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Numerical Solution of Optimization Test-Cases by Genetic Algorithms

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(1)Numerical Solution of Optimization Test-Cases by Genetic Algorithms Nathalie Marco-Blaszka, Jean-Antoine Desideri. To cite this version: Nathalie Marco-Blaszka, Jean-Antoine Desideri. Numerical Solution of Optimization Test-Cases by Genetic Algorithms. RR-3622, INRIA. 1999. �inria-00073055�. HAL Id: inria-00073055 https://hal.inria.fr/inria-00073055 Submitted on 24 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. Numerical solution of optimization test-cases by Genetic Algorithms Nathalie Marco-Blaszka and Jean-Antoine Désidéri. N° 3622 Février 1999 THÈME 4. ISSN 0249-6399. apport de recherche.

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(60) •t•. 

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(62) $&%' (()*+ Initialize population gen = 0. front = 1. Is population classified ?. No. Identify non dominated individuals. gen = gen + 1 Yes Reproduction according to dummy fitness. Assign a dummy fitness. Apply the sharing method (niche method). Crossover Mutation. Yes. Is gen < gen_max ?. Assign a new dummy fitness Ignore temporarily the current non dominated individuals. No Stop front = front + 1. /’ ij—tk7ƒ~eŒ „’/lnr_¤¦Yb%mzƒ†o‰rp© oxb7e ½ rts  rtd)ijs%mpoqe { hWrtƒ~oxins7—5­ ®± ijyuijs”teƒ~oqe { ijsoxb7e|Fmzy~erp©/rts7y~oxƒqmzijsoqy± ab7e|v%mzƒxmzd)eoxe ƒxy"©ªrtƒ oxb7e­|® mzƒ~e  v;dRrtmÉv7OyxinijdR« mzlsHk7d¬Qeƒ‰rp©/—tes7e ƒxmpoqijrts7y fŽ GG Gw±à–  ¬7ijs%mzƒ†}^oqrtkwƒxs%mzd)esHo de •lj²³i´kOvQoqoYijrty~mpijousoxino"©ªrtƒxs‘ rtƒ~srtmpo yxo"y~¬7r9”zi´o~e ²³ƒ lne”tel e kw lji { inmzs { iny†oYmzsw e Ê { { A s / ’ n i z — 7 k x ƒ  e ‡ y D f = ¯  m µ ° 0 ² = ¯ D ¬ 0 ° Q £ q o w b  e µ o ¦ ¤  r z m % s z m ´ l H } q o j i F  z m  l ª © 7 k 7 s   x o n i z r 7 s | y z m x ƒ e. e w v n i   q o e ª © t r ƒ J L MO£;©ªk7s7oxinrzs rz©$'&"I L(±„’ in—tkwƒxe=;Kƒxev7ƒxeyxe soxy‰oqbweƒxmzs7—te|rz©oqbwempv7v7ljin mpoqijrts $'&"I L( $ # / I # 1 (± > >#- ^à š  ó›9š 8 Ç œ¢› Q à žt%š Ã§Æ 0ó1› Q Ç ›¢ž Ç / š›_œ9W 0 / ›9š  es\”pmz’/ljk%ij—tmpk7oxinƒxrzes7y yŽ%©ª¯=rtmƒ°µ²0eF¯=mz¬DY°³b‘²¯=dRF°µmɲ0¯O{ ind)°0£Qijoq«Fbwmpe¸oxinrtrtsZvw± oqijdRmzl‚ mzƒxeoqr©ªƒxrzsHoxy mzƒ~e { e v7ijoxe { ±   o oqrWrtÁ2(f  G5rty~o Ñ!Ñ-íXwæç ÔÔ.

