## A Coordinate Gradient Descent Method for Nonsmooth ... - MIT

Paul Tseng. Department of Mathematics. University of .... is nonconvex, then an example of Powell [47] shows that coordinate descent methods can cycle among ...

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 ? JcCKJ  J`eQJmH     U   U   OÈ Z¥:ÝBBsmSL.   \$  b : .(&\$s %   U   U b :5P. &îs %   U   U DUABC XbeQe  8iSRKA¥:ÝE[sXE=j:T. &îs % ( ;>WYfe>Z 6   U   U >7\$- XE=j¢?7JM=7\cJ 6   >7\$-   ~ÀeQR A7S R ;Q=7\MJ 6    U   U >7;>R!PoAB_7=7j7JMj\$S7P[ZYfXERKR ;Q=7hkHKA°X±R _7P7RKJMÁ_7Jc=7\MJ±;{D =7Jc\MJcR R^XEC5ZBS  J3\[email protected] RK_7WiJnHK?XbH  6    U   U >7M- RKABWiJ   vXb´I;>=7h°H^?7JeQ;>WY;{HVXER  8 ' - 0 ;[email protected]^?7J ;Q=JcÁ_XEe>;>H|Z¤:=  ; :K:E . " sc:E .&î"s ! &\$s % K   : g s    .   : g s     ` ' X°\MeQJdXbCV\cAB=[H^CKXEj7;>\MH^;>AB=  vV?_7R 6   >7 -   FT_f7fgAERKJ H^?XbHcS7;Q=XEjj7;>HK;QAB=\$S  0  &   0iDUABCXEe>e  vV?7JM=\$SDUABCVJdXE\µ,? gS    +8 % 0  :    s  :     s + 8 &   :     s + 8  :     s + 8  %  0 V vV?Jc=¥:[s!;>[email protected]½JMWYWYX`  kZI;>JceQjR     U #" :  OK  s U K S !  % !  .    % !:  % s % U   U V :ÝTs FI;Q=7\MJ 6   >7 - STH^?7;>R³;>WYfeQ;QJMR 6  " :   OK  Ds >7 -   :U\sFT_7f7foABRKJH^?7XbH"  ;QR1\µ?7ABR Jc=rP[Z°H^?JÉ XE_7R R  FIAB_H^?  Jce>e  *3C _7eQJ³XE=j  0  &   0±DUABCXEe>e  FI_7f7foABRKJ±HK?Xb+H ¤ ;QRÀXa\ce>_7R HKJcCÀfgAB;>=[HÀAbD 6   >  ½ JMH 6   >7·PoJkXYRK_7P7R JcÁ_7JM=7\cJ`AED 6   >°\cAE=LEJcC hB;Q=7h HKA    vV?7JM=\$S P[Z¥:UPtsmS 6  " :   OK  3s >7 -  3È ZHK?7J`É XE_7R R  FIAB_H^?  Jce>e  *iCK_eQJ]: Å s(SoHK?7;QRÀ;Q=yHK_7CK= ;>WYfeQ;QJMR 6 *  >7 - S  ?7JMCKJ  Jj7JM=7AEH^J3DUABCVR ;QWif7eQ;>\c;{H|Z(*    " :   s 1È Z£:RK\3AELD % ZI;QJMeQj7R #&%Y: g < s K   : g )s :  . g "s !  :  . î )s  &¼ : . g "s ! #f%Y: gs Z  * , '  ?JcCKJ &  ;QRÀXE=[Z\ce>_7R HKJcCnfoAB;>=HÀAED 6 &  >7  FT;>=7\cJ &   0rDUABCÀXEe>e  8iS &    ,  vV?7;QR³RK?A  R HK?XbH  "  : g s  `R AiHK?XbHcS7P[Z½JMWYWYX°  BS y ;QRX°R H^XbH^;>AB=XEC5ZrfoAB;Q=[HAED   

