SELECTION Deterministic Stochastic Proportionate selection: - - PDF document

SELECTION

• Deterministic
• Stochastic

– Proportionate selection: Roulette Wheel Selection – Rank based selection – Tournament Selection

COMPETITION

• µ (mu) denotes the size of the (parent)

population

• λ (lambda) denotes the number of offspring

produced

• (µ, λ) competition: new population is formed

exclusively from offspring (aka Generational EA)

• (µ + λ) competition:

new population is formed from old population (parents) and

• ffspring (called Steady State EA when λ <<

µ)

Travelling Salesperson Problem (TSP)

• Problem: given x cities, what is the short-

est tour in which each city is visited once and only once?

• NP-hard: no known algorithm which is a

polynomial function of the number of cities

• Example applications

– Circuit board drilling (17,000 cities) – X-ray crystallography (14,000 cities) – VLSI fabrication (1.2 million cities)

• Trial solution representation: permutation
• f integers

• Fitness function: straightforward
• Genetic operators: not obvious

Boolean Satisfiability Problem (SAT)

• Given a compound Boolean statement, find

a set of truth assignments such that the statement evaluates to TRUE.

• NP-complete: NP-hard and NP
• Trial solution representation: binary string
• Fitness function: not obvious
• Genetic operators: crossover and mutation

Theoretical Foundations of Genetic Algorithms

• Schema - a template allowing exploration
• f similarities among individuals (binary strings)
• A schema consists of 0’s, 1’s and *’s (don’t

care symbol)

• One particular schema represents all strings

(a hyperplane or subset of the search space) which match it on all positions other than ‘*’

• Every schema matches exactly 2r strings,

where r is the number of *’s

• Each string of length m is matched by 2m

schemata

• For length m there are 3m possible schemata
• The order of schema S (denoted by o(S))

is the number of fixed positions (non-don′t care positions) in S (= m − r)

• The defining length of schema S (denoted

by δ(S)) is the distance between the first and the last fixed string positions (it de- fines the compactness of information con- tained in a schema)

• The number of strings in a population at

time t matched by schema S is denoted by ξ(S, t)

• The fitness of a schema at time t, eval(S, t),

is defined as the average fitness of all strings in the population matched by the schema S

• Population consists of strings {v1, · · · , vpopsize}
• Given p strings {vi1, · · · , vip} in population

matched by schema Si, then: eval(Si, t) =

p

• j=1

eval(vij)/p (1)

• Total fitness of population F(t) = popsize

i=1

eval(vi)

• Assume generational model with propor-

tional (roulette wheel) selection

• Single string selection chance: eval(vi)/F(t)
• Selection chance for average string matched

by schema S: eval(S, t)/F(t)

• Combining the above we get:

E[ξ(S, t+1)] = ξ(S, t)·popsize·eval(S, t)/F(t) (2)

• Average population fitness F(t) = F(t)/popsize
• Reproductive schema growth equation:

E[ξ(S, t + 1)] = ξ(S, t) · eval(S, t)/F(t) (3)

• If schema S remains above average by ǫ%,

in other words eval(S, t) = (1 + ǫ) · F(t), then we obtain the following geometric pro- gression equation: E[ξ(S, t)] = ξ(S, 0)(1 + ǫ)t (4)

• Now assume 1-point crossover with crossover

chance pc; a crossover point is selected uni- formly among m − 1 possible locations

• Probability of schema destruction:

pd(S) ≤ pc · δ(S) m − 1 (5)

• Consequently, probability of schema sur-

vival: ps(S) ≥ 1 − pc · δ(S) m − 1 (6)

• New reproductive schema growth equation:

E[ξ(S, t+1)] ≥ ξ(S, t)·eval(S, t) F(t)

• 1 − pc · δ(S)

m − 1

• (7)
• Finally, add mutation with bit mutation

chance pm; single bit survival is 1 − pm

• Schema survival ps(S) = (1 − pm)o(S)

• Since pm ≪ 1, schema survival can be ap-

proximated as ps(S) ≈ 1 − o(S) · pm

• Combined reproductive schema growth equa-

tion: E[ξ(S, t + 1)] ≥ ξ(S, t) · eval(S, t) F(t)

• 1 − pc · δ(S)

m − 1 − o(S) · pm

• (8)
• Schema Theorem: Short, low-order, above-

average schemata receive exponentially in- creasing trials in subsequent generations of a genetic algorithm

• Building Block Hypothesis: A genetic algo-

rithm seeks near-optimal performance through the juxtaposition of short, low-order, high- performance schemata, called the building blocks

• Consequence: the manner in which we en-

code a problem is critical for the perfor- mance of a GA - it should satisfy the idea

• f short building blocks

GRAY CODING

• Desired: points close to each other in rep-

resentation space also close to each other in problem space

• This is not the case when binary numbers

represent floating point values

• m is number of bits in representation
• binary number

b = (b1, b2, · · · , bm)

• Gray code number

g = (g1, g2, · · · , gm)

binary gray code 000 000 001 001 010 011 011 010 100 110 101 111 110 101 111 100 PROCEDURE Binary-To-Gray g1 ⇐ b1 for k = 2 to m do gk ⇐ bk−1 XOR bk end for

PROCEDURE Gray-To-Binary value ⇐ g1 b1 ⇐ value for k = 2 to m do if gk = 1 then value ⇐ NOT value end if bk ⇐ value end for

Evolution of Evolution Strategies

• Earliest ES had popsize = 1 and the sole

genetic operator employed was mutation

• EC literature often refers to the (1+1) ES

as “two-membered evolution strategy”

• Individual represented as pair of float-valued

vectors (x, σ), with x representing a point in search space and σ a vector of standard deviations

• Mutation: xt+1 = xt+N(0, σ) where N(0, σ)

is a vector of independent random Gaus- sian numbers with a mean of zero and stan- dard deviations σ

• Competition:
• ffspring (xt+1, σ) replaces

its parent (xt, σ) iff fitness(xt+1) > fitness(xt)

• If all components of σ are identical and the
• ptimization problem is regular, it is pos-

sible to prove the Convergence Theorem: For σ > 0 and a regular optimization prob- lem, p{ lim

t→∞ f(xt) = fopt} = 1

• Optimize convergence rate with Rechen-

berg’s “1/5 success rule”: The ratio ϕ of successful mutations to all mutations should be 1/5. Increase the vari- ance of the mutation operator if ϕ is greater than 1/5, otherwise decrease it.

• Applying the 1/5 rule every k generations

can be performed as follows: σt+1 =

    

cd · σt if ϕ(k) < 1/5 ci · σt if ϕ(k) > 1/5 σt if ϕ(k) = 1/5 where ϕ(k) is the mutation success ratio during the last k generations. Schwefel used in a number of his experiments the following values: cd = 0.82 and ci = 1.22(= 1/0.82)

• For some classes of functions this rule lead

to premature convergence; solution: in- creased population size

• EC literature often refers to an (µ + 1)-ES

as “multi-membered evolution strategy”

• All individuals have equal mating probabil-

ities

• Recombination can be added in the form
• f uniform crossover
• Further ES evolution lead to the (µ+λ)-ES

and the (µ, λ)-ES

• The deterministic 1/5 rule was replaced by

a stochastic process: σt+1 = σt · eN(0,∆σ) where ∆σ is a parameter of the process