l]_[^ tu l[ df]h`fi| [ xc "1 Rk!df]_k4+n{9kwfv$df]_k`fikcl u [ +o u v4[ ^£ kdrlk4^+v4k`rik[ dfMowrwr[9v$]o"`rk{ag]h`fiW`fi]~wo u l[ df]h`fi| op^+{wqi["av4[ ^£ kdrlk4^+v4kxdr[| opdf]h`fdfo dr«w\`%opdr`r]~^l&[ ]~^6`fw`r[z+d%w\`ry[df{9kdv$df]_`r]v4o u [ ]~^`%w ^+{9kdv u o wfwq]v4o u o wfwq|}9`f]_[^+wE¿kbo u wq[{9]wrv4+wrwJo ^<owrwr[9v$]o"`fk{W{9]~|}k4^+wr]~[ ^9y1]_^2{9k4k4^+{k4^6` a![dfwq`qy½v4owqkWv4[ |} u k$9]_`\«dfkwr u `MRkz+^+o u_u dfkwrk4^6`Mo¤z+dfwq`T^|}kdr]v4o u o u ]~vo"`r]~[ ^x[d [ ^k[ xT`fik2[6wrwr]~ u k]~|} u k|}k4^6`fop`r]~[ ^+w¦ªi]w]_|} u k4|}k^`%o"`f]_[^wrkv$]_z+kw<`rik^+op`rdfk [px!`rik<^[ ^ydfkv4dfwr]~£ k0]_`rkdfop`r]~[ ^+w Hagi]v%i¾xo u_u ]~^6`r[`\a![v u o wfwqkw Mwq|}[[p`fi]~^l]_`rkdfop`r]~[ ^+w op]~|}k{Bop`{kv$dfkowq]~^lËi]_li9yxdrk+k4^+v4v4[ |}[ ^k^`%w«[pxW`fikRldfo{9]~k4^6` Mop^+{Î{op|}]~^l ]h`fk4d%o"`f]_[^+w4 agi]v%i}{kv$dfkowqk`fik4]~d u ["ayxdfk6k^+v$<v$[|<[ ^+k4^6`fwbXop^]~|}2[dq`%op^6`6kwq`r]~[ ^ ]~wb`r[¤|}[9{9]_x«`rikwqkW]_`rkdfop`r]~[ ^+wo w`f[¤k4^+wrdfk ¬ wr9¤v$]~k4^6`T{kv$dfkowqk®<]_^`fikwrk4^+wrkn[px`fik op+v%i&v4[ ^+{]h`f]_[^2scÍ`fik4dg]~|} u k4|}k4^6`fop`r]~[ ^6kwq`r]~[ ^2wgopdfkv4[ ^+v4k4df^k{ag]_`ri`ri+kx[ df| [px2`rikdfkv$+dfwr]_£k]_`rkdfop`r]~[ ^+w pd%op^l]_^lxdr[| xdfk4k!x[ df| Xagikdrkc`fik[ 9`f]_|}]~o"`f]_[^<op` u ["ack4d u k£ k u wc]wl ["£ kdr^+k{}dfk u ¤¤o vv$d%o v4drk6]_dfk4|}k^`%wfsJ`r[<z9k{«v$9v u kwMXwq2v%i«ow`fik op^+{ v4v u kw]~^+wr]~drk{¾|0 u `r]~l df]{`rkv%i^]6kw%s" !Tk4|}[^+w\`fdfop`r]~[ ^¾[ x`ri+k<k4¤v$]~k4^+v4 [pxW`fik¿|<k4`ri[9{Î]~w«{9[^kR[^ÎoÌ|<]~^]~|0| wqdrxo v4kdf[ u k| ag]h`fiBi]~l i u [wfv$] u~u op`r[ df 2[^+{o drv4[ ^+{9]_`r]~[ ^2w4 #df[ u k4|wb[ x`ri+]~w`\2k<]_^£[ u £6]~^l¤+¾`r[R ~0|}] u_u ]_[^¾£"opdf]~o u kw a!kdrknwr[ u £ k{&`fikW^kao u l[ df]h`fi| ]_^¾± t ªg³ t B[ ôp^«[ u {9]wqi u op9`f[ $ # %X #k^`f]_+| ¡&±[] u k

