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RMA-LSCh-CMA, presentation for WCCI'2014 (IEEE CEC'2014)

The document discusses a region-based memetic algorithm (RMA-LSCh-CMA) for real-parameter single objective optimization. It compares RMA-LSCh-CMA to a memetic algorithm without regions (MA-LSCh-CMA) on 30 benchmark functions. The results show that RMA-LSCh-CMA has better computational complexity and obtains better results than MA-LSCh-CMA, especially for higher dimensions. For a dimension of 100, RMA-LSCh-CMA is statistically significantly better than MA-LSCh-CMA.

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Inuence of regions on the memetic algorithm on Real-Parameter Single Objective Optimisation Daniel Molina1 Benjamin Lacroix2 Francisco Herrera2 1Computer Science, University of Cádiz 2CCIA Deparment, University of Granada CEC'2014, 9 July 2014 http://sci2s.ugr.es
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Outline 1 Introduction 2 Niching technique: Regions 3 MA-LSCh-CMA 4 RMA-LSCh-CMA 5 Comparisons and results
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Real-parameter single optimisation Optimisation problems Global Optima f (x∗ ) ≤ f (x)∀x ∈ Domain Real-parameter Optimisation Domain ⊆ D, x∗ = [x1, x2, · · · , xD] Single objective optimisation An objective/tness function to optimise. Interested in obtaining only one optimum.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Memetic Algorithms Memetic Algorithms for continuous optimisation Algorithm responsible of the Global Search, exploration. Local Search Method that renes solutions obtained for accuracy. +
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MAs and diversity
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MAs and diversity
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MAs and diversity
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MAs and diversity To improve the search LS method should explore in nearly area. GS algorithm should be focused in exploration.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Niching technique: Clearing
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Niching technique: Clearing Problems related with clearing Expensive (Euclidean distance). Very sensitive to the niche radio.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Niching Technique: Regions Proposal Divide the domain search in hypercubes, regions. Only one solution (best) is accepted in one hypercube. Euclidean distance niches Regions
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Advantages Regions Region vs distance niching Simple and quick (no euclidean distance).
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Advantages Regions Region vs distance niching Simple and quick (no euclidean distance). Easy to divide them in smaller hypercubes.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Region Based Algorithm Global Search Only best solution for each region is accepted. Local Search Initial LS depth in function the region size. To explore mainly inside the region. Regions size Size of regions is getting smaller during the search.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results In this work Multimodal optimisation: obtain many optima RMA-LSCh-CMA, a MA based on regions. Good results in niching competition (CEC'2013). Questions for simple optimisation: only one optimum Improve results for simple optimisation? Improvement in computer time?
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results In this work Multimodal optimisation: obtain many optima RMA-LSCh-CMA, a MA based on regions. Good results in niching competition (CEC'2013). Questions for simple optimisation: only one optimum Improve results for simple optimisation? Improvement in computer time? MA-LSCh-CMA vs RMA-LSCh-CMA CEC'2014 benchmark with RMA-LSCh-CMA. Compare RMA-LSCh-CMA vs MA-LSCh-CMA (no regions).
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MA-LSCh-CMA MA-LSCh-CMA Adapt the LS depth for each solution. Good results in continuous optimisation. Features An Genetic Algorithm as Global Search algorithm. CMA-ES as Local Search algorithm. It can apply the LS several times on the same solution. More promising solutions ⇒ plus LS runs ⇒ ⇒ Greater LS depth. Reference D. Molina, M. Lozano, C. García-Martínez, F. Herrera. Memetic Algorithms for Continuous Optimization Based on Local Search Chains. Evolutionary Computation, 18(1), 27-63, 2010.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MA-LSCh-CMA A promising solution It is maintained more time in population. It can be improved by LS more times.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results MA-LSCh-CMA LS Chaining LS continues (LS parameters values) ⇒ more LS depth.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Global Search algorithm Apply crossover and mutation Select two Parents Ioff Add Current Population New Random Solution ¿Stopping Criterion? ¿loff better than worst? Replace worst individual - Stop No No Yes New Random Solution Region Occuped? No No Improve previous? Replace worst individual Yes Yes Mute new solution Ioff Ioff Ioff Region Occuped? Region Occuped?