U: jtsÀFI_7f7foABRKJ H^?XºH  ;>R\µ?7ABRKJM=¼P[Z]H^?7J±É XE_7RKR  FTAE_H^?  JceQe  °C _7eQJbStXE=7j  0  &   0aDUABC³XEe>e  FT_7f7foABR JHK?XbH Y  ;QRX3\MeQ_7R5H^JcC1foAB;Q=[HAbD 6   >  ½ JMH 6   >[email protected]X3RK_P7RKJMÁI_Jc=7\MJnAED 6   >³\cAE=LEJcC hB;Q=7h  HKA   1È Z£:7 - iXb=7j 6    >7 - S  ?JcCKJ  JOjJc=7AEHKJ      " :  OK  s  FT_f7fgAERKJ¼DU_7C5H^?7JMCKWiABCKJH^?XbHrJc;>HK?7JcC % ;>Rr\cAE=HK;Q=_7AB_RrAB=#j7ABW % ABC  (+ * (  ¼DUABCrXbeQe- ÂABC ;>=D    (  }·J  ;>eQeîR ?7A  H^?7XbH :ÝBBs 6 " :   OK  Ds >7 -  V vV?Jc=\$S\$P[Z:5EsmS 6 " :   3s >7 -   FT;Q=\cJESgP[Z¥:BsmS:5d[smS:Ý[s;Q=½ JMWYWYXa  BSîXE=7j&   0tS  J XbeQRKA`?XdLEJ " :   7 -   vV?7Jc=îSgXbCKhB_7;>=7h°XERV;>=rH^?7J3fCKAIAEDAED:=HAbD  }¢J f7CKALBJa:ÝBBs!P[[email protected] \cAB =[H^CKXEj7;>\MH^;>AB=  FI_7f7foABRKJ H^?XºHO:Ý[Bs!;>RDe rH^?XºH         !\$#&%a:   !  Ls .c#&%Y:   s  \$  !: . &îs         #     !    K #      ! :    s       ! #f%Y:   !    L s .c#&%a:   s   :    )s       .¦:    )s  &     ' K   ?7JcC J!H^?7J¿e¯XER5HR5H^JMfk_7RKJMR:ÝBs;>=k½ JcWiWaXÀ    FI;Q=7\MJ 6   >7 - ÀXE=7j 6   >V;>R/PoAB_7=j7Jcj\$SBHK?7J eQJMD¹H  ?Xb=7jOR ;Qj7JH^JM=7j7R/HKAnqMJcCKAXER  8  ' - 0  FT;>=7\cJ 6    >7 - nXb=7j 6 &  >V;QRPoAB_7=7j7JMj\$S H^?7JC ;QhB?[H  ?XE=jrRK;>j7JnH^JM=7j7R!HKA°qcJMCKA`XER  8 [' - 0  vV?I_RH^?7JÁ_Xb=HK;>H|ZYPgJmH  [email protected]^?7JMW XEeQR AOH^JM=7j7R!HKAkqcJMCKA`XER  8 ' - 0  FT;>=7\cJ  ;QR¿RKWiATAEHK?aR AkH^?XbH 6 #  >7 -   : g s(S :ÝBBs DUABeQe>A  R  O Ï  ¡ FT_f7fgAERKJ  (+ * (  ÀDUABCVXEe>e 1È ZYDU_C H^?JcCfXbRKRK;>=7h±HKA`X±RK_7PRKJcÁ_7JM=7\cJ;>D=Jc\cJMRKRKXEC ZES  J \dXE=·XERKR _7WYJ`H^?7XbHJc;>HK?7JcC    iDUABCXEe>e  8 /ABC   ; iDUABCXEeQe  8   Ê|=¤H^?Ji»C R H RK_7Po\dXER JESH^?JaR^XbWYJaXECKhB_WYJM=HkXER±;>= ÇXER J¤:Bs f7CKALEJcR:ÝBBs  Ê|=·H^?7JaRKJc\MAB=7j¥R _7Pg\cXERKJbS  J ?XdLBJ DUCKABW,H^?7J ~nC WY; l A°CK_7e>JH^?XºH :   !  _s ( :   "s ! " \$  ABCcSJcÁ_7;>LbXEe>Jc=[H^e{ZBS    s .  :   "s !:5 . " ^s #b: %Y:   !  Ls . %Y:   sK_s ( " : #     ! &        s  :   !  L  

