NSGA-II算法的python实现（包含详细注释案例）

# NSGA-II Python Implementation of NSGA-II algorithm in form of a python library. This implementation can be used to solve multivariate (more than one dimensions) multi-objective optimization problem. The number of objectives and dimensions are not limited. Some critical operators are chosen as: Binary Tournament Selection, Simulated Binary Crossover and Polynomial Mutation. Note that we didn't start everything from scratch but modified the source code from [wreszelewski/nsga2](https://github.com/wreszelewski/nsga2). We are very thankful to Wojciech Reszelewski and Kamil Mielnik - authors of this original version. The following items are modified: * Fix the crowding distance formula. * Modify some parts of the code to apply to any number of objectives and dimensions. * Modify the selection operator to Tournament Selection. * Change the crossover operator to Simulated Binary Crossover. * Change the mutation operator to Polynomial Mutation. ## Usage 1. Class **Problem** Defined in _problem.py_. Using to define a multi-objective problem. Arguments: * `objectives`: A list of functions, representing the objective functions. * `num_of_variables`: An integer, representing the number of variables. * `variables_range`: A list of tuples of two elements, representing the lower and upper bound of each respective variables. * `same_range`: A boolean argument, default = `False`. If true, the range of all variables are the same (this case `variables_range` has only one element), otherwise each variable has its own range. * `expand`: A boolean argument, default = `True`. If true, input of functions are treated as list of respective variables (e.x. `f(x,y,z)`), otherwise they are treated as a vector (e.x. `f([x1, x2, x3])`). 2. Class **Evolution** Defined in _evolution.py_. Using to run NSGA-II. 1. Arguments: * `problem`: An object of class Problem. * `num_of_generations`: An integer, default = 1000, representing the number of generations. * `num_of_individuals`: An integer, default = 100, representing the number of individuals, a.k.a the population size. * `num_of_tour_particips`: An integer, default = 2, representing the number of participants in tournament selection operator. * `tournament_prob`: A real number, default = 0.9, representing the probability used in tournament selection. * `crossover_param`: An integer, default = 2, representing the parameter used in simulated binary crossover. * `mutation_param`: An integer, default = 2, representing the paramenter used in polynomial mutation. 2. Methods: * `evolve`: * Arguments: None. * Return: List of the best individuals in the last generation. ## Example Examples about SCH problem and KUR problem are showed in _sch.py_ and _kur.py_. In addition, the results of some popular multi-objective problems is demonstrated as belows: * SCH  * KUR  * ZDT1  * ZDT4  * VIENNET  ## Authors * Pham Ngo Gia Bao, Ho Chi Minh University of Technology * Tram Loi Quan, Ho Chi Minh University of Technology * A/Prof. Quan Thanh Tho, Ho Chi Minh University of Technology (advisor) * A/Prof. Akhil Garg, Shantou University (advisor) We are very thankful to A/Prof. Tho and A/Prof. Akhil for helping and guiding us to finish this work.