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: LS Application Original Model LS Intensity in function of the solution. Improvement solutions always replaces previous one. Region model LS Intensity in function of the solution. If LS result in one region, maintain only the best solution. A randomly new solution is generated.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Regions division Region division Each dimension is divided in ND regions. After each 25% evaluations, ND = 2 · ND.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Regions division Region division Each dimension is divided in ND regions. After each 25% evaluations, ND = 2 · ND.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Regions division Region division Each dimension is divided in ND regions. After each 25% evaluations, ND = 2 · ND.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Experimentation Experimental test suite Used the proposed benchmark in CEC'2014, 30 functions. Tree unimodal function (f1-3). Thirteen simple multimodal functionS: f4 − f16. Six hybrid functions: f17 − f22. Seven composition functions: f23 − f30. Dimensions 10, 30, 50, 100. Two stopping criterions: Error lower than 10-8. when MAXFEs is achieved. MaxFEs = 100000 · D.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Experimentation Experimental test suite Used the proposed benchmark in CEC'2014, 30 functions. Tree unimodal function (f1-3). Thirteen simple multimodal functionS: f4 − f16. Six hybrid functions: f17 − f22. Seven composition functions: f23 − f30. Dimensions 10, 30, 50, 100. Two stopping criterions: Error lower than 10-8. when MAXFEs is achieved. MaxFEs = 100000 · D. Computational complexity Error obtained
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Parameters Parameters values in common Parameter Name Value Population size 100 Crossover Operator BLX − 0.5 NNAM 3 Istep 100 δLS 10−8 Parameter values for RMA-LSCh-MA Parameter Name Value Initial division number (ND) 10 Number of divisions update 4 Multiplier for each update 2
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Computational Complexity MA-LSCh-CMA Dim T1 ˆT2 ˆT2−T1 T0=70 10 198 415 3.01 30 607 4061 51.55 50 1097 14443 199.19 100 3394 80267 1130.48 RMA-LSCh-CMA Dim T1 ˆT2 ˆT2−T1 T0=70 10 205 1660 20.49 30 654 2914 32.28 50 1350 7097 82.10 100 3821 14312 152.04 Conclusion: RMA-LSCh-CMA is better Dimension Time RMA/MA Complexity RMA/MA 50 49% 41% 100 17% 13%
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Results for dimension (average for function) Dimension Criterion MA-LSCh-CMA RMA-LSCh-CMA 10 Median 8.024321E+01 7.645782E+01 Mean 7.961767E+01 7.649017E+01 Std 2.015156E+01 2.032637E+01 30 Median 3.485442E+02 3.031459E+02 Mean 3.151299E+02 2.871957E+02 Std 2.007272E+02 1.026800E+02 50 Median 1.038021E+03 7.876310E+02 Mean 8.066302E+02 7.551120E+02 Std 6.773991E+02 1.817873E+02 100 Median 3.023720e+03 1.418016e+03 Mean 4.538572e+03 1.434800e+03 Std 4.185062e+03 2.869823e+02 Conclusions RMA-LSCh-CMA obtain better results. This improvement increases with dimensionality.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Results for dimension (statistical test) Wilcoxon's test Dimension p-value 10 0.7662 30 0.1148 50 0.0778 100 0.0362 Conclusions For dimension ≤ 100, no detected signicant dierences. For dimension 100, RMA-LSCh-CMA is statistically better than MA-LSCh-CMA.
