Convergence of interior point algorithms for the monotone linear complementarity problem
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Convergence of interior point algorithms for the monotone linear complementarity problem J. Fre´de´ric Bonnans , Clovis C. Gonzaga. N˚ 2074 Juillet 1994. PROGRAMME 5 Traitement du signal, automatique et productique. ISSN 0249-6399. apport de recherche. 1994.
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(165) {JZ]jk(bY[\^bjt{Jb xyY[Z^xtb/Yuh[fibnZg¯n@b/(b/Y`_q._qp\]Z^Y[fih[}=anY[b/« §b/Yub/jt{JblfijQh[}yb+fpjzh[Y[Zzlzmt{Jh[fiZ^j1r_^{{JZ^Ylkfpj=\
(166) h[ZCh[}=b+¡tb/}t_/zfpZ]m=YZ2h[}=b
(167) n@bvmyb/jt{JbnV± ) ´_jtl,!± , ) ´`¥°sj h[}ybnb-{JZ^aaCb/jkhn$}=bjR b+Y[bW§b/YhuZonxtb/blQZ{WZ^jz(b/Y[\]b/jt{Jb b+aCb_gjoh[}=b-nxtb/blRfphu}R}=fw{`}Qhu}=b nbvm=bjt{Jb , ) {JZ^jz(b/Yu\^bn huZR¦/b/Y[Zy¥ O {Wh[mt_qpqp¢&h[}yb/Y[bfn+_{WqpZ(n@b{JZ]j=j=b{JhufpZ]j ¡tb/h@2b/b/j,hu}=bonxtb/bl,Zg {JZ]jk(b/Yu\^b/jP{JbZ , ) _jtl¨hu}=b xt_gY_gab/hub/Y`n ) _gjtl ) r=_]n"2b x=Y[Z(bVfij¨qib/aa_ cz¥ ¡tb/qpZ h[}t_h , ) : _ ±u % ) [ ) ) ´ , ) [ ) L ~ B± , L) ´ q u&b lzblkmt{Jb§Y[Z^a e }=b/Z^Yub/a z¥Ó+h[}t_h R ±²f§´ O qp\]Z^Y[fih[}=an"§Z^Y}=fw{`} ,± Y ) a2V ) +´ c _gjPl ) ) fwn ¡Z]m=jtlkbl _/¯_¢Q§YuZ^a ¦/bY[ZQfij,n@Z^abCnmy¡=« nbvm=bjt{Jb Rzh[}=bn@b+_qp\^Z]Y[fih[}=an \]b/j=b/Y_ghub+nbvm=bjt{Jbnn@mt{}oh[}t_oh Y ) Y Wr¡tb{/_mtnbh[}=b-nb/Yufpbn fijQhu}=b qpbaa_Vfn m=jk¡tZ]m=jtlkbl¥te }=fwn{/_^n@b fpjt{Jqimtlkbn7h[}=bV§Z]qpqpZfij=\+ab/hu}=Zzl=n R d e }=bx=Y[blkfw{JhuZ^Yu«s{JZ]Y[Yub{Jh[Z]Y._qp\]Z^Y[fih[}=a fpj Rh[}=fn._gqi\^Z]Y[fphu}=a _qphub/Y[jt_h[bnI_ wCj=b/«¬n[{_gqpfij=\ n@h[b/xtn_gjtl {Jb/jzh[bY[fpjy\+nhub/xtn fih[} ) _ _gjtl ) _ ^¥te }=b_gqi\^Z]Y[fphu}=afng«¬nh[bxRv(mt_^lkY_ghufw{/_qpqp¢C{JZ]jk(b/Yu\^b/jzh/¥ d (}=Z]Y[h³nhub/xtn_gqp\]Z^Yufph[}yan Rzh[}=bn@b abh[}=Zzl=n mPnb ) 7 \ rtªzblErt_gjPl ) ¡tZ^m=jPlkblR_/_/¢§YuZ^a ¦/b/YuZ ±²bfph[}yb/Y ) _ Z]Y ) Yubnm=qih`n$§YuZ^a _+qpfij=b nb_gY`{`}E´¥=e }=b-_gqp\]Z^Yufph[}yan _Y[b qpfij=b_gYuqp¢C{JZ^jz(b/Yu\^b/jzh/¥ ±²fif²´ O qi\^Z^Yufphu}=an"§Z]Y }=f{} ,± Y ) a2V ) +´ c r ) ]r ) R®h[}yfwnfijt{Jqimtlkbn _gqiq «¬nmyxbY[qpfij=b_Y[qp¢ {JZ]jk(b/Yu\^b/jzh_gqi\^Z]Y[fphu}=an ¡t_]nblQZ]j&©b/huZ^jRn@h[b/xPnV_gjtlQqifpj=bn@b_gY{}=bnJ¥te }=b
(168) n@bvmyb/jt{Jbn_Y[b-{JZ^jz(b/Y[« \^bjkhr¡ym=h hu}=bVqpfiafphIxtZ^fijkh2a_/¢ j=Z]h ¡tbVh[}yb {Jb/jzh[Y_gqtZ^x=hufpa
(169) m=aRr]lkb/xtb/jtlzfpj=\Z^j}=bh[}=b/Y ) ) _ [ Z]Yj=Z^h¥ ±²fifpf§´7e }=bqpZ^jy\^bn@hVn@h[b/xxt_h[}Q§Z^qiqpZfpjy\
(170) _gqp\]Z^Yufph[}ySa RE,± Y ) a2V ) 1´ c r 3 q ^ckr ) _ ^r_gjtl ) nmP{}&h[}t_h U± Y ) : a2V ) :J´ _ 7¥ ==YuZ^a h[}ybhu}=b/Z^Yub/aRrth[}=bCfph[bY`_ghubnV{JZ]jk(bY[\^b !¥ Vnfij=\h[}=fn¤¬_]{Jh_gjtl h[}ybY[bn@m=qphn Zs IR { (}t_j=b(±u/^zg´r ab/m R IR { (}P_gj=b(r b³{JZ]jt{JqpmPlkbh[}t_h hu}=bV_qp\]Z^Y[fih[}=a fn7nm=xtb/Yuqpfpjyb_gYuqp¢ {JZ]jk(b/Yu\^b/jzh/¥ze }=fn ab_jtn h[}t_h )
(171) _gjtl bV¬_gqiqEfijQ{_^nbC±²fif²´7_¡tZg(b ¥ ±²fi´³¼1_Y[\^bnh[bxtnxt_h[},§Z^qiqpZfpjy\Q_qp\]Z^Y[fih[}=an R§YuZ^a ±²fpfif²´rEfph fwn{Jqpb_gY h[}t_h h_g£zfpj=\ ) m rPªkblr ) _ _jtlQU± Y a2V 7´ c nm w{Jfib/jzh[qp¢ j=b_gY7h[}=bZ]x=h[fia_gq(¬_^{Wb(r^h[}yb}=Z]qpb³nbv(m=b/jP{Jb fiqpq=¡tbfpj =¥ ä§å ã äwË.
(172) G 41( 4 & !#"%$)' !)."+ U)7 /)0 (2"3) "5 , ,6(8 0: ". /. e }=b/j Y ) Y ._]n§Z^Y n}=Z]Y[h7n@h[b/x-_gqp\]Z^Yufph[}yanJ¥==Z^Y_gj
(173) _Y[¡=fih[Y_gY[¢ fij=fphufw_gq(xtZ]fpjzh.fpj rgb_]{}fphub/Y`_h[fpZ]j {/_j¡tb {WZ^jtn@h[Y[mP{Jh[blC_^nfpjhu}=bVx=Yufpa_qq_gYu\^bnhub/xtn _qp\]Z^Y[fih[}=a §Z^Yqpfij=b_Y$xyY[Z^\]Y`_aafpjy\+lkb/(bqpZ^xtbl ¡z¢ zZ]j=¦_g\ _±u/^zg´³_gjPl ¡z¢o"Z(Z n _gjPl#"Vfw_q ±u/]´ RmPnb
(174) _nbv(m=b/jP{Jb Zly_gaxtblo©b/h[Z]j&nhub/xtn h[Z
(175) Z^¡yh`_gfij,,± Y ) :Ga2V ) :J´¯nmt{`}Qhu}t_gCh 1 Y ) :2V ) : ) , ) %
(176) ;1 3 7¥te }=b-_gY[\]m=ab/jzh_g¡tZ(b+n@}=Znh[}P_gh §Z^Yq_gYu\^b ]_gqpmybn"Z. /
(177) Z]j=qp¢Z]j=b+©b/h[Z]jQn@h[b/xfiqpqE¡tbj=b/blkblE¥ -'7 !
