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A Global Algorithm for Nonlinear Semidefinite Programming

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(1)A Global Algorithm for Nonlinear Semidefinite Programming Rafael Correa, Héctor Ramírez C.. To cite this version: Rafael Correa, Héctor Ramírez C.. A Global Algorithm for Nonlinear Semidefinite Programming. [Research Report] RR-4672, INRIA. 2002. �inria-00071913�. HAL Id: inria-00071913 https://hal.inria.fr/inria-00071913 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A Global Algorithm for Nonlinear Semidefinite Programming Rafael Correa — Héctor Ramírez C.. N° 4672 Decembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4672--FR+ENG. THÈME 4. apport de recherche.

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(61) § ¡HVJ¦ ™›c5p$_j^\^  X –bdš ž”_jd k abPRT d *_ ¤ G c * › x 4 c²N G)M

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(63) § s Ÿ¢¡G 4 S!G c * › x 4 c N G+M#J/J š s N G)M

(64) J V J &RŸ¢¡G 4 S  s N G)M

(65) J V J s *_ ¤ s«N G+M#J ¤ ª¦

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(67) § s Ÿ¡2G 4 S G c * › x 4 c N G+M ( JŠJ š s«N G+M ( J V J'&aQmW P”T d”m T )j (|G+M ( W 4 W J'& G)M ( JbT)j (|G)M ( W 4 ( W ( JW 4 X W X[Z \ AmW RQ ¶ ¬ L ¬ 1 –d 1¬ 1 ‚éTa‚]”h$eRpicœ5TWdRc• G+M#abPR1J T* hiT;mG+c5)M d (k J [NdRT<¤J]”_j^\[˜asl ‚éTa ` @  G)M0(J M ƒ3T*X _# ` m  ^NG)cJ0M h1(T J k ƒ0M _jT 0^\^—• 0M *([˜abP«mT dJabT p M0( _jd k p_ k [N]” h #h1]”mPKaiP”_ja™L L ™›c5pw_f^N^™›T;_5h1[\ƒR^\T*e3cf[\dJabh @ ž F T8eRpbcœfTabP”_a ™›c5G+p‚)M _j(^\J ^ * 6Q hi] B©m[\T dJab^Nl^\_fpiˆ5k Tfž 0M ( [`h aiP”T]RdR[`¤J]RTe3cf[\d5awhi_jai[`hs™›l=[\dRˆ

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(72)   GJI K L )j (|G)MœW 4 ( W ¬ ( J ¶ ¡H5~j¦ k T5žf•ŸT‚hiPRc• abP”_a G))M (W0QmW 4 ( W0Q2J [`h¥_e3cf[\dJaqaiP0_a–hb_ab[\h1”T;hAaiPR T /_fpi]0h1Pg0§ /W]RPRdR§ßO]”m ~fT;p  O * c  m 5 c ” d  m \ ^ ] _fd k aiPRTŸh1T<mc5d k §ßcfp k T p·hi] B©m[\T dJamc5d k [Nai[\cfd”h·™›cfp·abPRT¥c5egai[\SU[nK<_ai[\cfdWeRpbcfƒ”^NT;S ¡£f}5¦ O(PRTU‚_jˆ5pb_fdRˆf[`_jd _5hihicgm[`_abT k • [NaiPSU[\dR[NSU[K;_jai[\cfdeRpbcfƒR^\T S"¡Hj}J¦¥[`h jƒ1 G+MyW  3W 2e3W 4J ¯ T L G)M#J s Ÿ¢¡G 4 (  J s ¬ p( S G+M#J s *_ ¤ S G)M

(73) J ¤ ª s *_ 5¤  ¤ ª¦

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(78) J-p ¬ ( s * S G)M

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(80) J c8 JŠJbTQ 4;\RQ W 2 [ Q ¤ M c G+M#)M J ( ¤@c Oa&aQmQ ¶ N 8c  ’ a‚m;_jd ƒ0TT<_fhi[N^\l.j 1hiT T daiP0_a G)MœW  W 2e,W 4bJ,T G)M ( W0QmW 4 G+My( WW0Q2J J hb_ai[`h1G)”M T;( h(W0Qm_j^\W ^·4 ab( PRW0T;Q2hiJ Tmcfd k [˜ab[Nc5d”h O(PRT € T;hbh1[`_jd cj™ • [˜abPpiT<h1e3T;maŸabc8aiPRTœ_jpb[`_jƒR^\T;h  _ja [`h(ˆf[\œfT d ƒ=l A .