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 10 Function Mean Median Std 1 0.000000E+00 0.000000E+00 0.000000E+00 2 0.000000E+00 0.000000E+00 0.000000E+00 3 1.025485E-07 0.000000E+00 7.251275E-07 4 8.500798E-02 0.000000E+00 6.010972E-01 5 1.365196E+01 1.999868E+01 9.235427E+00 6 1.478613E-04 0.000000E+00 3.869960E-04 7 0.000000E+00 0.000000E+00 0.000000E+00 8 0.000000E+00 0.000000E+00 0.000000E+00 9 3.316530E+00 2.984877E+00 1.632705E+00 10 7.677946E+00 3.414961E+00 2.324503E+01 11 2.013497E+01 1.182953E+01 3.230385E+01 12 1.646457E-02 1.124002E-02 2.378510E-02 13 3.292333E-02 2.886623E-02 1.567129E-02 14 1.264898E-01 1.207352E-01 3.356309E-02 15 4.714944E-01 4.476892E-01 9.598157E-02 16 1.054166E+00 1.123691E+00 4.701500E-01 17 7.833846E+01 4.022057E+01 9.486355E+01 18 5.220721E+00 3.346637E+00 4.586417E+00 19 7.660733E-02 6.187272E-02 4.399772E-02 20 8.056691E+00 3.121138E+00 1.262373E+01 21 4.928617E+01 1.736043E+01 7.047081E+01 22 8.474625E+00 6.321621E-01 1.120228E+01
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 10 (II) Function Mean Median Std 23 3.294575E+02 3.294575E+02 2.842171E-13 24 1.084430E+02 1.087211E+02 2.946063E+00 25 1.750708E+02 1.977504E+02 3.232990E+01 26 1.000364E+02 1.000324E+02 1.402073E-02 27 1.847796E+02 3.000632E+02 1.547445E+02 28 3.887168E+02 3.600612E+02 8.073141E+01 29 2.270654E+02 2.288594E+02 1.275042E+01 30 5.851143E+02 5.640864E+02 6.482633E+01
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 30 Function Mean Median Std 1 0.000000E+00 0.000000E+00 0.000000E+00 2 0.000000E+00 0.000000E+00 0.000000E+00 3 2.619492E+01 0.000000E+00 6.591081E+01 4 0.000000E+00 0.000000E+00 0.000000E+00 5 1.999971E+01 1.999979E+01 2.525192E-04 6 1.135849E+00 1.042735E+00 1.002887E+00 7 1.932801E-04 0.000000E+00 1.366697E-03 8 1.953497E-02 0.000000E+00 1.380693E-01 9 1.792877E+01 1.790926E+01 3.972464E+00 10 8.124660E+01 8.580182E+00 1.016626E+02 11 1.549521E+03 1.610068E+03 5.819634E+02 12 1.597474E-02 1.323069E-02 8.122876E-03 13 1.376759E-01 1.365498E-01 2.045458E-02 14 2.216354E-01 2.274716E-01 3.444128E-02 15 2.450783E+00 2.472471E+00 4.281959E-01 16 9.647416E+00 9.659988E+00 9.328604E-01 17 6.978608E+02 7.524421E+02 3.226947E+02 18 5.669142E+02 2.333781E+02 6.910744E+02 19 5.822519E+00 5.824098E+00 1.339573E+00 20 1.988780E+02 1.837007E+02 8.390115E+01 21 5.732908E+02 5.407018E+02 2.634281E+02 22 1.588346E+02 1.463604E+02 6.165145E+01
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 30 (II) Function Mean Median Std 23 3.152442E+02 3.152442E+02 8.012869E-05 24 2.222347E+02 2.240828E+02 6.341092E+00 25 2.056072E+02 2.049766E+02 2.532965E+00 26 1.020888E+02 1.001303E+02 1.384855E+01 27 3.284447E+02 3.099193E+02 3.377058E+01 28 8.233830E+02 8.466024E+02 1.129816E+02 29 1.146583E+03 1.098135E+03 2.377853E+02 30 2.040669E+03 1.984265E+03 4.929734E+02
