# pyMultiobjective
## Introduction
A python library for the following Multiobjective Optimization Algorithms or Many Objectives Optimization Algorithms: **C-NSGA II** (Clustered Non-Dominated Sorting Genetic Algorithm II); **CTAEA** (Constrained Two Archive Evolutionary Algorithm); **GrEA** (Grid-based Evolutionary Algorithm); **HypE** (Hypervolume Estimation Multiobjective Optimization Algorithm); **IBEA-FC** (Indicator-Based Evolutionary Algorithm with Fast Comparison Indicator); **IBEA-HV** (Indicator-Based Evolutionary Algorithm with Hypervolume Indicator); **MOEA/D** (Multiobjective Evolutionary Algorithm Based on Decomposition); **NAEMO** (Neighborhood-sensitive Archived Evolutionary Many-objective Optimization); **NSGA II** (Non-Dominated Sorting Genetic Algorithm II); **NSGA III** (Non-Dominated Sorting Genetic Algorithm III); **OMOPSO** (Optimized Multiobjective Particle Swarm Optimization); **PAES** (Pareto Archived Evolution Strategy) with Fast Non-Dominance Sorting); **RVEA** (Reference Vector Guided Evolutionary Algorithm); **SMPSO** (Speed-Constrained Multiobjective Particle Swarm Optimization); **SMS-EMOA** (S-Metric Selection Evolutionary Multiobjective Optimization Algorithm); **SPEA2** (Strength Pareto Evolutionary Algorithm 2); **U-NSGA III** (Unified Non-Dominated Sorting Genetic Algorithm III).
## Usage
1. Install
```bash
pip install pyMultiobjective
```
2. Import
```py3
# Import NSGA III
from pyMultiobjective.algorithm import non_dominated_sorting_genetic_algorithm_III
# Import Test Functions. Available Test Functions: Dent, DTLZ1, DTLZ2, DTLZ3, DTLZ4, DTLZ5, DTLZ6, DTLZ7, Fonseca-Fleming, Kursawe, Poloni, Schaffer1, Schaffer2, ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, Viennet1, Viennet2, Viennet3
from pyMultiobjective.test_functions import dent_f1, dent_f2
# OR Define your Own Custom Function. The function input should be a list of values,
# each value represents a dimenstion (x1, x2, ...xn) of the problem.
# Run NSGA III
parameters = {
'references': 5,
'min_values': (-5, -5),
'max_values': (5, 5),
'mutation_rate': 0.1,
'generations': 1500,
'mu': 1,
'eta': 1,
'k': 2,
'verbose': True
}
sol = non_dominated_sorting_genetic_algorithm_III(list_of_functions = [dent_f1, dent_f2], **parameters)
# Import Graphs
from pyMultiobjective.util import graphs
# Plot Solution - Scatter Plot
parameters = {
'min_values': (-5, -5),
'max_values': (5, 5),
'step': (0.1, 0.1),
'solution': sol,
'show_pf': True,
'show_pts': True,
'show_sol': True,
'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front
'custom_pf': [], # Input a custom Pareto Front(numpy array where each column is an Objective Function)
'view': 'browser'
}
graphs.plot_mooa_function(list_of_functions = [dent_f1, dent_f2], **parameters)
# Plot Solution - Parallel Plot
parameters = {
'min_values': (-5, -5),
'max_values': (5, 5),
'step': (0.1, 0.1),
'solution': sol,
'show_pf': True,
'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front
'custom_pf': [], # Input a custom Pareto Front(numpy array where each column is an Objective Function)
'view': 'browser'
}
graphs.parallel_plot(list_of_functions = [dent_f1, dent_f2], **parameters)
# Plot Solution - Andrews Plot
parameters = {
'min_values': (-5, -5),
'max_values': (5, 5),
'step': (0.1, 0.1),
'solution': sol,
'normalize': True,
'size_x': 15,
'size_y': 15,
'show_pf': True,
'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front
'custom_pf': [] # Input a custom Pareto Front(numpy array where each column is an Objective Function)
}
graphs.andrews_plot(list_of_functions = [dent_f1, dent_f2], **parameters)
# Import Performance Indicators. Available Performance Indicators: GD, GD+, IGD, IGD+, Maximum Spread, Spacing and Hypervolume
from pyMultiobjective.utils import indicators
parameters = {
'min_values': (-5, -5),