(81) . ª ) G )M (J s / ¬ ( ­ ª S ­ G)0M (J L s  . y v x . ­ Y Y . Q W ¡HA5¦ ªGJI   L j¢1 G)M0(W/QmW 4 ( W/Q2JT  s / A S ­ G)0M ( J ª S ­ G)0M (;J s S G))M ( J p S G))M ( J Y * ­ vyx *Y Y Y  Q Z* d • [NaiP ¯ T S Ÿ¢¡G 4 ( D DE I F N DGJICIKM L J V u  ­ ž”_jd k aiP”Tmcfd”Twcf™m pi[Nai[`m _f^ k [\piT<m¾ai[\cfd0h¥™›c5p eRpbcfƒR^\T S"¡Hj}J¦Ÿ[\h >'1 G))M (W0Q2JbT 2G+±>W —J@XyZ \ ] ? – d¯ L G)0M (J p ± s Ÿ¢¡G 4 (—J s°¬ p( S G)0M ( J-±'TQmW Y Y / ] ± u * N G+M)u ( J c X – › G VZX¡ N G+)M ( JŠJ " ¶ d uvœx * M ’ d• P”_ja(™›cf^\^Nc•$h(•ŸTW• [N^\^éh1ab_abTwaiPRThiT;m cfd k §ßcfp k T;pŸhi] B©m[\T dJa‚mc5d k [Nai[\cfd GI± p W —J ª j¢1 G+M ( W/Q W 4 ( W/QJ8  ± : RQ W Y 1 ™›c5p$_jd=l©dRc5dmK T;pic8œfT<m¾abcfp G+±>W —J@X >'G+)M (W/QJ O(PR[`h mc5d k [Nai[\cfd«m _fdƒ3T• pi[Na1abT d_fh ± p ª L G)M ( J“± s ­/ vœA x ¬ ( ­ ± p ª S ­ G)M ( J-± s ± p  ± s * ¤ S G)M ( J-± ¤ ª s * ¤  ¤ ª¦

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(91)

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(93) § \    

(94)   W ™›cfp$_f^N^ U TV0 Q ž=aicfˆ5TabPRT p • [NaiP ¡ 3 ‡<¦–•ŸTwc5ƒgab_f[Nd U p /uwvy] x ±  u * N G+M M u ( J U &  ^ ¤ ±  ¤¤ U ¤ ª W ™›c5p$_j^\^ U X VYX¡ N G+)M ( J _jd k _f^N^  W * • P”[\mP [NSUeR^\[\T;h(abP”_a / ] ±  u * N G)M u ( J X – › G VZX¡ N G)M ( JŠJ¶ d uwvyx * M.

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(155) G)k M   /± J ˆ5[Nœ5T d [Nd k ¡›V0‡<¦ O(PRTTng[`hsabT d”m Twcf™ 4  [\m h¥_jˆfd]”ƒ3_fTpb_fmdJc5aiS8T T e”]gƒ=aiTl*k [\dR_5T<hŸ¤5™›]0cf_j^\^N^\c[˜as•$lh;¡›? V{f¦A_fd aiPRT ™»_5m¾aAabP”_a [`hAe3c5hi[˜ab[Nœ5T T0dR[˜abT «c5piT eRpbT;m [\hiT ^\lfž Y=aiT e« ²® ¯ TVQ W ¡ ­ ¯ T ^ Y=aiT e ‡ ’ ™*¡›V{f¦–[\h hb_ab[\h1”T k • [NaiP 4gTV¡ ­WaiP”T d 4 ™TV¡ ­_jd k h1aicfeaiP”T^N[\dRT;_fp$h1T<_jpmP Y=aiT eK ’ ™Ÿd”cja;ž—ab_~5T ¡ ­ š x T q¡ ­ žé[\d”mpbT;_5h1T%®«ƒ=lKc5dRT8_fd k ˆfc aicKYJabT e ‡ • [˜abP XVG+Q W ^ J _RngT k mc5d”hsa_jdJa 4  ­  vtT”d”T M  š x T6M  s 4  ±  S R ­=± Ž ­     "& WE))  &"¢ m  L W S |( N /" > x M I) .ƒ |(™ m y m"&¢("&6  +6  "W /"   )        . "  1)("&™ " n0 ‘   ("  &) " ( g " -"4)  ? i g #(=WE))  &"   m"¢£ +( "  k  /"k50 › &x "- )mC+6  " (")*U -"W. ' œ "&"M"  "&  " k  " J¢0`|("0(—i02" m"& .   4  m"Zx &"M"  x"&  " G+M W  š ¶ ¬  š J  "  m"&  |" Š  "£ n . 2 ! I) . E# M1g m"Z&"M"  "&  " ? 2 m"&"#"  "&  "W  " |( 2G 4  š x W ¬  š x J " /"| C#(" (  * :  2 m"&/" J ²|("   ª W. Y m   J= 1i  q£   \  ª   ²J= &" |" “' m"  n . 2 ! I) 1EM M * . . . . . . . .

(156) ‡}. 