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 50 Function Mean Median Std 1 0.000000E+00 0.000000E+00 0.000000E+00 2 0.000000E+00 0.000000E+00 0.000000E+00 3 2.344839E+02 1.771750E+01 3.389534E+02 4 1.866664E+01 0.000000E+00 3.799332E+01 5 1.999991E+01 1.999994E+01 1.196615e-04 6 6.341466E+00 6.121990E+00 2.302482E+00 7 5.316288e-04 0.000000E+00 2.199802e-03 8 1.935430e-05 1.981726e-05 1.409291e-05 9 3.564299E+01 3.681346E+01 6.519047E+00 10 2.239043E+02 2.424665E+02 1.479644E+02 11 3.457559E+03 3.623177E+03 6.444440E+02 12 1.558060e-02 1.438951e-02 5.331830e-03 13 2.207196e-01 2.192767e-01 2.348579e-02 14 2.459198e-01 2.467165e-01 2.176523e-02 15 4.798497E+00 4.686589E+00 7.357523e-01 16 1.874715E+01 1.872406E+01 1.113784E+00 17 1.988571E+03 1.943020E+03 6.507009E+02 18 7.011082E+02 4.790563E+02 5.980367E+02 19 1.398320E+01 1.394770E+01 1.928223E+00 20 1.031268E+03 7.641425E+02 7.514658E+02 21 1.275634E+03 1.249797E+03 4.111007E+02 22 3.325134E+02 3.107434E+02 1.752374E+02
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 50 (II) Function Mean Median Std 22 3.325134E+02 3.107434E+02 1.752374E+02 23 3.440050E+02 3.440050E+02 2.551582e-04 24 2.638528E+02 2.589911E+02 7.121144E+00 25 2.126556E+02 2.099881E+02 6.896996E+00 26 1.001966E+02 1.001966E+02 2.873123e-02 27 4.811120E+02 4.884886E+02 7.218094E+01 28 1.370109E+03 1.308381E+03 2.477899E+02 29 1.632789E+03 1.622501E+03 3.665546E+02 30 9.860507E+03 9.589913E+03 9.844969E+02
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: results for dimension 100 Function Mean Median Std 1 2.096037e-07 1.338985e-01 2.148479e-01 2 0.000000e+00 0.000000e+00 0.000000e+00 3 2.047680e+03 1.918758e+03 8.327843e+02 4 1.056709e+02 1.055354e+02 7.303576e+01 5 1.999996e+01 1.999996e+01 3.002242e-05 6 2.612602e+01 2.686612e+01 5.038756e+00 7 0.000000e+00 4.327280e-04 1.420433e-03 8 4.418525e-05 4.696169e-05 1.717164e-05 9 9.452115e+01 9.539947e+01 1.426078e+01 10 9.563256e+02 9.839455e+02 2.804663e+02 11 8.529045e+03 8.549727e+03 7.114195e+02 12 1.303121e-02 1.355434e-02 3.805194e-03 13 3.006908e-01 3.065968e-01 2.713187e-02 14 1.248828e-01 1.240539e-01 9.999619e-03 15 1.042165e+01 1.060084e+01 1.545311e+00 16 4.286827e+01 4.268038e+01 1.160471e+00 17 5.194470e+03 5.183200e+03 7.980310e+02 18 5.891675e+02 9.787416e+02 1.273552e+03 19 1.036788e+02 9.951786e+01 1.373116e+01 20 2.714441e+03 2.975314e+03 1.101745e+03 21 3.516025e+03 3.516743e+03 7.521412e+02 22 7.056086e+02 7.793208e+02 3.230215e+02
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results RMA-LSCh-CMA: results for dimension 100 (II) Function Mean Median Std 23 3.483111e+02 3.483129e+02 1.373773e-02 24 3.592935e+02 3.590149e+02 7.535504e+00 25 2.341050e+02 2.368728e+02 1.413945e+01 26 2.000780e+02 2.000692e+02 2.536428e-02 27 9.033352e+02 9.112618e+02 1.235201e+02 28 3.179931e+03 3.288493e+03 5.832432e+02 29 3.275740e+03 3.165691e+03 7.553812e+02 30 9.383213e+03 9.247364e+03 9.434189e+02
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Conclusions Conclusions Based Memetic Algorithm improves results also with only one global optima. Improvement in time and errors increases with dimensionality. Good results in CEC'2014 benchmark?
Introduction Regions MA-LSCh-CMA RMA-LSCh-CMA Results Any question?