(178) $#% ' (
(179) )R !R + 2
(180) % % *. ¼Eb/h >d ]3h
(181) ¡tb+_jQZ^xtb/j&n@b/h/r_gjPl {R d ] _lk/f PEbY[b/jzh[f_g¡=qib§m=jt{JhufpZ]jQfphu}QqiZz{/_gqiqp¢o¼Efpxtnu{}yfph[¦ {JZ]jkhufpjzm=Z^mtn \]Y`_]lkfpb/jz*h Yjc ®,± Y ´¥ ub{JZ]jtnflkb/Yhu}=b m=jt{JZ]jtnhuY`_gfij=blj=Z]j=qpfij=b_gYxyY[Z^\]Y`_aafpjy\ x=YuZ^¡=qib/a afij=
(182) fp
(183) a
(184) fp¦/b ®U± YE´ q u&bVn@}t_gqiqEn@h[mtlz¢\]b/j=b/Y_gqtx=Y[Z]xtb/Y[hufpbn7Zglkbnu{Jb/jzh2_gqp\]Z^Yufph[}yan§Z^Y2h[}=fn$x=YuZ^¡=qib/aR¥]e }=b§Z]Y[a_h §Z]Y hu}=b _gqi\^Z]Y[fphu}=an7fwn h[}yb¤§Z]qpqiZgfij=\=rk§Z]Y_\]fp(b/j Y cSd ] h R tºG Dw l 2 B !#" / R/_ ´ _
(185) lkZ ®,± YE4 Y. ) 3: R/_ Y ) [
(186) )) / RQ_A/ [ ]r. . }=b/YubV§Z^YVb_]{}6/tr ) cld ]3hC_gjPl
(187) ) c ±sa[ ´_Y[b-nmt{`}Qhu}t_ghY ) : c .¥ O jo_gqi\^Z^Yufphu}=a fpqpq1¡tb {/_qpqpbl#_ `lkbn[{Jb/jzh_gqi\^Z]Y[fphu}=a fÐIh[}=b/Yub b/ªkfnhn 7
(188) nmt{`}h[}t_h §Z^Y³_gqiq / c d 0r ±²f§4´ 1 ) 1 ? 1 ®,± Y ) ´ 1r ±²fif²´ IU± Y )([
(189) ) ) ´ 3 ®U± Y )/´ %
(190) ) 1 ) 1 1 ®,± Y )´ 1¥ O nbvm=bjt{JbU± Y ) ´7fi j Rfpqpqt¡tbn[_fwl{JZ]axt_]{Jh fÐ1fph2fwn{JZ]jkh_gfij=blfijnZ]abV{WZ^axt_]{Jhn@b/h .r _gjPl¨_+xtZ^fijk*h Y c!fwnn@h`_ghufpZ]jt_gYu¢¨fÐ" ®,± YE(´ _ k¥ e }=bj=b/ªzh$qib/aaC_n@}=Zn$hu}t_gh2h[}=b/YubV_Y[bh@ Z+_qphub/Y[jt_h[fi(bn§Z^Y2_lkbn[{Wb/jkh2_qp\^Z]Y[fih[}=Sa R]b/fphu}=b/Y.hu}=b nhub/xtn_gY[bon@ m w{Wfpb/jzh[qi¢Qqw_Y[\]bh[ZRYub_^{`},h[}ybZ^x=hufpa_q2nb/hr®_jtl,h[}=bj _qpq2_^{{Jm=a
(191) m=qw_h[fpZ]jxtZ]fpjzh`n+_gYub nh_ghufpZ^jP_gY[¢ r=Z^Yh[}=bnhub/xtn_Y[b+n@}=Z^YuhV_jtl¨hu}=bnbv(m=b/jP{Jb {JZ]jk(b/Yu\^bn huZ
(192) _jyZ^j=«¬nh_gh[fiZ^jt_Y[¢oxZ]fpjzh/¥ u&b _gYub m=jt_/_Y[b Zg7hu}=b b/ªzfwnhub/jt{JbZg h[}=fn Y[bn@m=qphfij¨hu}=b qpfih[b/Y_gh[myY[b(¥ ãIã Ú WìJî. .