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DUABC XEeQe  8   FT;Q=\cJ " ; B S 6   >7 -  S 6   >7 - SXE=7j 6   >7 ;QR3PoAB_7=j7Jcj\$SH^?;QR RK?7A  RH^?XbHe>;QW;>=D : %a:  !  s. %a:  sKs b   FI;Q=7\MJ

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 b \$   #     ! &        ! #f%Y:  !  sL. #&%a:  s % ' H^?7;>R³;>=[email protected];QJMeQj7RH^?XºH 6 \$  >7 -   vV?7Jc=\$StH^?JORKXEWYJXEC hB_7WiJc=[HÀXER³;Q=yÇXERKJa:Bsf7C ALBJcR :Ý[BsXbeQRKA  }¢J ?XdLBJ DUCKABW':5Es¿H^?XbH " :   OK  s  #      !  :    s  &     ! #f%Y:  !    sL.c#&%a:  s %  8  V FI;Q=7\MJ  ;QR1R WYAIAEH^?îS 6   >7 -  S 6    >7 - SIXE=7j 6 &  >7 ;QR1PoAB_7=7jJcj\$SH^?7;>RGH^ABhEJMH^?JcC  ;>H^?y:ÝBBs ZI;>JceQjR 6 " :   O  3s >7 - S\cAE=HKC^XEj;Q\MHK;Q=7h:Y;QRn\µ?7ABRKJM=¢P[Z¼HK?7J°hBJM=7JcCKXEeQ;>qcJcj£É XE_7R R  FTJc;>j7JceGC _7eQJ]:5Bs(S"% ;>R3P7eQAI\µ´  R JcfXbC^XEP7e>J  ;>HK?kCKJMRKfoJc\mHH^A  DUABCXEeQe gS[XE=j°RK_7f    ;G0  vV?7JVe¯XbH H^JMC/;>WYf7e>;QJMR 6   >;>RPoAB_7=7j7JMj  FI_7f7foABRKJ±HK?Xb+H ¤ ;QRÀXa\ce>_7R HKJcCÀfgAB;>=[HÀAbD 6   >  ½ JMH 6   >7·PoJkXYRK_7P7R JcÁ_7JM=7\cJ`AED 6   >°\cAE=LEJcC hB;Q=7h HKA   1È ZaDU_C H^?JcCfXbRKRK;>=7hkH^AkXkRK_7P7R JcÁ_7JM=7\cJO;>D=7JM\cJcR R^XEC5ZBS  J3\dXE=XER RK_7WiJnHK?XbH 6   >7 - RKABWiJ   XE=7j   ¶DUABC±XEeQe  8  FT;Q=7\MJ    % 0DUABCkXEe>e gS  J]?XdLBJ    % 0   , yÈ Z¢HK?7J jJM»=7;{H^;>[email protected]D   XE=j   GS  J?XdLBJ DUCKABW':Ts¿HK?XbH #     !  :   )s       !\$#&%T:   !  s  K #   :  .9  s  ! : .9  )s     :  .   s  !\$#&%T7: Bs Z  * , V  FI;Q=7\MJ 6   >7 - ±P[Z:UPtsmSTfXERKR ;Q=7hkHKA±H^?JeQ;>WY;{HXER  8 ' - 0 XE=7jr_RK;Q=h±H^?7JnRKWiATAEHK?7=7JMRKRAED D %TkZI;QJce>j7R  XE=je>RK\3AEL #&%T7 :  [_ s K   : g s  : . g s  !  :  . g s    7:  . g s E ! #f%H7: Es Z  * , V vV?;QR±R ?7A  R±H^?7XbH    :  OK s  R AyH^?XºHdSGP[Z·½JMWYWYX  ES   ;QR±XR5HµXbHK;QAB=XbC Z¤foAB;Q=[HOAED  ;{H^? C JcR fgJM\MHH^A]H^?7J`\MABWYfoAB=7JM=[H^R ;Q=7jJm®TJcj£P[Z GS;  J  S  :  OIs b @DUAEC XEeQe   * ,  ;{H^?  B   rDUABC       FT;>=7\cJ 6   >M7 - SHK?7J PgAE_7=7j7JMj7=7JcR RÀAbD 6   >7¤;>WYf7e>;QJMR 6   8 >7 -    vV?7;QRV;>[email protected][email protected];>WYf7e>;QJMR 7 - ¤P[Z¦:\ 7 -    Ç!AB=[H^;>=I_;Q=7h¤;Q=¥HK?7;QR`WYXE=7=7JMCcS  [email protected]H^XE;Q=¥HK?XbH 6   8 >\ .  vV?_7RMS  J°\dXb=¢Xbf7f7e>ZH^?J`XEPgALEJ`XECKhB_WYJM=HnH^A 6  65 >7H^A 6   65 >7 -  SîDUABCÀ²   ' V V5V ' Q AEPHµXE;>=  :  O Ts b     * ,  ;{H^?  B        5 ' ²   'c ' V5V V ' .# '    ?JcCKJ 3 'K 8 ' V5V V 'K  + 8 XECKJi=7AB=7JMWYfTH|Z¤RK_7P7R JMHKRkAED 6  ' V V5V ' a >  ?7ABRKJi_7=7;>AB=£JcÁ_XEe>R 6  ' V V5V ' a >FOR JcJ :Bs  FT;Q=\cJ  ;QR³j7;54îJMCKJc=[HK;¯XEP7e>JXE=7jM% ;>R³PeQAI\µ´  RKJMfXEC^XbP7eQJ  ;>HK?C JcRKfoJc\mHH^A 3 'K 8 ' V5V V 'K  + 8 SH^?;QR ;>=]H^_7C =;>WYf7e>;QJMR¿H^?7XbH  :  OIs b kDUABCXEe>e   *,TS7RKA ¼ ;>RX`R5HµXbHK;QAB=7XEC ZafgAB;>=[HAED      

