3 35 - shree Uversty Joural or Research a Scetc Stues - Basc Sceces Seres Vol. 35 No. 3 3/ 4 / 9. / / 6..-.... :. - - - - -. - - - - -. - - - - - 9
3 35 - shree Uversty Joural or Research a Scetc Stues - Basc Sceces Seres Vol. 35 No. 3 A Cojuate Graet Metho or Solv Ucostrae Optmzato problems Dr. Sulma M. Mahmoo Dr. Mohama Al Mahra Motawe Receve 6 / /. Accepte 9 / 4 /3 ABSRAC I ths paper, we preset umercal metho or solv ucostrae optmzato problems. he metho s base o a set o cojuate search rectos, a the ths set s upate repeately by eerat ew cojuate raet rectos so that steepest escet coto a Wole- Powell cotos are satse. he metho s teste o set o staar problems. Numercal expermets show that the propose metho ca exact soluto or quaratc uctos, so t ca hh accurate soluto or over quaratc uctos. Moreover, the comparsos wth other avalable results llustrate the applcablty a ececy o the presete metho. Keywors: Search Drectos, Cojuate Drectos, Le Search, Cojuate Graet Metho, Ucostrae Optmzato. Assocate Proessor, Dept. o Mathematcs, Faculty o Scece, shree Uversty, Lattaa, Syra. Assocate Proessor, Dept. o Mathematcs, Faculty o Scece, shree Uversty, Lattaa, Syra. Postrauate stuet, Dept. o Mathematcs, shree Uversty, Lattaa, Syra.
shree Uversty Joural. Bas. Sceces Seres 3 35 :.. M R. [-5] x, x,, ] [ K,. x : R R : :. :.. -.... [] LI [5,4,3] ANDREI. - YUAN [6].
. - [-7].. Graet Vector : R,, K, x x. : x R,,, : x. 4. Lear Fucto : : C 4. :.. x c,,..., C c x, C C : [] essa Matrx : 3 : R : x x x M x x x x x L x x R L x x L x x O L M x
shree Uversty Joural. Bas. Sceces Seres 3 35 Quaratc Fucto : : 4 : C 4.3. C : j j x x. j x j j x j L j x j, j j j : 4.3 C, 4.4. j x j c,,..., x j : 4.3 j,, j,...,, x x j, j j,..., :. :[] j, 5 A δ A. : δ A δ >, δ., δ δ Aδ A Local Mmum : 6 : R ε >, 4.5 <. E Global Mmum < ε : 7 3
. [] Statoary Pot : 8 A { :. C [] Sale Pot : 9. [] :, Y R. AY he Numercal Soluto : }... : Lm < ε. ε > < ε., Y R : -.. : :, Y A AY : A, A A A : R,...,, R 4
shree Uversty Joural. Bas. Sceces Seres 3 35,...,, A A R 5.b λ 5.a 5.a λ λ L A [ λ ] A λ : A A. 5.b λ A A. : :,,...,,,,..., 5. : Φ,,,..., 5.3 : Φ Φ,,,..., 5.4 Φ Φ,,,... : m.,,,..., :. 5. : 5. :,,,..., 5.5 : 4.3 C C, C : 5.4 5
C C 5.6 : 5.6 5.7. > : - 5.5 5.7 5.8 : A 5.a 5.8 C C 4.3 R.. : 5.9.,...,,... -. 6
shree Uversty Joural. Bas. Sceces Seres 3 35 ε >, Φ Φ : : R :. :,, M M, M :,,,- :.:, - - :3 :4 :5,,, < ε :6. : :3 e, e,..., e. : e e e, 3,...,,... 4.3. : ψ,, :., :,, R 7
U W U : ψ ψ W U,,. W, ψ U, U U U C U 4.3 j, Φ j., U U C [ U U : : ]. : : j e e j,...,, :.- Φ : : :,..., :,,...,,, 5. 5. 5. 5. :. : : <, > :[] -a 5. 8
35 3 Sceces Seres shree Uversty Joural. Bas. 9 -b [] : c>, > < c 5.3 -c Wole-Powell :[6] δ 5.4 σ 5.5 - Wole-Powell :[6] δ 5.6 σ 5.7 /, δ.σ δ, a : : < : a b c c,, c :c :, : 5.8, /, δ < < δ : δ.5.4 5.5 5.8 : 5.9, σ δ, < < σ : σ