(193) J. *#v$
(194) } /XQ$. ¶ , $2$"5@!# + 9$' U± Y ) ´ !) !) )
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(196) ) _ [ ± ´ 9G ¥ Z^jtn@fwlkbY n@bv(m=b/jt{Jbn ±UY ) ´2a/± ) ´2a±
(197) ) ´\^b/jyb/Y`_h[blQ¡z¢Qhu}=b_qp\]Z^Y[fih[}=a*rE_jtl_]n[n@m=ab+h[}P_gh ±UY ) ´7fwn{JZ]jkh_gfij=blfpjo_+{JZ]axt_^{Wh .¥ ±²f§´I°² )
(198) ) [ r(hu}=b/j )
(199) ) 1 ) 1 [ r(¡tb{/_mtnb 1 ) 1 ? 1 ®±UY ) ´ 12fwn.¡tZ^m=jtlzblfpj ¥ e }=b/j*±UY ) ´fwn2_ _gmP{}z¢n@bv(m=b/jt{Jb³_gjPlhu}kmPn7Y ) YE rzfphu} Y c ¥ubVa
(200) mtn@h.x=Y[Z(bVhu}t_ghY fwn$jyZ^h. n h_ghufpZ^jP_gY[¢ ¥ ¼Eb/h _'GYjc ®±,YE´}_ *¡bh[}=bn@b/hZg2n@h`_h[fpZ]jt_gYu¢xZ]fpjzh`nfpj ¥t°² S _ zr=h[}yb/jh[}=b/Yub fwn7j=Z^hu}=fpjy\huZ+¡b x=YuZ(blE¥-|hu}=b/Y[fnb(rzqpb/h ¡tb_\]qpZ]¡t_gqI¼Efixtn[{`}=fphu¦ {JZ]jtnh_gjzh §Z^Y Qfij ¥ (fpjP{Jb 1 ) 1 ? 1 IU± Y ) ´ 1r 1 ) 1K3 - 1 ®,± Y ) ´ 1r=}=b/Y[bH- 7 +fwn_
(201) {JZ^jPnh`_jkh¥t°sh §Z^qpqiZn h[}t_h 1 Y ) : % Y ) 1F3
(202) ) 1 ) 143
(203) ) - 1 ®U± Y ) ´ 1Gq O §hub/Y_jQfphub/Y`_h[fiZ^j1r 1 ®U± Y ) : ´ 1 b 1 ®,± Y ) ´ 1 % 41 Y ) : % Y ) 1Ga b ±[ % -S
(204) ) ´ 1 ®U± Y ) ´ 1Gq (fijt{Jb
(205) ) kr=§Z^Y / n m w{Jfpbjkhuqp¢
(206) qw_gYu\^bC % -S
(207) )7
(208) _gjPlQqpZ^\t±u % -S
(209) ) ´ b % > -S
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(211) ´ 1 ) 9 : ±u % -S
(212) 2´ a )
(213) _gjPl qiZ^\ 1 IU± Y ) ´ 1ob qiZ^H\ 1 ®,± Y )
(214) ´ 1[ ) 9;: qiZ^\t±[ % -S
(215) ´ b qpZ]\ 1 ®,± Y )
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(219) °sh"§Z]qpqiZgnh[}t_hVqpZ] \ 1 ®U± Y ) ´ 1 }t_]nV_qpZ b/Y¡tZ]m=jtlErI_gjPlQh[}zmtMn 1 IU± Y ) ´ 1 fwn¡Z]m=jtlkbl_/_/¢§Y[Z^a ¦/b/YuZ=rzbnh_g¡=qifwn}yfpj=\±§f²´`¥ ±²fif²´ O n[n@m=abVj=Z h[}t_h )
(220) ) _ [ ¥ ==Z^Y³_gjo_gYu¡=fphuY`_Y[ ¢ 7 krtlk b rtj=b _'GY c 1 ®,± YE´ 1 7 *q ä§å ã äwË.
