Package: nsde
Non-dominated Sorting Differential Evolution (NSDE)The Non-dominated Sorting Differential Evolution (NSDE) algorithm combines the strengths of Differential Evolution [1]
with those of the Fast and Elitist Multiobjective Genetic Algorithm NSGA-II [2], following the ideas presented in [3],
to provide an efficient and robust method for the global optimization of constrained and unconstrained, single- and
multi-objective optimization problems.InstallationNSDE is available onPyPi, so it can be installed usingpip install nsde. It can
also be installed usingpython setup.py installfrom the root of this repository.Note that several methods of NSDE are written in C++ to accelerate the code. Therefore, in order to install NSDE from
source, a working C++ compiler is required. For Windows, this has only been tested using Visual Studio.UsageTo solve an optimization problem using NSDE, define a function which takes a single input argument,x, which
represents the design vector, and outputs a list of objective values,f, and constraints,g(optional). For example:defunconstrained(x):return[x**2,(x-2)**2]defconstrained(x):returnsum(x*x),1-xThe first represents an unconstrained problem with two objectives. The second represents a constrained problem with a
single objective.It is important to note that constraints are expected to be in the formg(x) <= 0. It is the user's responsibility to
transform constraints into this form.Once formulated, problems can be solved using NSDE as follows:importnsdeopt=nsde.NSDE()opt.init(constrained,bounds=[(-100,100)]*2)opt.run()x_opt=opt.pop[0]f_opt=opt.fit[0]In the last two lines, the optimal design vector and objective function value are retrieved from the optimizer.
As you can see, they correspond to the first elements of the optimizer'spopandfitarrays.
These are multi-dimensional arrays which store the population's design vectors and objective function values for each
individual in the population (1 row per individual).
At each new generation, these arrays are sorted such that the first rows correspond to the best individual and the last
to the worst.For multi-objective problems, it is more useful to look at the pareto front:opt=nsde.NSDE()opt.init(constrained,bounds=[(-100,100)])opt.run()pareto=opt.fit[opt.fronts[0]]When calling.run()on an instance of theNSDEclass, the problem is solved until convergence or the maximum number
of generations is reached. Alternatively, it is also possible to solve problems one generation at a time by treating
the instance of theNSDEclass as an iterator:forgenerationinopt:print("f_opt = ",generation.fit[0])OpenMDAOThe NSDE algorithm can also be used inOpenMDAOusing theNSDEDriverclass.ReferencesStorn, R., and Price, K. "Differential Evolution – A Simple and Efficient Heuristic for global Optimization over
Continuous Spaces." Journal of Global Optimization, Vol. 11, No. 4, 1997, pp. 341–359.doi:10.1023/a:1008202821328.Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II.”
IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, 2002, pp. 182–197.doi:10.1109/4235.996017.Madavan, N. K. "Multiobjective Optimization Using a Pareto Differential Evolution Approach." Proc. of IEEE Congress
on Evolutionary Computation. Vol. 2, 2002, pp. 1145-1150.doi:10.1109/CEC.2002.1004404.