. 5.4 5.6 5.5 : < σ < : 5. 5.9 5.7 σ δ, σ,,,,. : PW.5 PROGRAM Rosebroch_Fucto ; Uses Wcrt; ype termextee; M Array [..3,..3] of extee; VAR,Y: Array [..5] of extee; s,s: Array [..3] of extee; Det, orm,orm, S,S,e :extee;,j,,,,:teer; IM:M ; l:text; FUNCION x,y:extee:extee; Be :sqry-xxsqr-x; E; FUNCION x,y:extee:extee; Be :4x-yxxx-; E; FUNCION x,y:extee:extee; Be :--yxx; E; FUNCION x,y:extee:extee; Be :4-y3xx E; FUNCION x,y:extee:extee; Be :-4x e; FUNCION x,y:extee:extee; Be :-4x E; FUNCION x,y:extee:extee; Be : E; Proceure vrvar IM:M; be Det:x[],y[]x[],y[]-x[],y[]x[],y[]; IM[,]: x[],y[]/det; IM[,]: -x[],y[]/det; IM[,]:-x[],y[]/Det; IM[,]: x[],y[]/det; 3
shree Uversty Joural. Bas. Sceces Seres 3 35 e; FUNCION hx,y:extee:extee; Be h:-s-es-xs-sesxes-sqresxy E; FUNCION hx,y:extee:extee; Be h:sssqrs-sesx-4sses-sqresxy E; BEGIN {ma} wrtel'rosebroch Powell ew metho to solve m'; wrtel''; assl,'e:\roseb.at'; rewrtel; x[]:.5 ;y[]:.5; :; x[3]:x[];y[3]:y[]; s[]:;s[]:; s[]:;s[]:; :; :; REPEA x[]:x[3]; y[]:y[3]; or : to o be s[]:s[];s[]:s[]; s:s[];s:s[]; e:; repeat e:e-hx[],y[]/hx[],y[]; : utl Abshx[],y[]<.; wrtel'ala',,'',e::5; []:[] es; Y[]:Y[]eS; e; INVRIM; s[]:-m[,]x[],y[]m[,]x[],y[]; S[]:-M[,]x[],y[]M[,]x[],y[]; wrtel' Det',,'',et::5; wrtel''; wrtel' M ',,'',M[,]::5,' M',,'',M[,]::5; wrtel' M ',,'',M[,]::5,' M',,'',M[,]::5; s:s[]; s:s[] ; e:; Repeat e:e-hx[],y[]/hx[],y[]; : utl Abshx[],y[]<.; wrtel'ala',,'',e::5,' ',; [3]:[] es; Y[3]:Y[]eS; :; orm:sqrtsqrx[3],y[3]sqrx[3],y[3]; wrtel' ',,'',[3]::8,'; Y',,'',Y[3]::8,' ',x[3],y[3],' ',orm:; wrtel' Iterato ',,' '; real ; utl orm<.e-8 ; closel END. : SD Itr. NKK :. K, N K 3
PW.5 tr. tr... Math 5 K K tr. 9,8,5,3,.4 [7,8].7,6,5, [9] [] K K 3.,,3,3,4 4 [] SD tr. K.,,,,3,4 9,8,7,6,3, 5,3,4 9,8,3,3,4 [] tr. K K tr. K K 9,8,6,3, 6. 9,8,3, [3] [6] K K 3,,3,,3,4 9,8,7,5,3,.,,3,4 7.,,,3,4 8 [4] 3.. 4,3,4 9 8 : [8] Coc : x x, x, :[9] Rosebroc : [ / ] [ x, x x ] 3
shree Uversty Joural. Bas. Sceces Seres 3 35.,...,.> Extee Rosebroc :4 :[8],m3 Beale : 3 m [ ], y.5, y.5, y3 y x x,,..,m,..65 3,.5 : [8] roometrc : 4 j [ ], cos x cos x s x,,, [ / ] j.,..., : [9] Cube : 5 [ x, 3 x x ].,..., : [9] Freueste a Roth : 6 { 3 x x [5 x x ] x} { 9 x [ x x 4] }. : m3, [9] Brow Baly Scale m [ ], 5,4 : 7 6, x, 3 xx x 6 x x x x,., 6 6 : 3 [5] Box hree-dmesoal :[5] [exp.x exp.x [exp.x exp.x [exp.3x exp.3x. x x 3 x 3 3 6 exp. exp ] exp.3 exp 3] 8 exp. exp ],,, 33
[ x [ x x : 4 [5] Woo :9 x ] 4. x ] [/ [ 9 x x 4 x x 4 3 ] ],,, x 3, [7,8] : Quas-Newto Quas-Newto rust Reo rust Reo ء Metho [8] Methos [7] Itr. Itr. Itr..359E-6 5.4378E-6 9 4.6355E-9 -,.937E-6 9 7.35E-6 3 7.473E-, 5.47E-7 8 3.37E-5 6 8,- 3.584E-6 3.58E-6 4 3-3.6,5.6 4.857E-5 5 8.56E-7 9 5.937E-8 8, - 8.333E-7 6.639E-7 4.43E-8 5,.834E-7 59.943E-7 48 5.8483E-4 7 /,/ 6.33E-6 47.863E-6 33.87E-4 8 /5,/5 3 4.[9] : ن rust Reo Methos[9] Itr. Itr..864E-.94E- 36 5.3849E-34 7.863E-6 9 -., 9.547E-6.58E- 43 6 -., 5 6.995E-7 7.95E- 5 7.876E-3 3.379E- 4 -.5, 6.E-.778E- 4.65E-33.68E- 4, 7.[] :3 rust Reo Methos[] K K N K K N. 99E- 37 59 3.387E-4 5 34 39. 4E- 9 8 37.3558E-37 7 33 4 3 7. 55E-6 8 5 8 7 4 48 5. 993E-4 5 35 46 9.667E-9 6 36 8 7. 788E- 39 9 58 9.667E-3 3 3 9,3.5,.5 3 :5. -,,-,, 3,, -.5 34
