Evolutionary Computation
Genetic Algorithms
Sources
• Material in this lecture comes from, “Handbook of
Natural Computing,” Editors Grzegorz Rosenberg,
Thomas Back and Joost N. Kok, Springer 2014.
In set S
In set X
Coding function c: X  S
Decoding function is the inverse of c
• GAs manipulate coded versions of the problem parameters instead of the
parameters themselves, i.e., the search space is S instead of X
• Conventional methods search from a single point while GAs operate on a
whole population (reduces the risk of becoming trapped in a local extreme)
• No use of auxiliary information about the objective function such as
derivatives. GAs do need a meaningful decoding function
• GAs use probabilistic transition operators while conventional methods for
continuous optimization apply deterministic transition operators
312=52*6 evaluations of the fitness function
Conventional method likely gets stuck
in a local maxima
Roulette wheel selection
One-point crossing over
Bitwise mutation
𝑟𝐻,𝑡 𝑓 − 𝐻, 𝑡
∗ 1−
𝑚 𝑓 − (𝑡)
𝑟𝐻,𝑡 𝑓 − 𝐻, 𝑡
∗ 1−
𝑚 𝑓 − (𝑡)
𝑟𝐻,𝑡 𝑓 − 𝐻, 𝑡
∗ 1−
𝑚 𝑓 − (𝑡)
Schemata with above-average fitness and short defining length tend to produce more
offspring than others: Called building blocks. The number of strings belonging to
building blocks grows exponentially.
2-armed bandit problem: left arm produces an outcome with mean mu_1 and variance
sigma_1^2, right arm produces ….
The optimal strategy is to allocate slightly more than an exponentially increasing number of
trials to the observed best arm
Consider the two schemata of order one which have their specification in the same
position. The Schema theorem states that the GA implicitly decides between these two
schemata based on incomplete data (observed average fitness values).  GA solves a lot
of 2-armed bandit problems in parallel
Schemata of higher order  GA is solving an enormous number of k-armed bandit
problems in parallel. Also here the observed better alternatives should receive an
exponentially increasing number of trials and this us exactly what a GA does!
Criticism:
• Schema theorem is an inequality
• It only applies to one generation into the future: how do hyperplanes get sampled in the
future?
• 11*** and *00** have an inconsistency in the 2nd bit: over multiple iterations both
cannot receive increasing trials. The Schema theorem does not predict how such
inconstistencies are sorted out.
• Problems with a great deal of consistency (i.e., the most fit schemata tend to agree) 
GA effective
• GAs often use very small populations  only smaller order schemata are sampled in
reality (search relies more on hill climbing, which can be very effective)
• Schema information up to any fixed order can be computed in polynomial time for some
NP complete problem 