Exploiting problem structure in pattern search methods for unconstrained optimization
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(137) @#_ Kg=Q75vE9;%E3bNy<I.v 0Z&KAbgKg=@CD&1 P7NICqC7%=bNbFvn@Cb]r@CD&=C48687%ICqC7N=bNbIJBmKNE==>fKg=BD&=DÕ IIfDb_ZB;>KN9kIJBrvn@Cbn]Z;:EmbNKg=@CDf 9k=Q7hCv(9kKNE+jC=Q7N1 qCIIfD+P7NICqC7N=b%b89<B9kKN9@Cy<yk1Ch#@CBD^@#_ Kg=Q7]r@UPÎ # # ÑÇ7%bNK_=y<yf0=ykIv VQR @"K 4 ½ B ;>IJBKg7%@CbgKQh_IC7pG]r9<BbNZ7N_ KNE&=r@CykqCIC7%9kKNE] E@CD IC0KN@C9<B&=D KNE=^ICPKN9<]r@Cy_ZB;>KN9kIJBYj#@Cy<Z&= 0,1 KNE&=KN9<]^=m]a@UP Î # # ÑÇ7%bgK_ =y<yF0=ykI.v R <V#4 Å B uQ_ VaKNE=@CykqCIC7:9kKNE] ]r@CD&=qCIIfD bgKg=@CD&1mP7NICqC7N=b%b@#Kt@Cy<yKN9<]^=b482nIJy<yk=;>KN9kjC=yk1^KNE=bg=7N=bNZykKNbbNE&Iv KNE@#KnKNE&=(7:@#Kg=5IC_7N=> DZ;>KN9<IJB[IC_ ]r@UP Î # # ÑA9bAbgKg7NIJB&qJy<1mP7%IC0yk=] D&=QP=BD&=BKQhv9<KNE[9<y<y ;>IJBD9kKN9kIJB9<Bqp0=9<B&q @rbN9<qJB9 Ç;Q@CBKn_@C;>KgIC74 . ¸µ» a º ¼Z¸µ. Cq 7%9<D 0@Cbg=D P@#KgKg=Q7%B bg=@#7%;:E @CykqCIC7:9kKNE] vE9<;%E =>fPykIJ9kKNbP@#7NKN9<@Cy8bN=QP@#7%@#09<y9kK 1YE@Cb 0=Q=BD=bN;>7%9k0=D3@CBDY9<]^Pyk=]^=BKg=D4Au(Z]^=Q7%9<;Q@Cy7N=bNZykKNb(bNE&Iv KNE@#K5KNE&=+Zbg=IC_ P@#7g KN9<@Cybg=QP@#7%@#09y<9kK 1+D&7%@C]a@#KN9<;Q@Cy<yk1+7N=DZ;>=b8KNE&=5;>IJ]^PZ&KN@#KN9kIJB@Cy=VÖIC7NK7N=*Z9k7N=D^KgI ]r9<Bf 9<]r9<Q=^P@#7NKN9<@Cy<y<1 bg=QP@#7%@#0yk=^_ZB;>KN9kIJBbQ4mHAE9<b@Cy<ykIv=D P7NIC0y<=]rbv9kKNE KNE&IJZbN@CBDbIC_ j#@#7%9<@#0yk=bKgI0=rbgIJykjC=D 9<B @[7N=@CbgIJB@#0yk=r@C]^IJZBKIC_nKN9<]^=C4aHAvI3j#@#7%9<@#KN9<IJBbIC_nKNE&= @CykqCIC7%9<KNE] v=Q7N= =>&@C]r9<B&=D{mZbg= IC_5@qC7%=Q=D&1 @#PP7NIJ@C;%E 9<B-;:E&IIJb%9<B&qYvE9<;:E bNKg=QPb KgI KN@ ¾C=9<B @CB 9kKg=Q7%@#KN9kIJBh@CBD 7%=QjC=Q7%bN9<B&q[KNE=abgKg=QP D9k7N=;>KN9kIJBb4mHAE&=^_IC7%]^=Q7 9<bIJBy<1 @#PPy9<;Q@#0yk=rKgIYP@#7NKN9@Cy<yk1 bg=QP@#7%@#0yk=a_ZB;>KN9kIJBbQh@CBD vn@Cb+bNE&I.v(B KgI ]r@#7NqJ9<B@Cy<yk1 9<]+ P7NIjC=(KNE=@CykqCIC7%9kKNE][4 ½ K9<b8PIJbNbN9<0yk=nKgI9<]rP7NI.jC=(IJBaKNE=qC7N=Q=D&1rbNKg7%@#Kg=QqC1r0, 1 ÇBD9<B&q KNE&=;>IJ]09<B@#KN9kIJBIC_B&IJBf 9<BKg=Q7%@C;>KN9<B&q+bgKg=QPbnv(E9<;%Em1f9k=y<DbKNE&=qC7N=@#Kg=bgKA7%=DZ;>KN9kIJB9<B KNE&=rIC<0 =;>KN9kjC=_ZB;>KN9kIJBh0Z&KKNE9b7N= *,Z9k7N=bbNIJykj,9B&q3@ ;>IJ]09<B@#KgIC7:9<@Cy ICPKN9<]a9k@#KN9kIJB P7NIC0yk=] =QjC=Q7N1!9kKg=Q7%@#KN9<IJB4HAE&= 7N=y<@#KN9kjC=yk1-]^IfD&=bgK^qJ@C9<BbrIC0KN@C9<B=D 01-ZbN9B&q KNE&= qC7N=Q=D&1 bgKg7%@#Kg=QqC1[bNZqCqC=bgKtKNE@#KtKNE&=qJ@C9<BbnIC0KN@C9<B&=D 0,1b%Z;%E[@CB[@#PP7NIJ@C;%E[vIJZy<D 0= ]r@#7NqJ9B@Cy|4aHAE=rbgKg7%@#Kg=QqC1 IC_n7N=QjC=Q7%bN9B&q[KNE&=abgKg=QP D9k7N=;>KN9<IJBbvn@CbjC=Q7N1 V= Ö=;>KN9kjC=a=>, ;>=QPKvE&=B @Cy<y,bNZ&0bgP@C;>=bv=Q7N=IC_D9<]^=BbN9kIJBIJB&=C4 Å BKNE&=bg=nP7NIC0yk=]rbKNE=F7N=QjC=Q7%b%9<B&q bgKg7%@#Kg=QqC13]a@CD&=B&IrD9 Ö=Q7N=B;>=@Cb(9kKAD9<D B&ICK(@CykKg=Q7AKNE&=pbN=QKAIC_bN=@#7%;%ED9k7N=;>KN9kIJBbQ4 6 @#KgKg=Q7%B bg=@#7%;%E @CykqCIC7%9kKNE]abE@jC=ay9<]r9kKg=D= a;Q9k=B;>1Ch0Z&K@#7N=^jC=Q7N1 7NIC0ZbgK@CBD @CD@#PKN@#0yk=C4 i bN9B&qpP@#7%KN9<@Cybg=QP@#7%@#09<y<9<K 1+]r@ ¾C=bKNE&=] @#PPy<9<;Q@#0y<=tKgI y<@#7%qC=P7NIC0yk=]rbQh 9<B;QyZD9<B&qP7NIC0y<=]rbt@#7%9<bN9B&q_7NIJ] D9<bN;>7N=QKN9k@#KN9<IJB4 ®H!ЫH° «
(138) W« ± 6FE4#G84#HIJ9BLK vIJZy<D y<9 ¾C=KgI5=>fP7N=b%bE9<bqC7:@#KN9kKNZD&=KgIKNE&= i B9kjC=Q7%bN9<K 1 IC_52n@CBKg=Q7N0Z&7%1 z 7%b ¾f9<B&= &=yykI.vb%E9kP _IC7+9kKNbpbNZPPIC7NKpDZ&7:9<B&q3@3jf9<bN9<KKgI u(=Qv w
(139) =@Cy<@CBD^9<B^KNE&=AbgP7%9<B&qW#RC)R Ì,hfvE9<;:E^;>7N=@#Kg=DrKNE=AICPPIC7NKNZB9<K 1p_IC7FKNE=t7N=bg=@#7%;:E P7N=bN=BLKg=DmE&=Q7N=C48HAE@CB ¾,b8@#7%=@Cy<bgI DZ&=tKgI ½ @CB 2IICP=A@CBDmoIC0o7NIJZ&qJEKgIJBm_ IC7KNE&=9k7 9<BKg=Q7N=bgK@CBD _ 7%9<=BDyk1aD9<bN;QZbNb%9kIJBbQ4. . 9·9µ» À VAÁ o542IJybgIJB @CBD 6E4G84HIJ9<BLKQh 5 )1 !# ! "1
(140) 1
