Theoretical Analysis of Jumping Gene Operations - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 69

4.3 TheoremsofEquilibriumandDynamicalAnalysis
Based.on.the.schemata.evolution.equations.of.copy-and-paste.and.cut-and-
paste,.(4.30).and.(4.52),.the.following.two.theorems.can.be.derived.

Theorem4.1:TheoremofEquilibriumforCopy-and-Paste

For.any.primary.schemata.competition.set. S .with.order. o (ξ),.all.the.sche-
mata.in. S .will.globally.asymptotically.converge.to. 1/2 ( )

o ξ .under.a.copy-
and-paste.operation,.despite.the.initial.proportion.of.the.schemata.in.the.
population.

Theorem4.2:TheoremofEquilibriumforCut-and-Paste

For.any.primary.schemata.competition.set. S ξ . with.order. o (ξ),.all.the.sche-
mata. in. S ξ . will. globally. asymptotically. converge. to. 1/2 ( ξ . under. a. cut-

and-paste.operation,.despite.the.initial.proportion.of.the.schemata.in.the.
population.

o

The.proofs.of.the.theorems.are.given.next.

4.3.1 Distribution Matrix for Copy-and-Paste

Consider.the.copy-and-paste.operation.and.let. p copy 1 = .for.which . Equation.

(4.30) . can.be.rewritten.as

∑ ∑

( ) i

P ( ξ , t + 1) = a × P ( ξ , ) t × P ( ξ , ) t .for. S . i = , 1,.2 ,…M ,. (4.53)

ξ ∈ ξ ,

i mn m n i

ξ ∈ S ξ ∈ S

.

m ξ n ξ

where

L L − L L −

g g

∑ ∑

( ) i

[ ]

a = K ∆ ξ ( , V ; ξ , G ) ( ∆ ξ , V ′ ξ ; , V ′ ) . (4.54)

mn i k m c k , i k n k

.

c = 0 k = 0

with

1

K = .

2

( L L g − + 1)

.