'max_values': (5, 5),
'step': (0.1, 0.1),
'solution': sol,
'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front
'custom_pf': [] # Input a custom Pareto Front(numpy array where each column is an Objective Function)
}
gd = indicators.gd_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)
gdp = indicators.gd_plus_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)
igd = indicators.igd_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)
igdp = indicators.igd_plus_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)
ms = indicators.ms_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)
sp = indicators.sp_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)
print('GD = ', gd)
print('GDP = ', gdp)
print('IGD = ', igd)
print('IGDP = ', igdp)
print('MS = ', ms)
print('SP = ', sp)
parameters = {
'solution': sol,
'n_objs': 2,
'ref_point': [], # A Reference Point. If empty, an arbitrary Reference Point will be Used
}
hypervolume = indicators.hv_indicator(**parameters)
print('Hypervolume = ', hypervolume)
```
3. Try it in **Colab**
- C-NSGA II ([ Colab Demo ](https://colab.research.google.com/drive/1sXxCWV6dDmNXmes7RDka4OqKOtM0t9YX?usp=sharing)) ( [ Original Paper ](https://open.metu.edu.tr/bitstream/handle/11511/69040/12625931.pdf))
- CTAEA ([ Colab Demo ](https://colab.research.google.com/drive/1IC5m7JfmhT0ihWBhziQdfyq1PAHrmW1p?usp=sharing)) ( [ Original Paper ](https://doi.org/10.48550/arXiv.2103.06382))
- GrEA ([ Colab Demo ](https://colab.research.google.com/drive/1H2w77kCGUj33qI7uIE-e68999zy1L8tf?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/TEVC.2012.2227145))
- HypE ([ Colab Demo ](https://colab.research.google.com/drive/1cpIWZTECKfyf9jp_iiSuOJaWcWjy_NCr?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1162/EVCO_a_00009))
- IBEA-FC ([ Colab Demo ](https://colab.research.google.com/drive/1BBD0nWaE5SqL5n2Jpa_fDYgkWGSpy8xu?usp=sharing)) ( [ Original Paper ](https://www.simonkuenzli.ch/docs/ZK04.pdf))
- IBEA-HV ([ Colab Demo ](https://colab.research.google.com/drive/1XoiEAR3xpx0DbivrSp_QEFA32xm_R1lk?usp=sharing)) ( [ Original Paper ](https://www.simonkuenzli.ch/docs/ZK04.pdf))
- MOEA/D ([ Colab Demo ](https://colab.research.google.com/drive/1BP2qM9coiOTq28ZYeQEqxHSCHBeh3-Io?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/TEVC.2007.892759))
- NAEMO ([ Colab Demo ](https://colab.research.google.com/drive/1ctVjjOKhLQ1DqQJ0ozcvp2pClmbwBg8O?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1016/j.swevo.2018.12.002))
- NSGA II ([ Colab Demo ](https://colab.research.google.com/drive/1aD1uiJOCezCG6lotMAQENGas4abEO3_6?usp=sharing)) ( [ Original Paper ](http://dx.doi.org/10.1109/4235.996017))
- NSGA III ([ Colab Demo ](https://colab.research.google.com/drive/18zcEdU3NNplFiXAqH8g-oSrEhWB-uqQN?usp=sharing)) ( [ Original Paper ](http://dx.doi.org/10.1109/TEVC.2013.2281535))
- OMOPSO ([ Colab Demo ](https://colab.research.google.com/drive/1cvSZllLYhU6UvuFM7KgDvb1YaNLZVU32?usp=sharing)) ( [ Original Paper ](http://dx.doi.org/10.1007/978-3-540-31880-4_35))
- PAES ([ Colab Demo ](https://colab.research.google.com/drive/1iz5Q9CYiLpyYEKJzd0KwQrGrZykr49TX?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/CEC.1999.781913))
- RVEA ([ Colab Demo ](https://colab.research.google.com/drive/1KYYAsMM52P6lxHRk5a9P8yrnRhwCgT5i?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/TEVC.2016.2519378))
- SMPSO ([ Colab Demo ](https://colab.research.google.com/drive/17m9AT9ORHvVqeqaRjBga1XCEuyG1EPzz?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/MCDM.2009.4938830))
- SMS-EMOA ([ Colab Demo ](https://colab.research.google.com/drive/1hCAW70vVRC-NXmkHPUdX_gK2aADyliQS?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1016/j.ejor.2006.08.008))