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(65)  !W#$!W ' (!"   . Function F1(x,y) Function F2(x,y) -60 -50 -40 0. -30 -10. -20 -20. -10 -30. 0 3. -40. 2 1. -3. 0 -1 -2 -3. -3. -2. -1. 0. 1. -50. 3. 2. -2 -1 -3. -2. Wjë@]#ò9Ø9ípÞµÙ?ßà×í 

(66). 0 -1. W>']#ò9Ø9íÉÞ¾Ù?ßà×í  

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(68) •_Œ. 

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(70) $&%' (()*+. Multi-criteria optimisation on Poloni functions - Without constraints (n=2). Multi-criteria optimization on Poloni functions - With constraints (n=4). 0. 0 Optimal pareto front. -5. -5. -10. -10. F2. Criterion F2. Optimal Pareto front. -15. -15. -20. -20. -25. -25. -30 -18. Wjë@]  

(71)  -16. -14. -12. ë0íÉÛ. -10 -8 Criterion F1. -6. -4. -2. -30 -18. 0. -16. -14. -12. ëíÉÛ #$%&'. -10. -8. -6. -4. -2. 0. F1. W > ] . Multi-criteria optimization on Poloni functions - With constraints (n=8). 

(72)  "!. Multi-criteria optimisation on Poloni function - Without constraints (n=16). 0. 0 Optimal Pareto front. Optimal Pareto front. -5 -5. -10. F2. F2. -10 -15. -15 -20. -20 -25. -30 -18. WjÞ]( ! . Ñ!Ñ-íXwæç ÔÔ. -16. -25 -18. \ë0ípÛ    WjÛ ](*),+ -  

(73)  )\ëíÉÛ &  ’ in—tkwƒxe|ŽR Ê vwoxindRmzl!‚ mzƒxeoqr)©ªƒxrtsoqy¤"b7es§s7rpo"k7yxijs7—R rts7y†oqƒxmzinsoxy ± -14. -12. -10. -8. F1.  

(74) " ). -6. -4. -2. 0. -16. -14. -12. -10. -8. F1. -6. -4. -2. 0.

(75) •Ff. () 

(76)   '   !

(77)  !W#$!W ' (!"   . Multi-criteria optimisation on Poloni function - With constraints (n=2). Multi-criteria optimisation on Poloni function - Without constraints (n=4). -8. -8 Optimal Pareto front. Optimal Pareto front. -12. -12. -14. -14. -16. -16 F2. -10. F2. -10. -18. -18. -20. -20. -22. -22. -24. -24. -26 -30. Wjë[]. -25. \ëíÉÛ #$%&'. -20. -15 F1. -10. -5. -26 -25. 0. -20. \ë0ípÛ ' . -15. -10. -5. 0. F1. W > ].     

(78) ). Multi-criteria optimisation on Poloni function - Without constraints (n=8).  - 

(79) ). Multi-criteria optimisation of Poloni function - With constraints (n=16). -8. -5 Optimal Pareto front. Optimal Pareto front. -10 -10. -12. -14 -15 F2. F2. -16. -18 -20 -20. -22. -25. -24. -26 -25. WjÞ]( ! . -20. -30 -25. ë0ípÛ    WjÛ ](*),+ -  

(80)  )\ëíÉÛ &  ’/ij—tk7ƒ~e=F  Ê vwoqijdRmzl!‚/mzƒ~eoxrR©ªƒ~rtsoqy¤"b7e s2k7y~ins7—R rzs7y~oxƒqmzijsoqy ± -15. -10. F1. -5. 0. -20. -15. -10. -5. 0. F1.  

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(87) indReFmzmws7± e  yR©ªk7s7oqijrts ijy { e v7ijoxe { rzs¨’/ij—tk7ƒ~e  ª© rtƒ $'& , ( ., -/A@B@B@. OFI GO9J. OE  P . O±   o5e0wbwin¬7i´oqy¢sHk7d)e ƒxrzk7y5ljrO mzl. J L M. Keane’s function for n=2. 0.36 0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0. 0 2. 2 4. 4 6. 6 8. 8. /’ in—zk7ƒxe R

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(93) 

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