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RMA-LSCh-CMA, presentation for WCCI'2014 (IEEE CEC'2014)

  • 1.
    Inuence of regionson the memetic algorithm on Real-Parameter Single Objective Optimisation Daniel Molina1 Benjamin Lacroix2 Francisco Herrera2 1Computer Science, University of Cádiz 2CCIA Deparment, University of Granada CEC'2014, 9 July 2014 http://sci2s.ugr.es
  • 2.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Outline 1 Introduction 2 Niching technique: Regions 3 MA-LSCh-CMA 4 RMA-LSCh-CMA 5 Comparisons and results
  • 3.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Real-parameter single optimisation Optimisation problems Global Optima f (x∗ ) ≤ f (x)∀x ∈ Domain Real-parameter Optimisation Domain ⊆ D, x∗ = [x1, x2, · · · , xD] Single objective optimisation An objective/tness function to optimise. Interested in obtaining only one optimum.
  • 4.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Memetic Algorithms Memetic Algorithms for continuous optimisation Algorithm responsible of the Global Search, exploration. Local Search Method that renes solutions obtained for accuracy. +
  • 5.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MAs and diversity
  • 6.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MAs and diversity
  • 7.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MAs and diversity
  • 8.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MAs and diversity To improve the search LS method should explore in nearly area. GS algorithm should be focused in exploration.
  • 9.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Niching technique: Clearing
  • 10.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Niching technique: Clearing Problems related with clearing Expensive (Euclidean distance). Very sensitive to the niche radio.
  • 11.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Niching Technique: Regions Proposal Divide the domain search in hypercubes, regions. Only one solution (best) is accepted in one hypercube. Euclidean distance niches Regions
  • 12.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Advantages Regions Region vs distance niching Simple and quick (no euclidean distance).
  • 13.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Advantages Regions Region vs distance niching Simple and quick (no euclidean distance). Easy to divide them in smaller hypercubes.
  • 14.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Region Based Algorithm Global Search Only best solution for each region is accepted. Local Search Initial LS depth in function the region size. To explore mainly inside the region. Regions size Size of regions is getting smaller during the search.
  • 15.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results In this work Multimodal optimisation: obtain many optima RMA-LSCh-CMA, a MA based on regions. Good results in niching competition (CEC'2013). Questions for simple optimisation: only one optimum Improve results for simple optimisation? Improvement in computer time?
  • 16.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results In this work Multimodal optimisation: obtain many optima RMA-LSCh-CMA, a MA based on regions. Good results in niching competition (CEC'2013). Questions for simple optimisation: only one optimum Improve results for simple optimisation? Improvement in computer time? MA-LSCh-CMA vs RMA-LSCh-CMA CEC'2014 benchmark with RMA-LSCh-CMA. Compare RMA-LSCh-CMA vs MA-LSCh-CMA (no regions).
  • 17.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MA-LSCh-CMA MA-LSCh-CMA Adapt the LS depth for each solution. Good results in continuous optimisation. Features An Genetic Algorithm as Global Search algorithm. CMA-ES as Local Search algorithm. It can apply the LS several times on the same solution. More promising solutions ⇒ plus LS runs ⇒ ⇒ Greater LS depth. Reference D. Molina, M. Lozano, C. García-Martínez, F. Herrera. Memetic Algorithms for Continuous Optimization Based on Local Search Chains. Evolutionary Computation, 18(1), 27-63, 2010.
  • 18.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MA-LSCh-CMA A promising solution It is maintained more time in population. It can be improved by LS more times.
  • 19.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results MA-LSCh-CMA LS Chaining LS continues (LS parameters values) ⇒ more LS depth.
  • 20.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Global Search algorithm Apply crossover and mutation Select two Parents Ioff Add Current Population New Random Solution ¿Stopping Criterion? ¿loff better than worst? Replace worst individual - Stop No No Yes New Random Solution Region Occuped? No No Improve previous? Replace worst individual Yes Yes Mute new solution Ioff Ioff Ioff Region Occuped? Region Occuped?