:¹DµsFT;Q=\cJ   ;QR \µ?7AERKJc=¥P[ZHK?7JY~ÀCKWi;l [email protected]_eQJESJM;>H^?JcC        ABC Jce>RKJESP[Z¢½[email protected] Å := U  .   U Z  * , V  # . ]c    UÒ        ª  "    «ª  b WY;>=    : gs  )¨m«^¨¼¨  Q§ § KªE­{ªE«µ§  ( ª  " *E(¶§ )  )ª 

7j ;>R H:'   e £RK_ ei\c;>Jc=[H^e{Z·eQXECKhEJES  ?JcCKJ(  (,¼XE=7j \$   eQ;>W   :   &s O1RKJMJ Ä BSGfXbhBJyEºÆ  ABC HK?7J3=7AB=RKWiATAEHK?r\dXbRKJES  J3\dXb=7=7AEHVjJcCK;{LBJHK?7;QRRKXEWiJ;Q=7JMÁ_XEeQ;{H|ZiP7_ H;Q=R H^JcXEj  ABCK´  ;>H^[email protected]  JdXb´EJcC ;>=7JcÁ_XbeQ;>H|Z  ?7JcC JcP[Z]H^?JÁ_Xbj7C^XbHK;Q\3H^JMCKW U    8 .   U ;QRVCKJMf7e¯Xb\cJcjP[Z . \$   }¢J3»CKR5HÀ?7XLEJHK?7J3DUABeQe>A  ;Q=7h±HKJc\µ?7=;Q\dXbe eQJcWiWYX  ¨  )ª   §cª  Q§ ¨µ§`:ÝE[s   « §  a¨ b  %  6   >7 Q§kªa§7¬m§¨¨  µ¨  ª

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