188 Beam-based Correction and Optimization for Accelerators
where u is a random number drawn from the uniform distribution in (0, 1)
and µ c is a control parameter.
The mutation operation is performed by adding a random variation to
each parameter,
x
k = x k + L k δ k ,
(7.20)
where L k is the range of the x k parameter and δ k is a random number calculated with
δ(v <
1
2
) = (2v)
1
µm +1 − 1, or δ(v >
1
2
) = 1 − (2(1 − v))
1
µm +1 ,
(7.21)
with random variable v drawn from the uniform distribution in (0, 1) and µ m
is another control parameter.
For problems with a single objective function, the selection of solutions
to enter the next generation is done by simple sorting. Genetic algorithms
can be conveniently extended to multi-objective problems by applying nondominated sorting to select the fittest solutions. Solutions in the leading fronts
enter the next generation until the quota is filled up. When the last front with
qualified solutions yields more solutions than needed, the solutions can be randomly picked or chosen to maximize the diversity of the surviving population,
using the so-called crowding distance as the criterion. Constraints are easy to
implement for genetic algorithms as they can be considered as a part of the
fitness criteria; solutions that violate the constraints are given lower fitness.
The initial population can be generated randomly throughout the parameter space. However, it often works better if it is randomly generated in a small
neighborhood around a good solution.
Genetic algorithms are powerful methods for design optimization, yet they
also have limitations. The algorithms may have low efficiency as many of the
children solutions are similar to the parent solutions that have been previously evaluated. It may be difficult to apply the genetic algorithms for high
dimension problems as it would require a large population of solutions for the
algorithms to work and the corresponding computational cost for evaluating
these solutions could be prohibitive. The algorithms can converge prematurely
to a non-optimal region when all surviving solutions are similar. Improving the
percentage of mutation can alleviate the problem of premature convergence,
but it will also slow down the speed of convergence.
When used for the online optimization, genetic algorithms suffer an additional limitation. Negative values in the measurement noise will give some
solutions an advantage over the others. These solutions tend to survive the
selection operation, driving out the real good solutions and thus defeat the
working principles of the algorithm. Re-sampling for the population can mitigate the noise problem [14].
Particle swarm optimization (PSO) [71, 92, 59]: Particle swarm optimization is similar to the genetic algorithms in that it also manipulates a
where u is a random number drawn from the uniform distribution in (0, 1)
and µ c is a control parameter.
The mutation operation is performed by adding a random variation to
each parameter,
x
k = x k + L k δ k ,
(7.20)
where L k is the range of the x k parameter and δ k is a random number calculated with
δ(v <
1
2
) = (2v)
1
µm +1 − 1, or δ(v >
1
2
) = 1 − (2(1 − v))
1
µm +1 ,
(7.21)
with random variable v drawn from the uniform distribution in (0, 1) and µ m
is another control parameter.
For problems with a single objective function, the selection of solutions
to enter the next generation is done by simple sorting. Genetic algorithms
can be conveniently extended to multi-objective problems by applying nondominated sorting to select the fittest solutions. Solutions in the leading fronts
enter the next generation until the quota is filled up. When the last front with
qualified solutions yields more solutions than needed, the solutions can be randomly picked or chosen to maximize the diversity of the surviving population,
using the so-called crowding distance as the criterion. Constraints are easy to
implement for genetic algorithms as they can be considered as a part of the
fitness criteria; solutions that violate the constraints are given lower fitness.
The initial population can be generated randomly throughout the parameter space. However, it often works better if it is randomly generated in a small
neighborhood around a good solution.
Genetic algorithms are powerful methods for design optimization, yet they
also have limitations. The algorithms may have low efficiency as many of the
children solutions are similar to the parent solutions that have been previously evaluated. It may be difficult to apply the genetic algorithms for high
dimension problems as it would require a large population of solutions for the
algorithms to work and the corresponding computational cost for evaluating
these solutions could be prohibitive. The algorithms can converge prematurely
to a non-optimal region when all surviving solutions are similar. Improving the
percentage of mutation can alleviate the problem of premature convergence,
but it will also slow down the speed of convergence.
When used for the online optimization, genetic algorithms suffer an additional limitation. Negative values in the measurement noise will give some
solutions an advantage over the others. These solutions tend to survive the
selection operation, driving out the real good solutions and thus defeat the
working principles of the algorithm. Re-sampling for the population can mitigate the noise problem [14].
Particle swarm optimization (PSO) [71, 92, 59]: Particle swarm optimization is similar to the genetic algorithms in that it also manipulates a