  • 21.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: LS Application Original Model LS Intensity in function of the solution. Improvement solutions always replaces previous one. Region model LS Intensity in function of the solution. If LS result in one region, maintain only the best solution. A randomly new solution is generated.
  • 22.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Regions division Region division Each dimension is divided in ND regions. After each 25% evaluations, ND = 2 · ND.
  • 23.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Regions division Region division Each dimension is divided in ND regions. After each 25% evaluations, ND = 2 · ND.
  • 24.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Regions division Region division Each dimension is divided in ND regions. After each 25% evaluations, ND = 2 · ND.
  • 25.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Experimentation Experimental test suite Used the proposed benchmark in CEC'2014, 30 functions. Tree unimodal function (f1-3). Thirteen simple multimodal functionS: f4 − f16. Six hybrid functions: f17 − f22. Seven composition functions: f23 − f30. Dimensions 10, 30, 50, 100. Two stopping criterions: Error lower than 10-8. when MAXFEs is achieved. MaxFEs = 100000 · D.
  • 26.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Experimentation Experimental test suite Used the proposed benchmark in CEC'2014, 30 functions. Tree unimodal function (f1-3). Thirteen simple multimodal functionS: f4 − f16. Six hybrid functions: f17 − f22. Seven composition functions: f23 − f30. Dimensions 10, 30, 50, 100. Two stopping criterions: Error lower than 10-8. when MAXFEs is achieved. MaxFEs = 100000 · D. Computational complexity Error obtained
  • 27.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Parameters Parameters values in common Parameter Name Value Population size 100 Crossover Operator BLX − 0.5 NNAM 3 Istep 100 δLS 10−8 Parameter values for RMA-LSCh-MA Parameter Name Value Initial division number (ND) 10 Number of divisions update 4 Multiplier for each update 2
  • 28.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Computational Complexity MA-LSCh-CMA Dim T1 ˆT2 ˆT2−T1 T0=70 10 198 415 3.01 30 607 4061 51.55 50 1097 14443 199.19 100 3394 80267 1130.48 RMA-LSCh-CMA Dim T1 ˆT2 ˆT2−T1 T0=70 10 205 1660 20.49 30 654 2914 32.28 50 1350 7097 82.10 100 3821 14312 152.04 Conclusion: RMA-LSCh-CMA is better Dimension Time RMA/MA Complexity RMA/MA 50 49% 41% 100 17% 13%
  • 29.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Results for dimension (average for function) Dimension Criterion MA-LSCh-CMA RMA-LSCh-CMA 10 Median 8.024321E+01 7.645782E+01 Mean 7.961767E+01 7.649017E+01 Std 2.015156E+01 2.032637E+01 30 Median 3.485442E+02 3.031459E+02 Mean 3.151299E+02 2.871957E+02 Std 2.007272E+02 1.026800E+02 50 Median 1.038021E+03 7.876310E+02 Mean 8.066302E+02 7.551120E+02 Std 6.773991E+02 1.817873E+02 100 Median 3.023720e+03 1.418016e+03 Mean 4.538572e+03 1.434800e+03 Std 4.185062e+03 2.869823e+02 Conclusions RMA-LSCh-CMA obtain better results. This improvement increases with dimensionality.
  • 30.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Results for dimension (statistical test) Wilcoxon's test Dimension p-value 10 0.7662 30 0.1148 50 0.0778 100 0.0362 Conclusions For dimension ≤ 100, no detected signicant dierences. For dimension 100, RMA-LSCh-CMA is statistically better than MA-LSCh-CMA.