- SPEA2 ([ Colab Demo ](https://colab.research.google.com/drive/1OrxJxxAMSpKu_xSWc9UQlPOeM_mmVHmW?usp=sharing)) ( [ Original Paper ](https://kdd.cs.ksu.edu/Courses/CIS830/Handouts/P8.pdf))
- U-NSGA III ([ Colab Demo ](https://colab.research.google.com/drive/1-AO_S6OlqzbA54DlMFBDGEL-wHh9hayH?usp=sharing)) ( [ Original Paper ](https://www.egr.msu.edu/~kdeb/papers/c2014022.pdf))
4. Test Functions
- Dent ( [ Paper ](https://doi.org/10.1007/978-3-319-44003-3_12)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Dent.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1VTHJMmFUHw97tLu5jkTQCkupzj7VA2yp?usp=sharing))
- DTLZ1 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1ENkr2yDACfRwX1ZIidwC15T_YWJrDMtk?usp=sharing))
- DTLZ2 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1pmh6-4jWfQ2eXOzkUm2oydvKc5c0NALz?usp=sharing))
- DTLZ3 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ3.txt) ) ( [ Plot ](https://colab.research.google.com/drive/10pmPlqgrkwAcjA15jWSimgIH5rTWraZN?usp=sharing))
- DTLZ4 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ4.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1PbGRqbLI-wFWg1Orvr0X7Reh5er0vNdW?usp=sharing))
- DTLZ5 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ5.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1mF6UR_yXBSC3E4vHbgfAHFxA5z3rVCWx?usp=sharing))
- DTLZ6 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ6.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1U4E__uPcb5zdztFr8EiQqdegn53Xq7oX?usp=sharing))
- DTLZ7 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ7.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1mNK9WXdYTKP8NWYACXBeTZwWml1M4woZ?usp=sharing))
- Fonseca-Fleming ( [ Paper ](https://doi.org/10.1162/evco.1995.3.1.1)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Fonseca_Fleming.txt) ) ( [ Plot ](https://colab.research.google.com/drive/14LS3MNRwmgbq9ZqA6K7vfbBqMJE4hH4v?usp=sharing))
- Kursawe ( [ Paper ](https://doi.org/10.1007/BFb0029752)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Kursawe.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1bG15YQiRjVX9r4mtEpnt9D-IA2Cjx7Q_?usp=sharing))
- Poloni ( [ Paper ](https://www.researchgate.net/publication/243686783_Hybrid_GA_for_multi_objective_aerodynamic_shape_optimization)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Poloni.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1USBnNWf-UayqZERHq5PM6WWExcwOrOin?usp=sharing))
- Schaffer1 ( [ Paper ](https://www.researchgate.net/publication/236443691_Some_Experiments_in_Machine_Learning_Using_Vector_Evaluated_Genetic_Algorithms)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Schaffer1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1XIpcbZOHJq1xneYiyxCeb-ZhK9QZ3-81?usp=sharing))
- Schaffer2 ( [ Paper ](https://www.researchgate.net/publication/236443691_Some_Experiments_in_Machine_Learning_Using_Vector_Evaluated_Genetic_Algorithms)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Schaffer2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1X5-c9e5wAss9pZE6xOUGt-cVhZTeQ50K?usp=sharing))
- ZDT1 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1Cvzz5o1KWT9vNEyeirUa_pH8jsyOTp8d?usp=sharing))
- ZDT2 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1AvXi0fV7PEdcByDDaPCyHP1CyARf3hQV?usp=sharing))
- ZDT3 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT3.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1XTq9CRYKDu5KPKMAY7w9edF05wJHXhFB?usp=sharing))
- ZDT4 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT4.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1pEQxM18HD2ZnwU4E7hXY3IYIvq7QrJ0U?usp=sharing))
- ZDT6 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT6.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1xAG_8N-K-X83DNj24tWDNbb15q_KDi7V?usp=sharing))