(19) ¨&m¢bs& #kdfwr2kv`f]_£kwgopdfkz+^+o u_u {9kwfv$df]_k{= 2agi]v%i¾opdfkn[p`ri^6+|<kdr[+w op^+{]~^`fk4dfkwq`r]~^l+. ' G(GDHG;E?G*) + ,-/.10230547698:<;>=@?<ACBED;>=@? FG0<HI05HJ8KML9NO0<4P=QBED8RASD5L98RKTB>UWV<ACNSX%YBZAS?<8RAM[B?<8R6\[&;>=;>]Z8R[8E=XÛBZAM8R=<]ZL Y =<8R8RACL9=@]_BZ`@XSL9[L9aO;bXSL9BZ= 0dcWe\fhgijg9kOlZfmgnZo"loqp_rso5tZgoquvuEwgo5tZxy@z {<,E|O}{|Z~@x\yZ~Z~@,Z0 + yb-/.10230547698:<;>=@?<ACBEDx<F10@HI0<HJ8KML9NOx<70FG07@[q8RASXEx<HI0@HI0q7AS8E8R= x<;>=<? 0<4/0<.M8KP[&;>=0@4P`<`<ACBE:@L Y [&;bXSL9BZ= ;>=<?Q[B5?@8E6[&;>=;>]Z8R[8R=XML9= ;>8RASB5?@5=;>[L9GBZ`@XSL9[L9aE;bXCL9BZ=KML XCDZ;>ACL;>@698V?<8R69L X[B?<8R69NO0 nZ@woqlZnS77gw%EwSlSvfmxq|Z@hz9,E~ZZ|O}q,Z,E~@,Zx yZ~Z~@,Z0 + |b-/FG0P05^;>=<qx@ 0P0@7L9696mx5;>=<?F10^052;>ACRL;@0=XS8RACL9BZA`qBZL9=X^[8XC<B?<NUBZA^;GR6;>NCN^B>UJ8R6969L9`@XSL9sDZ;>ACL;bY XSL9BZ=;>6L9=<8R5;>69L XSL98RNO0O=110OL98R]Z698RAOxZ j0>7<;bXSXC;>NOx230O¡M8EL9=@Z8E=@NCv<69BZNSNOx;>=@?j70R¢I;>=69B8R[8R=£3;Z;>=5Y ?<8RASNOxq8R?<L XSBZACNOx ¤g9tb¥e5uOwnZwi_lZoqvu&lZ tnZwg¦f¥5i§jlZoqp§nSfhÏlZw%uPRnZw/oqno goquClw_nReqfmg¦ig kOlZfmgnZox`<;>]Z8EN y@,EO}5yZ|Z©@x\¡M8EL9?@8E69q8RAC]<x8EAS69L9= x@.P8K«ªIBZAS\x<yZ~Z~Z|@0\¬5`<ASL9=<]Z8RA^¢I8RAC6;>]<0 .

(20) + {>-j¬\0<0J8R=<NSBZ= xJHd0W702 R=<=<8RNOx<0 2 BZA8Z x ;>=<?<0 ¬@;>ASL9v 0J¬5E;>6;><698;>69]ZBZACL XS<[NL9=BZ`@XSL9[L9aE;bXCL9BZ=z BZ[`<@XC;bXSL9BZ=;>6J8:@`q8RACL9[8R=XCNE0@ dAC8R`<ASL9=Xs47.PH>2s¬Y M,Z,>©bY ~ZZ~>{<xq2;bXS<8R[&;bXCL9RNM;>=<? ^BZ[`@@XC8RA ¬5RL98R=<R8Zx<4PAC]ZBZ=<=<8.7;bXSL9BZ=;>6 HW;>qBZAC;bXCBZA%x@47AS]ZBZ=<=<8Zx<6969L9=<BZL9NEx ¬54/x<yZ~Z~>{<0@WB;>`<`q8E;>AsL9=XC<87 IACB>Y R8R8R?<L9=<]ZNGB>U*XC<8&,E~bXS"4P47 4 b%¬<¬@2 2 <6 XSL9?<L9NCRL9`<69L9=;>A%47=;>6 5NCL9N7;>=<?" 7`@XSL9[L9aO;bXSL9BZ=h2 4 1 G BZ=@U8RAC8R=<R8Zx47<]Z@NSX7|Z~jY^¬8E`5XC8R[q8RA1,ZxyZ~Z~>{<0 + ©b