(221) G 41( 4 & !#"%$)' !)."+ U)7 /)0 (2"3) "5 , ,6(8 0: ". /c. ¢Clkbrtj=fih[fiZ^jZg lkbnu{Jb/jzh_gqp\]Z^Yufph[}yaRr. (fijt{Jb R fwn ¡tZ^myjtlkblfpj . I±UY ) ´ 3 I±UY / ´ %
(222) ) 1 ®±UY ) ´ 1 1 ) 1Gq ) _jtl 1 ) 1 ? 1 I ±UY ) ´ 1r=2b {WZ^jt{JqimtlkbVhu}t_gh
(223) ) 1 ®±,Y ) ´ 1 L ) . [. . q. ©Z ^_ nunm=aCb¨¡z¢ {JZ^jzh[Y_^lkf{Jh[fiZ^j h[}t_h±UY ) ´ }t_^n
(224) _gj _^{/{Jmyam=q_ghufpZ^j&xtZ^fijkh Y _'/ cSd 0 Y ) c! *fn fpj rPj=fphub(¥ ==Y[Z^a ±sc´r=fphu} 1 ®±UY ) ´ 1*b $§Z]Y / c r
(225) ). g_ jPl2b{JZ^jt{Wqpmtlkbhu}t_gh7hu}=bnb/h7Z1fpjPlkfw{Jbn ¡tb h[}=bortYnhfpjPlkb/ªfij ¥te }=bj Vn@fpj=\o±UK(´`rt bnb/bh[}t_h _^n. ) . c. ¬± c´ ¥7e }yb/j h[}=bonbh ±,K´. [ a. fn._gqnZ fij rtj=fih[b(¥>==Z^Y7b_]{}. _ d0 %. 1 Y ) % Y ) 1K3 ) 9;:
(226) 1 1*_ ) /jc \]Y[ZnJr. 1 Y ) % Y ) 1 . ~. . ) 9;:
(227) ). / c. rqpbh./. . 7 /. ±@^´ q. k r_gjtlQhu}kmtnh[}yb+nmy¡tnbv(m=b/jP{Jbn ±UY ) ´ ) _gjPl ±,Y ) ´ ) }t_/(bhu}=bn[_ab_]{/{Jm=a
(228) m=qw_h[fiZ^j&xZ]fpjzh`nW¥ m=h+_qpq_]{/{Jm=a
(229) m=q_gh[fiZ^j&xtZ^fpjzh`n Z
(230) ±UY ) ´ ) n[_h[fnN§¢ 1 ®±,YE´ 1 3 W¥e }=fwnfpaxyqpfpbn Y c r{JZ^jzh[Y_^lkf{Jh[fij=\+h[}=b+}z¢zxZ]h[}=bnfwn_gjPlR{JZ^axyqpb/hufpj=\+hu}=b x=YuZ(Zgu¥ ¶k/ I$¶z»»I¶k( ¶ Ht tºtG D¿ Z]jtnflkb/Yh[}=bRaCZzlkb/q³_gqp\]Z^Yufph[}ya =¥p*_gjtl _ nbv(m=b/jt{WbQZg [xtb/Yuh[m=Yu¡t_ghufpZ]jtn Q± ) ´2fpp j d ]3hy¥ O xbY[h[myY[¡tblQ_qp\]Z^Y[fih[}=a fwn_gjR_qp\]Z^Y[fih[}=a h[}t_h"§Z]qpqpZnh[}=b aZzlzb/qsr ¡=m=h}P_^nh[}=bnhub/x\^fi(b/j¡k¢ . . .
(231)
(232) . . . Y. ) :. RQ_ Y ) [.
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(234) {/_gxP_^{Jfih¢Z_Clkbn[{Wb/jkh³_gqi\^Z]Y[fphu}=a huZ5rtjtl&nh_ghufpZ^jP_gY[¢oxtZ^fpjzh`nfwnj=Z]h _ Pb{Wh[blE¥ m=hjyZg h[}y7b rtY`n@h xt_Y[h7Zg h[}=bqpb/aCa__g¡tZ(bfn.j=Z^h7\^mP_gY`_jkhub/blErzn@fpjt{Jb'{Jqpb(b/Y xtb/Yuh[m=Yu¡t_ghufpZ]jtn$afi\^}zh.lzY[fp(b h[}yb n@bv(m=b/jt{Jb³h[Z
(235) _
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(246) nbvm=bjt{Jb(¥ e1Zx=YuZ(b±§fpf§´`rtlkbrPj=b 7_^n¡bW§Z^Y[b-_gjtlQjyZ^h[bh[}t_hmPnfpjy\h[}=b¼Efixtn[{`}=fphu¦{JZ^jtlzfph[fiZ^j§Z^Y fijQhu}=b {JZ]axt_^{Wh r=h[}yb/Y[bVbªkfnh`n_+\^qiZ^¡t_qI¼Efpxtnu{}yfph[¦¤{JZ^jtn@h`_jkh ¤§Z^Y Qfpj _jtl¨}yb/jt{Jb I±UY ) : ´ _ ®U± Y ) [
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