- Viennet1 ( [ Paper ](https://doi.org/10.1080/00207729608929211)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Viennet1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1JqtvAZw1Mh3VmEURZANtZBgYmd07j9vN?usp=sharing))
- Viennet2 ( [ Paper ](https://doi.org/10.1080/00207729608929211)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Viennet2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1tUFYPj9A3herBXg4rSylfBCP2jpDXIm8?usp=sharing))
- Viennet3 ( [ Paper ](https://doi.org/10.1080/00207729608929211)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Viennet3.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1okdsDM01DTuvg6VG_hozU42_KqScBcz7?usp=sharing))
5. Peformance Indicators
- GD ( [ Paper ](https://apps.dtic.mil/sti/pdfs/ADA364478.pdf))
- GD+ ( [ Paper ](https://doi.org/10.1007/978-3-319-15892-1_8))
- IGD ( [ Paper ](https://doi.org/10.1007/978-3-540-24694-7_71))
- IGD+ ( [ Paper ](https://doi.org/10.1007/978-3-319-15892-1_8))
- Maximum Spread ( [ Paper ](https://doi.org/10.1162/106365600568202))
- Spacing ( [ Paper ](https://doi.org/10.1109/TEVC.2006.882428))
- Hypervolume ( [ Paper ](https://scholar.afit.edu/cgi/viewcontent.cgi?article=6130&context=etd))
# Single Objective Optimization
For Single Objective Optimization try [pyMetaheuristic](https://github.com/Valdecy/pyMetaheuristic)
# TSP (Travelling Salesman Problem)
For Travelling Salesman Problems try [pyCombinatorial](https://github.com/Valdecy/pyCombinatorial)
# Acknowledgement
This section is dedicated to all the people that helped to improve or correct the code. Thank you very much!
* Wei Chen (07.AUGUST.2019) - AFRL Summer Intern/Rising Senior at Stony Brook University.
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"description": "# pyMultiobjective\n\n## Introduction\n\nA python library for the following Multiobjective Optimization Algorithms or Many Objectives Optimization Algorithms: **C-NSGA II** (Clustered Non-Dominated Sorting Genetic Algorithm II); **CTAEA** (Constrained Two Archive Evolutionary Algorithm); **GrEA** (Grid-based Evolutionary Algorithm); **HypE** (Hypervolume Estimation Multiobjective Optimization Algorithm); **IBEA-FC** (Indicator-Based Evolutionary Algorithm with Fast Comparison Indicator); **IBEA-HV** (Indicator-Based Evolutionary Algorithm with Hypervolume Indicator); **MOEA/D** (Multiobjective Evolutionary Algorithm Based on Decomposition); **NAEMO** (Neighborhood-sensitive Archived Evolutionary Many-objective Optimization); **NSGA II** (Non-Dominated Sorting Genetic Algorithm II); **NSGA III** (Non-Dominated Sorting Genetic Algorithm III); **OMOPSO** (Optimized Multiobjective Particle Swarm Optimization); **PAES** (Pareto Archived Evolution Strategy) with Fast Non-Dominance Sorting); **RVEA** (Reference Vector Guided Evolutionary Algorithm); **SMPSO** (Speed-Constrained Multiobjective Particle Swarm Optimization); **SMS-EMOA** (S-Metric Selection Evolutionary Multiobjective Optimization Algorithm); **SPEA2** (Strength Pareto Evolutionary Algorithm 2); **U-NSGA III** (Unified Non-Dominated Sorting Genetic Algorithm III).\n\n## Usage\n\n1. Install\n```bash\npip install pyMultiobjective\n```\n\n2. Import\n```py3\n\n# Import NSGA III\nfrom pyMultiobjective.algorithm import non_dominated_sorting_genetic_algorithm_III\n\n# Import Test Functions. Available Test Functions: Dent, DTLZ1, DTLZ2, DTLZ3, DTLZ4, DTLZ5, DTLZ6, DTLZ7, Fonseca-Fleming, Kursawe, Poloni, Schaffer1, Schaffer2, ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, Viennet1, Viennet2, Viennet3 \nfrom pyMultiobjective.test_functions import dent_f1, dent_f2\n\n# OR Define your Own Custom Function. The function input should be a list of values, \n# each value represents a dimenstion (x1, x2, ...xn) of the problem.