Processes 2019, 7, 163
2.3.3. Mutation
Mutated parameter values are generated using the following algebraic expression where boldface
denotes vector quantities:
v
i = m best 1
i + F(m best 2
i − m best 3
i ).
(4)
The expression shows that the trial vector element v i , where i designates the index of the parameter
value being mutated, is the sum of a population member element m best 1
i and the scaled difference
between two additional population member elements, m best 2,3
i . The three m best vectors correspond
to randomly selected members of the population that won a single round of tournament selection.
The winner of tournament selection is the parameter value vector that has a lower objective function
evaluation. While each tournament selection is between population members sampled without
replacement, the selection of m best vectors for mutation between rounds of tournament selection
allows sampling with replacement. As a result, m best vectors may be identical. F is the mutation
constant, and can be specified by the user.
2.3.4. Selection
Once the trial member is constructed, the fitness of the member is evaluated. If the trial member
has a lower objective function evaluation than the original member at the current population index,
the trial member is more fit and selected to replace the original member in the population.
2.3.5. Termination
After the entire population of parameter vectors has undergone recombination, the stopping
criteria are assessed to determine if the population has converged on a solution. Termination of
differential evolution is achieved when the maximum number of generations has been reached or
when the threshold value is met. The threshold value can be selected to consider the smallest eigenvalue
of the system, such that the eigenvalue must be sufficiently close to zero to have reached the bifurcation
behavior, or the threshold value will correspond to the best objective function fitness from all members
in the population. The fitness threshold is the default stopping criteria.
2.3.6. Conditions for Optimal Convergence
There are several input parameters to the differential evolution algorithm that can be manipulated
to shift the balance between fast and accurate convergence. Generally, increasing the population size
and mutation constant while decreasing the recombination constant will improve the chance that the
algorithm converges on a global minimum. However, this will result in computational costs that slow
convergence. A population size of 50, and mutation and recombination constants of 0.5, are assigned
as default values for the algorithm and typically enable rapid convergence on a suitable solution.
2.4. Local Optimization Algorithm
Following the differential evolution routine, the objective function can be minimized further
using an optional bounded Broyden–Fletcher–Goldfarb–Shanno algorithm to provide a final local
optimization step [15–18]. The algorithm uses approximated Hessian updates that are dependent
on the approximate gradient at the point in parameter space where the current parameter vector
rests, such that it minimizes in the direction of steepest descent. A one-dimensional line search is
implemented to determine the step size. This local optimization dramatically reduces the final objective
function evaluation for both oscillators and turning points, often yielding a fitness that is minimized
by multiple orders of magnitude. However, this step is not recommended for most Hopf bifurcation
optimization problems, as fitness values smaller than 1 × 10 −3 frequently correspond to damped
oscillatory models.
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2.3.3. Mutation
Mutated parameter values are generated using the following algebraic expression where boldface
denotes vector quantities:
v
i = m best 1
i + F(m best 2
i − m best 3
i ).
(4)
The expression shows that the trial vector element v i , where i designates the index of the parameter
value being mutated, is the sum of a population member element m best 1
i and the scaled difference
between two additional population member elements, m best 2,3
i . The three m best vectors correspond
to randomly selected members of the population that won a single round of tournament selection.
The winner of tournament selection is the parameter value vector that has a lower objective function
evaluation. While each tournament selection is between population members sampled without
replacement, the selection of m best vectors for mutation between rounds of tournament selection
allows sampling with replacement. As a result, m best vectors may be identical. F is the mutation
constant, and can be specified by the user.
2.3.4. Selection
Once the trial member is constructed, the fitness of the member is evaluated. If the trial member
has a lower objective function evaluation than the original member at the current population index,
the trial member is more fit and selected to replace the original member in the population.
2.3.5. Termination
After the entire population of parameter vectors has undergone recombination, the stopping
criteria are assessed to determine if the population has converged on a solution. Termination of
differential evolution is achieved when the maximum number of generations has been reached or
when the threshold value is met. The threshold value can be selected to consider the smallest eigenvalue
of the system, such that the eigenvalue must be sufficiently close to zero to have reached the bifurcation
behavior, or the threshold value will correspond to the best objective function fitness from all members
in the population. The fitness threshold is the default stopping criteria.
2.3.6. Conditions for Optimal Convergence
There are several input parameters to the differential evolution algorithm that can be manipulated
to shift the balance between fast and accurate convergence. Generally, increasing the population size
and mutation constant while decreasing the recombination constant will improve the chance that the
algorithm converges on a global minimum. However, this will result in computational costs that slow
convergence. A population size of 50, and mutation and recombination constants of 0.5, are assigned
as default values for the algorithm and typically enable rapid convergence on a suitable solution.
2.4. Local Optimization Algorithm
Following the differential evolution routine, the objective function can be minimized further
using an optional bounded Broyden–Fletcher–Goldfarb–Shanno algorithm to provide a final local
optimization step [15–18]. The algorithm uses approximated Hessian updates that are dependent
on the approximate gradient at the point in parameter space where the current parameter vector
rests, such that it minimizes in the direction of steepest descent. A one-dimensional line search is
implemented to determine the step size. This local optimization dramatically reduces the final objective
function evaluation for both oscillators and turning points, often yielding a fitness that is minimized
by multiple orders of magnitude. However, this step is not recommended for most Hopf bifurcation
optimization problems, as fitness values smaller than 1 × 10 −3 frequently correspond to damped
oscillatory models.
8