(21) - <0108X%XCNj;>=<? ¬\0J j0 dAC 0 7`@XSL9[&;>669BOK XS<ACN%XGXCAC; S8RXSBZASXSBXS<8_[BBZ= 0 S n5woqlZnZo êeq ghuvp ZoqlZigvlZ Z§vfuEi§vx<y@myvz9,R{Z{b}q, >~@xyZ~Z~Z|@0 + b-14/0Av;>=<?5XO0d2 <6 XSL Y698D8R6^;>?;>`@XC;bXCL D8QNCBZ69@XSL9BZ=<NXSBqBZ<=<?;>A%3D>;>69<8Q`@ACBZ<698R[NO0 "lZf¥@uOilfhghE§QnS nZi7eq@fmlfhghnZox\|@,Z%,E|Zz |Z|Z|O}5|ZZ~@x*,E 50 + O-G£0qHI0qASL9]Z]ZNOx¢0<P0q¡M8E=@NCBZ= x;>=<?¬q0qs02 bBZAS[L9v\0q @ fmg twgp @fnZwglZ 0¬54723x I<L96;>?@8E6 Y `<@L;@x ¬4jxy>=@? 8R?<L XCL9BZ=xqyZ~Z~Z~@0 + b-14/0FG0BZ=<= x./05v0230q7BZ@69? x;>=<?Q I 0@HI0<WBZL9=XE!0 w@§v#f "%$û tgnZ&o "uOf¦¥<nEpZ§v0q.M<[q8RAs~@,L9=Q2 ^¬Y ¬5472 ¬58RACL98RNsBZ= 7`5XCL9[L9aE;bXSL9BZ= 0q¬54723x< d<L96;>?<8R69`<<L;@x ¬54/xyZ~Z~Z~@0 + b-/230L9NC@8EAE02 L9=<L9[L9aE;bXSL9BZ=;>69]ZBZACL XS<[N/UhBZA/DZ;>ACL;bXSL9BZ=;>6s?;bXC;;>NCNCL9[L96;bXSL9BZ= 0W'= $ûvvuEo(f 1u )>uE n*e " iuOofm§Gg¦&o +G5i_uOwgvlZ "uEf¥@nOpZ§Inw77fmin§he¥@uEwgh, "nEpuO¦ g¦o5tx@`;>]Z8RN|Z>{b}5|ZZ©@0q^^2£ ^x ,EZZ@0 + ,E~b-14/07ASL98Ks;>=< ;>=<? I0IHI0dWBZL9=XO0dHJBO;>6MRBZ=D8RAS]Z8E=@E8";>=;>6 5NCL9N_UBZA_`;>A%XCL XSL9BZ=<8R? 5;>NSL Y .P8KsXSBZ= <`q?;bXS8RNO0 +@i_uEwg§vR¥@u "lZf¥@uEi_lZfm.g -Zxq|Z@z {yZO}{Z{@xW,EZZy@0 + ,Z,-/230*HJ8KML9N;>=<? ¬\0Ij0d.7;>NC0I dAv;>XCL9E;>6P;>NC`q8RXCN&B>U[j<6 XSL9NCE;>698QBZ`5XCL9[L9aE;bXSL9BZ= [8XS<B?<NUBZA¢H Y ¬5%4 /j0G = ;>NCBZ= ^BZ=@]T;>=< ? ZBZNC8R`< FG0/¬5@L9=<=<8RAC6mx8E?@L XCBZASNO0x @ fhg§Cl ucWeqfmg¦ig kOlZfmgno lZoqp 1xs`<;>]Z8EN yZZ©O}5yZ@,Zx /7BZAC?@AC8RvXOx7M@83.M8RXS<8RAC6;>=@?<NOxyZ~Z~Zy@0 469K^8RA47E;>?<8R[L9 I<@Y 12 %3 . 69L9NS<8RACNO0 + ,Eyb-/230EHJ8KML9NW;>=<?¬\0bj0>.P;>NC 0b2 B?<8R6`@ACBZ<698R[N UBZA XS<8[@6 XCL9]ZASL9?BZ`@XSL9[L9aO;bXSL9BZ=1B>U<N%@N%XC8R[NJ]ZBED8EAS=<8R? ?<L 5\8EAS8R=XSL;>6J8R5<;bXCL9BZ=<NE0 S n5woqlZdnZ6o OguOofhg 7I niPeq@fmg¦o5txJ¦XSB&;>`<`q8O;>AvxyZ~Z~Z©@0 + ,E|b

(22) - <0 <0q2 BZA 8Z 0<W8RAv;>NSO;>698GBZ`5XCL9[&;>6! /7 NCBZ6 D8RASNO0<;>69;bXsXS<81%^472 yZ~Z~Z|BZ=@U8RAC8R=<R81L9= ¬5?<=<8x yZ~Z~Z|@0. 8.

(23)