\n\n# Run NSGA III\nparameters = {\n\t'references': 5,\n\t'min_values': (-5, -5),\n\t'max_values': (5, 5),\n\t'mutation_rate': 0.1,\n\t'generations': 1500,\n\t'mu': 1,\n\t'eta': 1,\n\t'k': 2, \n\t'verbose': True\n}\nsol = non_dominated_sorting_genetic_algorithm_III(list_of_functions = [dent_f1, dent_f2], **parameters)\n\n# Import Graphs\nfrom pyMultiobjective.util import graphs\n\n# Plot Solution - Scatter Plot\nparameters = {\n\t'min_values': (-5, -5),\n\t'max_values': (5, 5),\n\t'step': (0.1, 0.1),\n\t'solution': sol, \n\t'show_pf': True,\n\t'show_pts': True,\n\t'show_sol': True,\n\t'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front\n\t'custom_pf': [], # Input a custom Pareto Front(numpy array where each column is an Objective Function)\n\t'view': 'browser'\n}\ngraphs.plot_mooa_function(list_of_functions = [dent_f1, dent_f2], **parameters)\n\n# Plot Solution - Parallel Plot\nparameters = {\n\t'min_values': (-5, -5), \n\t'max_values': (5, 5), \n\t'step': (0.1, 0.1), \n\t'solution': sol, \n\t'show_pf': True,\n\t'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front\n\t'custom_pf': [], # Input a custom Pareto Front(numpy array where each column is an Objective Function)\n\t'view': 'browser'\n}\ngraphs.parallel_plot(list_of_functions = [dent_f1, dent_f2], **parameters)\n\n# Plot Solution - Andrews Plot\nparameters = {\n\t'min_values': (-5, -5), \n\t'max_values': (5, 5), \n\t'step': (0.1, 0.1), \n\t'solution': sol, \n\t'normalize': True,\n\t'size_x': 15,\n\t'size_y': 15,\n\t'show_pf': True, \n\t'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front\n\t'custom_pf': [] # Input a custom Pareto Front(numpy array where each column is an Objective Function)\n}\ngraphs.andrews_plot(list_of_functions = [dent_f1, dent_f2], **parameters)\n\n# Import Performance Indicators. Available Performance Indicators: GD, GD+, IGD, IGD+, Maximum Spread, Spacing and Hypervolume\nfrom pyMultiobjective.utils import indicators\n\nparameters = {\n\t'min_values': (-5, -5), \n\t'max_values': (5, 5), \n\t'step': (0.1, 0.1), \n\t'solution': sol, \n\t'pf_min': True, # True = Minimum Pareto Front; False = Maximum Pareto Front\n\t'custom_pf': [] # Input a custom Pareto Front(numpy array where each column is an Objective Function)\n}\ngd = indicators.gd_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)\ngdp = indicators.gd_plus_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)\nigd = indicators.igd_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)\nigdp = indicators.igd_plus_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)\nms = indicators.ms_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)\nsp = indicators.sp_indicator(list_of_functions = [dent_f1, dent_f2], **parameters)\n\nprint('GD = ', gd)\nprint('GDP = ', gdp)\nprint('IGD = ', igd)\nprint('IGDP = ', igdp)\nprint('MS = ', ms)\nprint('SP = ', sp)\n\n\nparameters = {\n\t'solution': sol, \n\t'n_objs': 2,\n\t'ref_point': [], # A Reference Point. If empty, an arbitrary Reference Point will be Used\n}\nhypervolume = indicators.hv_indicator(**parameters)\nprint('Hypervolume = ', hypervolume)\n\n```\n\n3. Try it in **Colab**\n- C-NSGA II ([ Colab Demo ](https://colab.research.google.com/drive/1sXxCWV6dDmNXmes7RDka4OqKOtM0t9YX?usp=sharing)) ( [ Original Paper ](https://open.metu.edu.tr/bitstream/handle/11511/69040/12625931.pdf))\n- CTAEA ([ Colab Demo ](https://colab.research.google.com/drive/1IC5m7JfmhT0ihWBhziQdfyq1PAHrmW1p?usp=sharing)) ( [ Original Paper ](https://doi.org/10.48550/arXiv.2103.06382))\n- GrEA ([ Colab Demo ](https://colab.research.google.com/drive/1H2w77kCGUj33qI7uIE-e68999zy1L8tf?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/TEVC.2012.2227145))\n- HypE ([ Colab Demo ](https://colab.research.google.com/drive/1cpIWZTECKfyf9jp_iiSuOJaWcWjy_NCr?