shree Uversty Joural. Bas. Sceces Seres 3 35 3, [] :4 6 New quas-newto methos[] Itr. K K SD N Itr. K K SD N 9 5 3 8 5 3 5 56 5 5 6 4 4 4 6 4 3 33 3 8 3 33 6 3 38 4 58 8 8 8 7 3 4 3 7 5 4 5 49 8 53 93 54 47 8 93 8 36 47 9.5, 4 :6. -,,, -, 5,,, 4,3.[] :5 Alorthm QNP Methos[] Itr. K Itr. K.49E- -----.3E- -- 4 3 --- 77 7 --- 365 3.77E-9 3 4 8 3 7 73 68 7 97.97E-35 8 4 3 7. 5E- 3 53 6.7E-3 3 8. 5-4 4 8 38 9.667E-3 4 5 9,.5 5 :7.,3,4,3,,, < 5 Nomootoe rust Reo [3] [3] :6 /m Itr. K K N /m Itr. K K N / 39 4 38 78 / 7 68 7 75 /3 4 5 5 3 /3 5 4 5 9 3 3/ 8 83 83 66 3/ 3 84 3 97 8 4/6 4 4 4 84 4/6 6 4 6 48 9 3, 6 :8.,3,,3,-, 35
[6] :7 < 5 A Cojuate Graet Metho [6] Itr. K K N Itr. K K N 3 57 68 5 7 6 7 68 7 36 57 3 7 3 3 55 34 89 3 9 3 6 3 7 3 3 5 39 5 44 8 4 3 6 38 98 4 6 44 6 5 9,.5, 7 :9.,,,,, 5, 6 Nomootoe Globalzato Metho [4] [4] :8 K K K K 7.658E-6 8.9387E-37 3 5 8.43E-5 9.939E-38 3 5 3 9. 6635-3 3 7 3 3 4 6 5 8 6 7 7. 3734E-6 3 4.7E-3 3 43 6 8.36-3 4 33 3.983E-3 4 5 8 9,.5,.5 8 :. 3,3,3,3 -,,,,.5,.5 36
shree Uversty Joural. Bas. Sceces Seres 3 35.4,,, :.3,, 8 : 37
.4 3,3,3,3 9 :3 :. 8....3... :. MURRAY, W., Numercal Methos or Ucostrae Optmzato, Acaemc Press. Loo a New Yor, 97.. L G., a C., We Z., New Cojuacy Coto a Relate New Cojuate Graet Methos For Ucostrae Optmzato, Joural o Computatoal a Apple Mathematcs 7 53 539. 3. ANDREI N., Accelerate Cojuate Graet Alorthm wth Fte Derece essa/vector Prouct Approxmato or Ucostrae Optmzato, Joural o Computatoal a Apple Mathematcs 3 9 57-58. 4. ANDREI N., Accelerato o Cojuate Graet Alorthms or Ucostrae Optmzato, Apple Mathematcs a Computato 3 9 36 369. 5. ANDREI N., Accelerate Scale Memoryless BFGS Precotoe Cojuate Graet Alorthm or Ucostrae Optmzato, Europea Joural o Operatoal Research 4 4 4. 38
shree Uversty Joural. Bas. Sceces Seres 3 35 6. YUAN G., Lu., We Z., A Cojuate Graet Metho wth Descet Drecto For Ucostrae Optmzato, Joural o Computatoal a Apple Mathematcs 33 9 59-53. 7. u C., Ya Y., Coverece O Coc Quas-Newto rust Reo Methos For Ucostrae Mmzato, Math. Appl. 998 7 76. 8. Wa F., Zha K. a a., A Fractoal Proramm Alorthm Base O Coc Quas-Newto rust Reo Metho For Ucostrae Mmzato, Apple Mathematcs a Computato 8 6 6 67. 9. ZANG J., ZANG K., QU S., A Nomootoe Aaptve rust Reo Metho For Ucostrae Optmzato Base O Coc Moel, Apple Mathematcs a Computato 7 465 473.. FU J., SUN W., Nomootoe aaptve trust-reo metho or ucostrae optmzato problems, Apple Mathematcs a Computato 63 5 489-54.. WEI Z., LI G., QI L., New Quas-Newto Methos or Ucostrae Optmzato Problems, Apple Mathematcs a Computato 75 6 56-88.. Wu., Su L., A Quas-Newto Base Patter Search Alorthm or Ucostrae Optmzato, Apple Mathematcs a Computato 83 6 685 694. 3. QING-JUN W., Nomootoe trust reo alorthm or ucostrae optmzato problems, Apple Mathematcs a Computato 7 474-48. 4. ZOU Q., SUN W., QI L., A Nomootoe Globalzato Alorthm Wth Precotoe Graet Path For Ucostrae Optmzato, Apple Mathematcs a Computato 7 457-464. 5. FAN S.-K. S., ZAARA E., A hybr smplex search a partcle swarm optmzato or ucostrae optmzato, Europea Joural o Operatoal Research 8 7 57-548. 39