(157)   . “ M  (

(158) G)M JGB 4 c    “ I j¢G)M  W  š x W ¬  š x JNB QmW S G)M  JNB Q W ¬ x G N G)M  JŠJ š B Q #( Ÿ¢¡G 4  š x N G)M  J/JGB Q Y  Ž ±—±$#% ^ J O(P”T T;¤J]R[\œE_f^NT;d”mT ƒ3Tas•ŸT T;d8aiPRT ]Rd=ƒ3cf]Rd k T k d”T;hbhAcj™ *  "‚_jd k G 4  š x W ¬  š x J "$[`h k [\pbT;m¾a ™›pbcfS _2J ¡›VfV=¦ GI´J § G &J ‚—T at]”hth1]”eRe0cJh1TaiP”_ja  "*[`h‚ƒ3cf]”d k T k Ÿl ¡›VfV=¦ G+oJ •ŸT~=dRc• aiP”_ja$aiP”T pbT*[`h‚_jd€[Nd k Tn  hi]”mPOc#aiP”m_jcfa d0*m^\™] kT T* •Ÿ¯TwT e”*pic œ5 T‚™›cfabp$P”__f* a ^N^ [N™ GI\ ´J  [\ª h(dRcfa aipb]RTwaiPRT;d G &J PRc5^ k h ‡RpbcfS ¡›V2{5¦¾žR• [NaiP 4gT 4  žg•–Tª ~=dRc• aiP”_jaaiPRT©> hiT;¤J]RT;d”mT  (

(159) G)M J "U[`h k T<mpbT;_5h1[\dRˆ™›cfp_j^\4 ^   \ B  ª Qž¶ _fd k aiPRT;d  (

(160) G+MJ\T> ™›cfp hicfSU„‚T^Nm^(c5^\d”[NSUh1ab[N_jabh8dJa [Nd žRG &Jc5ƒgab_j_fpb[NT.dR[\mdRc5ˆUd”hi_jT;ˆJ¤J_j]R[\d.T d0™›mpbT;cfh*S"cj™ ¡»V2ab{5PR¦qT.abPRT n=T[`h1^\[NaiSUT d0[Na mT.cf™ 4  Q hi]”mP aiP0_a 4  \ 4 ™›c5p„_f^N^ @ @  \  ª žg• PR[`mP [\SUeR^\[NT<h(™›pic5S aiP”T^N[\SU[˜a‚_fƒ0cœ5T‚abP”_a  B Q ’ d k T T k žR[\dRT;¤J]”_f^N[Nai[\T;h  & c ± p k  ±  s°¬ x G N G)M JŠJ š Ÿ¢¡G 4  š x J s ¤&¬  š x ¤¤ S G)M J ¤c * G ¬ x G N G)M J/J š s ¤ S G+M J ¤ J & c ± p k  ±  s G * c * @ J G ¬ x G N G)M 2JŠJ š sl¤ S G)M J ¤ J c * G ¬ x G N G)M JŠJ š s ¤ S G+M J ¤ J e”picœ5T‚abP”_a  & c ± p k  ± c @ G ¬ x G N G)M JŠJ š s ¤ S G+MJ ¤ J&aQmW ¡»V5}5¦ * • P”[\mP [NSUeR^\[\T;h(abP”_a$[N™  B Q aiP”T d ¬ x G N G+M 2JŠJ š B Q _jd k S G+M NJ B Qm¶ ’ dRT<¤J]”_j^\[˜asl ¡›VJ}5¦Œ_j^`h1cWhiPRc•$hŒaiP”_jaƒc ± ± p kB  Q ±  B Q žf_fd k abcfˆ5TaiP”T p• [˜abP8aiPRT(™»_5m¾aaiP”_ja k5 › x " [\h ƒ3cf]” kd k  T k ž5[Na¥[`hAT;_5h1l*aicmc5d”m^\] k T$aiP”_ja O(P”[\h;žjabcfˆfT aiPRT;pq• [NaiPK¡›VJ f_J¦_jd k aiP”T$ƒ0c5]Rd k T k dRT<hih cf™ "5žg[\SUeR^N[\T;h aiP”_ja IY j¢G+M W 4  š x W ¬  š x NJ B Q ¶ ‡Œ[\d”_j^\^Nl5žRƒ=l k T ”dR[Nai[\cfdcj™  •ŸT%~Jd”c• aiP0_a L G+MJ p ±™T  s * G ¬ x G N G+M2JŠJ š sl¤ S G)M J ¤ JNB QmW Y _fd k ™›pic5S"¡›VJ f_5¦–•ŸTh1T;TtabP”_a Ÿ¡2G 4  š x /uwvy] x ±  u * N G+M M u J J s ¬ p š x S G+M  J-±  T c ± p k  ±  c L G+M  J p ±  B Qm¶ Y Y * O(P”[\h8^\[NSU[Na#_fd k aiPRTT;¤J]”_f^N[Nai[\T;h¡»V5 jTE¦*_fd k ¡»V5 5m;¦ž¥aicfˆ5TabPRT pU• [NaiP aiPRTƒ3cf]Rd k T k dRT<hih8cf™ ¬  š x "5ž _f^N^\c• ]”h aicU• pi[NaiT.  

(161) š Ÿ¢¡G 4  š x N G)M JŠJ T  

(162) š c Ÿ¢¡G 4  š x /uwvy] x ±  u * N G)M M u J J T  

(163) š ¬ p š x S G)M  J-±  T  

(164) * š c¬ p š x S G+M  JUT6Qm¶ Y . . . . . . . . . .

(165)      

(166) 

(167) #

(168)     

(169)  3    

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