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1162/EVCO_a_00009))\n- IBEA-FC ([ Colab Demo ](https://colab.research.google.com/drive/1BBD0nWaE5SqL5n2Jpa_fDYgkWGSpy8xu?usp=sharing)) ( [ Original Paper ](https://www.simonkuenzli.ch/docs/ZK04.pdf))\n- IBEA-HV ([ Colab Demo ](https://colab.research.google.com/drive/1XoiEAR3xpx0DbivrSp_QEFA32xm_R1lk?usp=sharing)) ( [ Original Paper ](https://www.simonkuenzli.ch/docs/ZK04.pdf))\n- MOEA/D ([ Colab Demo ](https://colab.research.google.com/drive/1BP2qM9coiOTq28ZYeQEqxHSCHBeh3-Io?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/TEVC.2007.892759))\n- NAEMO ([ Colab Demo ](https://colab.research.google.com/drive/1ctVjjOKhLQ1DqQJ0ozcvp2pClmbwBg8O?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1016/j.swevo.2018.12.002))\n- NSGA II ([ Colab Demo ](https://colab.research.google.com/drive/1aD1uiJOCezCG6lotMAQENGas4abEO3_6?usp=sharing)) ( [ Original Paper ](http://dx.doi.org/10.1109/4235.996017))\n- NSGA III ([ Colab Demo ](https://colab.research.google.com/drive/18zcEdU3NNplFiXAqH8g-oSrEhWB-uqQN?usp=sharing)) ( [ Original Paper ](http://dx.doi.org/10.1109/TEVC.2013.2281535))\n- OMOPSO ([ Colab Demo ](https://colab.research.google.com/drive/1cvSZllLYhU6UvuFM7KgDvb1YaNLZVU32?usp=sharing)) ( [ Original Paper ](http://dx.doi.org/10.1007/978-3-540-31880-4_35))\n- PAES ([ Colab Demo ](https://colab.research.google.com/drive/1iz5Q9CYiLpyYEKJzd0KwQrGrZykr49TX?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/CEC.1999.781913))\n- RVEA ([ Colab Demo ](https://colab.research.google.com/drive/1KYYAsMM52P6lxHRk5a9P8yrnRhwCgT5i?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/TEVC.2016.2519378))\n- SMPSO ([ Colab Demo ](https://colab.research.google.com/drive/17m9AT9ORHvVqeqaRjBga1XCEuyG1EPzz?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1109/MCDM.2009.4938830))\n- SMS-EMOA ([ Colab Demo ](https://colab.research.google.com/drive/1hCAW70vVRC-NXmkHPUdX_gK2aADyliQS?usp=sharing)) ( [ Original Paper ](https://doi.org/10.1016/j.ejor.2006.08.008))\n- SPEA2 ([ Colab Demo ](https://colab.research.google.com/drive/1OrxJxxAMSpKu_xSWc9UQlPOeM_mmVHmW?usp=sharing)) ( [ Original Paper ](https://kdd.cs.ksu.edu/Courses/CIS830/Handouts/P8.pdf))\n- U-NSGA III ([ Colab Demo ](https://colab.research.google.com/drive/1-AO_S6OlqzbA54DlMFBDGEL-wHh9hayH?usp=sharing)) ( [ Original Paper ](https://www.egr.msu.edu/~kdeb/papers/c2014022.pdf))\n\n4. Test Functions\n- Dent ( [ Paper ](https://doi.org/10.1007/978-3-319-44003-3_12)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Dent.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1VTHJMmFUHw97tLu5jkTQCkupzj7VA2yp?usp=sharing)) \n- DTLZ1 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1ENkr2yDACfRwX1ZIidwC15T_YWJrDMtk?usp=sharing)) \n- DTLZ2 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1pmh6-4jWfQ2eXOzkUm2oydvKc5c0NALz?usp=sharing)) \n- DTLZ3 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ3.txt) ) ( [ Plot ](https://colab.research.google.com/drive/10pmPlqgrkwAcjA15jWSimgIH5rTWraZN?usp=sharing)) \n- DTLZ4 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ4.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1PbGRqbLI-wFWg1Orvr0X7Reh5er0vNdW?usp=sharing)) \n- DTLZ5 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ5.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1mF6UR_yXBSC3E4vHbgfAHFxA5z3rVCWx?usp=sharing)) \n- DTLZ6 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ6.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1U4E__uPcb5zdztFr8EiQqdegn53Xq7oX?usp=sharing)) \n- DTLZ7 ( [ Paper ](https://doi.org/10.1109/CEC.2002.1007032)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/DTLZ7.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1mNK9WXdYTKP8NWYACXBeTZwWml1M4woZ?usp=sharing)) \n- Fonseca-Fleming ( [ Paper ](https://doi.org/10.1162/evco.1995.3.1.1)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Fonseca_Fleming.txt) ) ( [ Plot ](https://colab.research.google.com/drive/14LS3MNRwmgbq9ZqA6K7vfbBqMJE4hH4v?usp=sharing)) \n- Kursawe ( [ Paper ](https://doi.org/10.1007/BFb0029752)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Kursawe.