Genetic Algorithm (GA)

Modeled after the evolutionary process theory.

Genetic Algorithm starts with the creation of a population of designs (a generation). These designs are then ranked with respect to their fitness. Fitness is a measure of how good a design is and it is calculated as a function of constraint violation and objective function values. Selected designs are then reproduced through the application of genetic operators, typically crossover and mutation. The individuals that result from this process (the children) become members of the next generation. This process is repeated for many generations until the evolution of a population converges to the optimal solution.

Usability Characteristics

Genetic Algorithm differ from conventional optimization techniques in the following ways:
- They are classified as exploratory methods.
- They work on a population of designs at once.
- The design population can be run in parallel.
- They do not show the typical convergence of other optimization algorithms. You will typically select the maximum number of iterations (generations) to be evaluated. A number of solver runs is executed in each generation, with each run representing a member of the population.
Genetic Algorithm does a global search.
They are well suited for discrete problems.
Genetic Algorithm is computationally expensive as it requires a large number of runs. In HyperStudy, a local search algorithm (Hooke-Jeeves or a response surface method) is utilized to improve the efficiency of Genetic Algorithm.
In HyperStudy, population size is calculated automatically according to the optimization problem that you defined. It can also be modified manually.
In HyperStudy, both a binary and a real coded Genetic Algorithm exists. Default is the real coded Genetic Algorithm as it is more efficient than the binary coded Genetic Algorithm.
Genetic Algorithm terminates if one of the conditions below are met:
- The convergence criteria is satisfied. This occurs when the minimum number of allowable iterations (Minimum Iterations) are run, the current design is feasible (Constraint Violation Tol. (%)), and the change of objective during m successive iterations is small (less than 0.001).
- The maximum number of allowable iterations (Maximum Iterations) is reached.
- An analysis fails and the “Terminate optimization” option is the default (On Failed Evaluation).

Settings

In the Specifications step, change method settings from the Settings and More tabs.

Note: For most applications the default settings work optimally, and you may only need to change the Maximum Iterations and On Failed Evaluation.

Table 1. Settings Tab
Parameter	Default	Range	Description
Maximum Iterations	200	>0	Maximum number of iterations allowed.
Minimum Iterations	25	>0 <=Maximum Iterations	Processes at least Minimum Iterations iteration steps. Use this setting to prevent pre-mature convergence. By setting Minimum Iterations to be the same as Maximum Iterations, the defined number of iteration steps will be run.
Population Size	0	Integer > 1	If Population Size is 0, then population size is calculated according to the following equation, where N is the number of input variables. $P o p u l a t i o n S i z e = [3 + 37 e^{- {(\frac{N - 1}{5.806})}^{0.5}}] N$ If Population Size is greater than 0, then population size uses the user defined value. If the allowable computational effort is limited, set your own value. Tip: In general, it is better to process at least 25 iteration steps.
Global Search	2	Integer 1 2 3	Controls the global search ability. The larger value of Global Search, the larger probability that the global optimal solution can be reached, but more calculating time is needed.
Constraint Violation Tol. (%)	0.1	>0.0	Global maximum allowable percentage constraint violation. Constraints must not be violated by more than this value in the converged design. When Genetic Algorithm is converged, the minimum number of allowable iterations (Minimum Iterations) are run, the current design is feasible (Constraint Violation Tol. (%)), and the change of objective during m successive iterations is small (less than 0.001). $\begin{array}{l} c_{\max}^{k} < g_{\max} \\ {\begin{matrix} \| \frac{f^{k} - f^{k - m}}{f^{0}} \| < 10^{- 3}, if (\| f^{0} \| > 1) \\ \| f^{k} - f^{k - m} \| < 10^{- 3}, if (\| f^{0} \| \leq 1) \end{matrix} \end{array}$ Where, $f$ is the objective value; $k$ is the current iteration number; $m$ is proportional to the global search parameter; $c_{\max}$ is the maximum constraint violation; $g_{\max}$ is the allowable constraint violation.
Type	Real	Real or Binary	Real Real coded Genetic Algorithm is used. Note: When Type is Real, Discrete States, Number of Contenders, Penalty Multiplier and Penalty Power are grayed out. Binary Binary coded Genetic Algorithm is used. Note: When Type is Binary, then Distribution Index is grayed out. In general, real coded Genetic Algorithm performs better than binary coded Genetic Algorithm. For discrete optimization problems, binary coded Genetic Algorithm could be better.
On Failed Evaluation	Terminate optimization	Terminate optimization Ignore failed evaluations	Terminate optimization Terminates with an error message when an analysis run fails. Ignore failed evaluations Ignores the failed analysis run.