(141) ! ! 5 ! 1 ! . 1 J 1 '5 !! 41 5
(142) <1#" %$ J1 %$ :h Å PKN9<]r9k@#KN9kIJBY@CBD z B&qJ9<B&=Q=Q7%9B&q&hjCIJy|4W,h PP4UÌ S 0 LSVW,hW#RCR&V#4 }À WÁ o54L2IJy<bNIJB @CBD^6FE4#G84CHIJ9<BLKQh \ =Q7%9kj#@#KN9kjC=>_7N=Q= y<qCIC7%9kKNE]x_IC78dfP@#7%bg= i B;>IJBf bNKg7%@C9<B&=D Å PKN9<]r9k@#KN9kIJB[6F7NIC0yk=]abQh&9<Bdf9<DD9C*,9|h 4¿4@CBD ` 4'&I);>( jU@#7:@Î z DbaÑ:h H7N=BDb9B ½ BDZbNKg7%9<@CyL@CBD PPy<9k=D`@#KNE&=]r@#KN9;QbQh&yZLv=Q7h#PP4fVtÌfV V>S 0,hW#RCRJW,4.
(143) VtÌ o542IJy<bgIJBx@CBDx6FE45G84HIJ9BLKQh 1 %% ! 1
(144) ! 1 '5 ! )1 5
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(147) ! 5 ! 1 ! 5 1 J1 '5 !h&d ½ ` X&4 Å PKN9<]a9k@#KN9kIJBhUjCIJy|4fVCyV Î|W#RCR&V2Ñ:hPP^sC 0 s 0,4 À} Á \ @.j,9<bhC254k h
(148) 5 75 5 )1 %$ ! ! ! Lh ]^=Q7:9<;Q@CBXJIJZ&7%B@Cy#IC_`@#KNEf =]a@#KN9<;QbQh
(149) IJy|%4 Ê Î V 0CUS Ñ:hP P ÊÌjÌ
(150) ÊS 4 À Á \ =BB9<bQhLX&4 z 4kh@CBD
(151) 4JHIC7%;>QIJBh " 1 1 5 5 ! ! h d ½ ` X4 Å PKtjCIJy|4# V Î V 0 0f2V Ñ:hPP[SCSLs LSÊS&4 ÀËÊ2ÁÈ7%9k=Qvn@CB ¾h 4f@CBDm6E4,GF4,HIJ9<BKQh !1
(152) ! 5 ! 1 ! 5 1 %% 1 45 !85 1
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(154) XC=Q=QjC=bh " 1 5
(155) <1 '5 !
(156) 5 !
(157) ) ! 1 1 1 5 h
(158) X&4 bNbgIf;#4
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(160) PPWfVW ,WfV 0,4 À(0Á+` IC7 =C h(X4X&4khto54Ad
(161) *4 È@#7N0Ivht@CBD &r4 z *4 ¿(9<yy<bgKg7NIJ][h
(162) 1 !
(163) ! 5 ! 1 ! . 5 1 J1 '5 ! 5 143 Qh 2t` H7%@CBbQ4`@#KNE4
(164) d,IC_K vn@#7N=Ch
(165) IJy| 4 Ê|Î V 0Csf2V Ñ:h
(166) PP 2V Ê SV#4 À VQER Á u(=y<D&=Q7.hX4 4@CBD O 4` =@CD h 1 5 5
(167) ! 1 '5 ! !%% 1 45 !h 2nIJ]^PZ&Kg=Q7X4jCIJy| 4 Ê|Î V 0 ) Ñ:h
(168) P P Ì#RJs HÌftV Ì,4 À VCAV Á HIC7%;>QIJBh
(169) 4kh ! 1 5 ! $ ! 5 J1'1 ! H5 41 :h5d ½ ` X&4 Å PKQ4&jCIJy|4 ʨΠV 0 0)ÊjÑ:hPP V ,WC,4. À ÌÁ.
(170)
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