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1bG15YQiRjVX9r4mtEpnt9D-IA2Cjx7Q_?usp=sharing)) \n- Poloni ( [ Paper ](https://www.researchgate.net/publication/243686783_Hybrid_GA_for_multi_objective_aerodynamic_shape_optimization)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Poloni.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1USBnNWf-UayqZERHq5PM6WWExcwOrOin?usp=sharing)) \n- Schaffer1 ( [ Paper ](https://www.researchgate.net/publication/236443691_Some_Experiments_in_Machine_Learning_Using_Vector_Evaluated_Genetic_Algorithms)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Schaffer1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1XIpcbZOHJq1xneYiyxCeb-ZhK9QZ3-81?usp=sharing)) \n- Schaffer2 ( [ Paper ](https://www.researchgate.net/publication/236443691_Some_Experiments_in_Machine_Learning_Using_Vector_Evaluated_Genetic_Algorithms)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Schaffer2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1X5-c9e5wAss9pZE6xOUGt-cVhZTeQ50K?usp=sharing)) \n- ZDT1 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1Cvzz5o1KWT9vNEyeirUa_pH8jsyOTp8d?usp=sharing)) \n- ZDT2 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1AvXi0fV7PEdcByDDaPCyHP1CyARf3hQV?usp=sharing)) \n- ZDT3 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT3.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1XTq9CRYKDu5KPKMAY7w9edF05wJHXhFB?usp=sharing)) \n- ZDT4 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT4.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1pEQxM18HD2ZnwU4E7hXY3IYIvq7QrJ0U?usp=sharing)) \n- ZDT6 ( [ Paper ](https://doi.org/10.1162/106365600568202)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/ZDT6.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1xAG_8N-K-X83DNj24tWDNbb15q_KDi7V?usp=sharing))\n- Viennet1 ( [ Paper ](https://doi.org/10.1080/00207729608929211)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Viennet1.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1JqtvAZw1Mh3VmEURZANtZBgYmd07j9vN?usp=sharing)) \n- Viennet2 ( [ Paper ](https://doi.org/10.1080/00207729608929211)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Viennet2.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1tUFYPj9A3herBXg4rSylfBCP2jpDXIm8?usp=sharing)) \n- Viennet3 ( [ Paper ](https://doi.org/10.1080/00207729608929211)) ( [Pareto Front](https://github.com/Valdecy/Datasets/raw/master/Pareto%20Front/Viennet3.txt) ) ( [ Plot ](https://colab.research.google.com/drive/1okdsDM01DTuvg6VG_hozU42_KqScBcz7?usp=sharing)) \n\n5. Peformance Indicators\n- GD ( [ Paper ](https://apps.dtic.mil/sti/pdfs/ADA364478.pdf))\n- GD+ ( [ Paper ](https://doi.org/10.1007/978-3-319-15892-1_8))\n- IGD ( [ Paper ](https://doi.org/10.1007/978-3-540-24694-7_71))\n- IGD+ ( [ Paper ](https://doi.org/10.1007/978-3-319-15892-1_8))\n- Maximum Spread ( [ Paper ](https://doi.org/10.1162/106365600568202))\n- Spacing ( [ Paper ](https://doi.org/10.1109/TEVC.2006.882428))\n- Hypervolume ( [ Paper ](https://scholar.afit.edu/cgi/viewcontent.cgi?article=6130&context=etd))\n\n# Single Objective Optimization\nFor Single Objective Optimization try [pyMetaheuristic](https://github.com/Valdecy/pyMetaheuristic)\n\n# TSP (Travelling Salesman Problem)\nFor Travelling Salesman Problems try [pyCombinatorial](https://github.com/Valdecy/pyCombinatorial)\n\n# Acknowledgement \nThis section is dedicated to all the people that helped to improve or correct the code. Thank you very much!\n\n* Wei Chen (07.AUGUST.2019) - AFRL Summer Intern/Rising Senior at Stony Brook University.\n",
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