Table 2. More Tab
Parameter	Default	Range	Description
Discrete States	1024	Integer > 1	Number of discrete values uniformly covering the range of continuous variables including upper and lower bound. Tip: Select as a power of 2, for example 64 = 2^6, 1024 =2^10, and so on. A larger value allows for higher solution precision, but more computational effort is needed to find the optima.
Mutation Rate	0.01	0.0-1.0	Mutation rate (probability). Larger values introduce a more random effect. As a result, the algorithm can explore more globally but the convergence could be slower. Tip: Recommended range: 0.001 – 0.05
Elite Population (%)	10	1.0-50.0	Percentage of population that belongs to elite. The one with highest fitness value is directly passed to the next generation. This is a very important strategy, as it ensures the quality of solutions be non-decreasing. A larger value means that more individuals will be directly passed to the next generation, therefore new gene has less chance to be introduced. The convergence speed could be increased. The drawback is that too large of values could cause premature convergence. Tip: Recommended range: 1.0 – 20.0.
Random Seed	1	Integer 0 to 10000	Controlling repeatability of runs depending on the way the sequence of random numbers is generated. 0 Random (non-repeatable). >0 Triggers a new sequence of pseudo-random numbers, repeatable if the same number is specified.
Number of Contenders	2	Integer 2 to 5	Number of contenders in a tournament selection. For larger values, individuals with lower fitness value have less chance to be selected. Thus, the good individuals have more chance to produce offspring. The bad effect is that, diversity of the population is reduced. The algorithm could converge prematurely.
Penalty Multiplier	2.0	>0.0	Initial penalty multiplier in the formulation of the fitness function as exterior penalty function. Penalty multiplier will be increased gradually with iterating steps going on. In general, larger values allow the solution to become feasible with less iteration steps; but too large of a value could result in a worse solution. Tip: Recommended range: 1.0 – 5.0.
Penalty Power	1	>0.0 <10.0	Penalty power in the formulation of the fitness function as exterior penalty function. Tip: Recommended range: 1.0 – 2.0.
Distribution Index	5	Integer 1 to 100	Distribution index used by real coded Genetic Algorithm. Controls offspring individuals to be close to or far away from the parent individuals. Increasing the value will result in offspring individuals being closer to the parents. Tip: Recommended range: 3.0 – 10.0.
Max Failed Evaluations	20,000	>=0	When On Failed Evaluations is set to Ignore failed evaluations (1), the optimizer will tolerate failures until this threshold for Max Failed Evaluations. This option is intended to allow the optimizer to stop after an excessive amount of failures.
Hybrid Algorithm	Hooke-Jeeves method	Hooke-Jeeves method Meta-model based method No hybrid	Hybrid algorithm used in Genetic Algorithm. Note: This parameter is used in Genetic Algorithm real type. It is not available for binary type.
Constraint Threshold	1.0e-4	>0.0	Used for constraint value calculation. In general, constraint value is normalized to its bound value. One exception is that, constraint value is not normalized if its absolute bound value is less than this parameter. Tip: Recommended range is 1.0e-6 ~ 1.0.
Use Inclusion Matrix	No	No With Initial Without Initial	No Ignores the Inclusion matrix With Initial Runs the initial point. The best point of the inclusion or the initial point is used as the starting point. Without Initial Does not run the initial point. The best point of the inclusion is used as the starting point.