  • 31.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 10 Function Mean Median Std 1 0.000000E+00 0.000000E+00 0.000000E+00 2 0.000000E+00 0.000000E+00 0.000000E+00 3 1.025485E-07 0.000000E+00 7.251275E-07 4 8.500798E-02 0.000000E+00 6.010972E-01 5 1.365196E+01 1.999868E+01 9.235427E+00 6 1.478613E-04 0.000000E+00 3.869960E-04 7 0.000000E+00 0.000000E+00 0.000000E+00 8 0.000000E+00 0.000000E+00 0.000000E+00 9 3.316530E+00 2.984877E+00 1.632705E+00 10 7.677946E+00 3.414961E+00 2.324503E+01 11 2.013497E+01 1.182953E+01 3.230385E+01 12 1.646457E-02 1.124002E-02 2.378510E-02 13 3.292333E-02 2.886623E-02 1.567129E-02 14 1.264898E-01 1.207352E-01 3.356309E-02 15 4.714944E-01 4.476892E-01 9.598157E-02 16 1.054166E+00 1.123691E+00 4.701500E-01 17 7.833846E+01 4.022057E+01 9.486355E+01 18 5.220721E+00 3.346637E+00 4.586417E+00 19 7.660733E-02 6.187272E-02 4.399772E-02 20 8.056691E+00 3.121138E+00 1.262373E+01 21 4.928617E+01 1.736043E+01 7.047081E+01 22 8.474625E+00 6.321621E-01 1.120228E+01
  • 32.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 10 (II) Function Mean Median Std 23 3.294575E+02 3.294575E+02 2.842171E-13 24 1.084430E+02 1.087211E+02 2.946063E+00 25 1.750708E+02 1.977504E+02 3.232990E+01 26 1.000364E+02 1.000324E+02 1.402073E-02 27 1.847796E+02 3.000632E+02 1.547445E+02 28 3.887168E+02 3.600612E+02 8.073141E+01 29 2.270654E+02 2.288594E+02 1.275042E+01 30 5.851143E+02 5.640864E+02 6.482633E+01
  • 33.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 30 Function Mean Median Std 1 0.000000E+00 0.000000E+00 0.000000E+00 2 0.000000E+00 0.000000E+00 0.000000E+00 3 2.619492E+01 0.000000E+00 6.591081E+01 4 0.000000E+00 0.000000E+00 0.000000E+00 5 1.999971E+01 1.999979E+01 2.525192E-04 6 1.135849E+00 1.042735E+00 1.002887E+00 7 1.932801E-04 0.000000E+00 1.366697E-03 8 1.953497E-02 0.000000E+00 1.380693E-01 9 1.792877E+01 1.790926E+01 3.972464E+00 10 8.124660E+01 8.580182E+00 1.016626E+02 11 1.549521E+03 1.610068E+03 5.819634E+02 12 1.597474E-02 1.323069E-02 8.122876E-03 13 1.376759E-01 1.365498E-01 2.045458E-02 14 2.216354E-01 2.274716E-01 3.444128E-02 15 2.450783E+00 2.472471E+00 4.281959E-01 16 9.647416E+00 9.659988E+00 9.328604E-01 17 6.978608E+02 7.524421E+02 3.226947E+02 18 5.669142E+02 2.333781E+02 6.910744E+02 19 5.822519E+00 5.824098E+00 1.339573E+00 20 1.988780E+02 1.837007E+02 8.390115E+01 21 5.732908E+02 5.407018E+02 2.634281E+02 22 1.588346E+02 1.463604E+02 6.165145E+01
  • 34.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 30 (II) Function Mean Median Std 23 3.152442E+02 3.152442E+02 8.012869E-05 24 2.222347E+02 2.240828E+02 6.341092E+00 25 2.056072E+02 2.049766E+02 2.532965E+00 26 1.020888E+02 1.001303E+02 1.384855E+01 27 3.284447E+02 3.099193E+02 3.377058E+01 28 8.233830E+02 8.466024E+02 1.129816E+02 29 1.146583E+03 1.098135E+03 2.377853E+02 30 2.040669E+03 1.984265E+03 4.929734E+02
