Convergence

Convergence criteria can be specified for each optimization algorithm by setting metrics parameter with an array of ConvergenceMetric-derived objects, e.g.

Evolutionary.optimize( f, x0, CMAES(metrics=[Evolutionary.AbsDiff(1e-5)]) )

Use Evolutionary.converged(result) to check convergence of the optimization algorithm. It is possible to access a minimizer using Evolutionary.minimizer(result) even if all convergence flags are false. This means that the user has to be a bit careful when using the output from the solvers. It is advised to include checks for convergence if the minimizer or minimum is used to carry out further calculations.

Convergence Metrics

Evolutionary.AbsDiffType

Absolute difference convergence for single objective optimization.

This convergence metric allows to estimate an absolute difference between consecutive states of the optimization algorithm, and triggers convergence when,

• |f(x) - f(x')| < ε

where ε is a tolerance value, x and x' are previous and current minimizers found by the optimization algorithm.

Evolutionary.RelDiffType

Relative difference convergence metric for single objective optimization.

This convergence metric allows to estimate a relative difference between consecutive states of the optimization algorithm, and triggers convergence when,

• |f(x) - f(x')|/|f(x')| < ε

where ε is a tolerance value, x and x' are previous and current minimizers found by the optimization algorithm.

Evolutionary.GDType

(Inverse) generational distance convergence metric for multi-objective optimization.

This convergence metric allows to estimate a (inverse) generational distance between consecutive states of the multi-objective optimization algorithm, and triggers convergence when,

• $GD(S,R) = \frac{1}{|S|}\left(\sum_{s \in S} \min_{r \in R}||f(s) - f(r)||^2\right)^{1/2} < \varepsilon$

where ε is a tolerance value, S and R are discrete Pareto front approximations for previous and current state of the optimization algorithm.

The inverse generational distance calculated as IGD(S,R) = GR(R,S). To use, IGD metric pass true value to the metric constructor argument, e.g.

• GD() create a generational distance metric
• GD(true) create an inverse generational metric

Auxiliary Functions

Evolutionary.gdFunction
gd(A,R)

Calculate a generational distance between set A and the reference set R. This metric measures the convergence, i.e. closeness of the non-dominated solutions to the Pareto front, of a population.

Note: Parameters are column-major matrices.

Evolutionary.igdFunction
igd(S,R)

Calculate an inverted generational distance, gd, between set S and the reference set R. Parameters are column-major matrices.

Evolutionary.spreadFunction
spread(S,R)

Returns a diversity metric of a population of set S to the reference set R`.