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Optimizing with Genetic Algorithms
by
Benjamin J. Lynch
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Outline
•<= /span>What are genetic algorithms?
–<= /span>Biological origins
–<= /span>Problems with Newton-type optimizers
•<= /span>How do we apply genetic algorithms?
–<= /span>Options to include
•<= /span>Encoding
•<= /span>Selection
•<= /span>Recombination
•<= /span>Mutation
–<= /span>Strategies
•<= /span>What programs can we use?
–<= /span>How do we parallelize?
•<= /span>MPI, fork/wait
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What are genetic algorithms? (GAs)
•A class of stochastic search strategies modeled after evolutionary mechanisms •A major difference between natural GAs and our= GAs is th= at we do not need to follow the same laws observed in nature. •Although modeled after natural processes, we c= an design ou= r own encoding of information, our own mutations, and our own selection criteri= a.
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Why would we use genetic algorithms?  Isn’t there a simple solutio= n we learned in Calculus?
•Newton-Raphson and it’s many relatives and variants are based on the use of local information. •The function value a= nd the derivatives with respect to the parameters optimized are used to take a step in an appropriate direction towards a local maximum or minimum.
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5
Newton-Raphson
&= #8226;Perfect for a parabola
–H is the Hessian (2nd derivative with respect <= /span>to all parameters optimized)
–g is the gradient
–x is the step to minimize the gradient
•
x
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Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
6
Where Newton-Raphson Fails
•A local method will only find lo= cal extrema
If we start our search here
We’ll end up here
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How do we use GAs to optimize the parameters we̵= 7;re interested in?
•Choose parameters to optimize
•<= /span>Determine chromosomal representation of parameters
•Generate initial population of individual= s (chr= omosomes)
•<= /span>Evaluate fitness of each individual to reproduce
•Allow selection rules and random behavior to select next population
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Definitions for this talk
•Gene – encoded f= orm of a parameter being optimized
•Chromosome – the complete set of genes (parameters) which uniquely describe= an individual
•Locus – the posi= tion of a piece of data within a chromosome
•Fitness – A valu= e we are trying to maximize
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•1. Initialization of population&#= 13;
•2. Evalua= tion of fitness
•3. Selection
•4. Recombination
•5. Repeat 2-4
Steps in a GA
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Overview of a Genetic Algorithm
Eval= uation
Sele= ction
Reco= mbination
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11
Initialization
<= span style=3D'color:#333399;mso-color-index:5'>•Decide how to encode the parameters
•Create an initial population with randomized chromosomes
•
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12
Create Initial Population
•<= /span>Population size is chosen (1-10 individuals/parame= ter = optimized for most applications)
•<= /span>Parameters to be optimized are encoded.
–<= /span>Binary, Base 10
–Let’s say we have 2 parameters with initial = values of 32 and 13.  In binary and base 10 they would = look like: =
»
–
–
•
•
•
100000001101
3213<= /span>
Chromosome of the
individual
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13
How binary genes translate into other general parameters
•101011101001010111010000100110
698  <= /span>        = ;       349        = ;       38
You need to
understand the
system you are
optimizing in
order to determine
the proper
parameter range
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Presentation
14
Create Initial Population
•<= /span>After we choose a population size and encoding = method, we must choose a maximum range for each = parameter.  Ranges for parameters should be determined based on what would be physically = reasonable (if you’re interested in solving a physical problem).
•<= /span>The initial population can be generated by = = randomizing the genes for each chromosome of the <= /span><= span style=3D'font-size:88%'>initial population
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15
Evaluation
•Assign a fitness value to each individual based on the parameters derived from its chromosome
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Evaluate the Fitness
•Next we must evaluate the fitness.  The fitness function is somehow based on the genes of the individual and should reflect how good a given set= of parameters is.
–Ability to swim faster than other robotic fish
–Ability of a density functional to better predict chemical phenomena
–Power output of a chemical laser
•<= /span>A larger fitness value will give the individual a = higher probability of being the parent of one or <= /span>= more children.
•
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Evaluate the Fitness
•Evaluation of the fitness is the computationally-intensive portion of a GA optimizatio= n
•<= /span>Each chromosome holds the information that = = uniquely describes an individual.
•<= /span>Each chromosome/(parameters set)/individual can be evaluated separate from the other individuals.&= #13;
–<= /span>GA optimizations are typically described as embarrassingly parallelizable
•<= /span>The evaluation of the chromosomes reduces down to a fitness value for each individual which will = be = used in the next step
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Selection of the Parents
•<= /span>The parents must be selected based on their fitnes= s
•<= /span>The individuals with a higher fitness must have a = = higher probability of having offspring =
•<= /span>There are several methods for selection
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19
Roulette Wheel Selection
•<= span style=3D'font-size:86%'>Probability of parenthood is proportional to fitness. •The wheel until two parents are selected. •The two parents create one offspring. •<= span style=3D'font-size:86%'>The process is repeated to create a new population for the next generation.
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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
20
Roulette Wheel Selection
•This form of selection = has problems if the fitness changes by orders of magnitude.
•If two individuals have a much higher fitness, they could be the parents for every child in the next generation.
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file:///C:/74CBA921/OptimizingWithGA_files/slide0053.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" PowerPoint Presentation
21
Another Reason Not to Use the Roulette Wheel
•If the fitness value is very close, the parents will be chosen with equal probability, and the function will cease to optimize.
•<= /span>Roulette selection is very sensitive to the problem being solved and generally requires modifications to work at all.
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Rank Selection
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•All individuals in the population are ranked according to fitness •<= span style=3D'font-size:86%'>Each individual is assigned a weight inversely proportional to the rank (or other similar scheme).
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23
Tournament Selection
&#= 8226;Tournament Selection
–4 individuals are randomly selected from the population.  Two are eliminated and two= become the parents of a child in the next generation
A
B
C
D
A
D
Fitness(D) > Fitness(C)<= /div>
Fitness(A) > Fitness(B)<= /div>
A simple and very
robust method for
choosing parents of
the next generation
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24
Tournament Selection
•Selection of parents= = continues until a new population is completed. •Individua= ls might be the parent to several children, or no children.
A
B
C
D
A
D
Fitness(D) > Fitness(C)<= /div>
Fitness(A) > Fitness(B)<= /div>
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25
Tournament and Rank Selection
•Tournament selection= = is very similar to rank selection (when we = assign a weight =3D 1/rank). •In the limit of a la= rge p= opulations, both have similar statistics