  • 35.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 50 Function Mean Median Std 1 0.000000E+00 0.000000E+00 0.000000E+00 2 0.000000E+00 0.000000E+00 0.000000E+00 3 2.344839E+02 1.771750E+01 3.389534E+02 4 1.866664E+01 0.000000E+00 3.799332E+01 5 1.999991E+01 1.999994E+01 1.196615e-04 6 6.341466E+00 6.121990E+00 2.302482E+00 7 5.316288e-04 0.000000E+00 2.199802e-03 8 1.935430e-05 1.981726e-05 1.409291e-05 9 3.564299E+01 3.681346E+01 6.519047E+00 10 2.239043E+02 2.424665E+02 1.479644E+02 11 3.457559E+03 3.623177E+03 6.444440E+02 12 1.558060e-02 1.438951e-02 5.331830e-03 13 2.207196e-01 2.192767e-01 2.348579e-02 14 2.459198e-01 2.467165e-01 2.176523e-02 15 4.798497E+00 4.686589E+00 7.357523e-01 16 1.874715E+01 1.872406E+01 1.113784E+00 17 1.988571E+03 1.943020E+03 6.507009E+02 18 7.011082E+02 4.790563E+02 5.980367E+02 19 1.398320E+01 1.394770E+01 1.928223E+00 20 1.031268E+03 7.641425E+02 7.514658E+02 21 1.275634E+03 1.249797E+03 4.111007E+02 22 3.325134E+02 3.107434E+02 1.752374E+02
  • 36.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: Results for dimension 50 (II) Function Mean Median Std 22 3.325134E+02 3.107434E+02 1.752374E+02 23 3.440050E+02 3.440050E+02 2.551582e-04 24 2.638528E+02 2.589911E+02 7.121144E+00 25 2.126556E+02 2.099881E+02 6.896996E+00 26 1.001966E+02 1.001966E+02 2.873123e-02 27 4.811120E+02 4.884886E+02 7.218094E+01 28 1.370109E+03 1.308381E+03 2.477899E+02 29 1.632789E+03 1.622501E+03 3.665546E+02 30 9.860507E+03 9.589913E+03 9.844969E+02
  • 37.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: results for dimension 100 Function Mean Median Std 1 2.096037e-07 1.338985e-01 2.148479e-01 2 0.000000e+00 0.000000e+00 0.000000e+00 3 2.047680e+03 1.918758e+03 8.327843e+02 4 1.056709e+02 1.055354e+02 7.303576e+01 5 1.999996e+01 1.999996e+01 3.002242e-05 6 2.612602e+01 2.686612e+01 5.038756e+00 7 0.000000e+00 4.327280e-04 1.420433e-03 8 4.418525e-05 4.696169e-05 1.717164e-05 9 9.452115e+01 9.539947e+01 1.426078e+01 10 9.563256e+02 9.839455e+02 2.804663e+02 11 8.529045e+03 8.549727e+03 7.114195e+02 12 1.303121e-02 1.355434e-02 3.805194e-03 13 3.006908e-01 3.065968e-01 2.713187e-02 14 1.248828e-01 1.240539e-01 9.999619e-03 15 1.042165e+01 1.060084e+01 1.545311e+00 16 4.286827e+01 4.268038e+01 1.160471e+00 17 5.194470e+03 5.183200e+03 7.980310e+02 18 5.891675e+02 9.787416e+02 1.273552e+03 19 1.036788e+02 9.951786e+01 1.373116e+01 20 2.714441e+03 2.975314e+03 1.101745e+03 21 3.516025e+03 3.516743e+03 7.521412e+02 22 7.056086e+02 7.793208e+02 3.230215e+02
  • 38.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results RMA-LSCh-CMA: results for dimension 100 (II) Function Mean Median Std 23 3.483111e+02 3.483129e+02 1.373773e-02 24 3.592935e+02 3.590149e+02 7.535504e+00 25 2.341050e+02 2.368728e+02 1.413945e+01 26 2.000780e+02 2.000692e+02 2.536428e-02 27 9.033352e+02 9.112618e+02 1.235201e+02 28 3.179931e+03 3.288493e+03 5.832432e+02 29 3.275740e+03 3.165691e+03 7.553812e+02 30 9.383213e+03 9.247364e+03 9.434189e+02
  • 39.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Conclusions Conclusions Based Memetic Algorithm improves results also with only one global optima. Improvement in time and errors increases with dimensionality. Good results in CEC'2014 benchmark?
  • 40.
    Introduction Regions MA-LSCh-CMARMA-LSCh-CMA Results Any question?