Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 203

G c,k G c,k

~

ξ m ξ m

c c

~

ξ n ξ n

k k

~

ξ n ξ n

V k V k V k V k

ξ i ξ j

p 1 p 2 p 1 p 2

Figure a.1
Visualization.of.finding.

 ξ .and.  ξ .(logical.XOR.is.performed.in.striped.areas).

m n

   q

(

c k

)

q ( , , c k , )

Therefore,.for. = ξ , ξ , , ,.there.exists.a.unique. = ξ ξ .such.that

m n m n

( ) i

f ( ) q = δ ( f ( ξ , V f ), ( ξ , G )) × δ ( f ( ξ , V f ′ ), ( ξ , V ′ ))

T i k T m c k , T i k T n k

 

( f ( , V f ), ( , G )) ( f ( , V f ), ( , V ))

= δ ξ ξ × δ ξ ′ ξ ′

T j k T m c k , T j k T n k

j



= f q ( )

. .

Based.on.the.logic.speciied.in . Equations.(A.4) ,. it.can.also.be.proved.that.if.
(

( ) j 

f

( ) i

q ) f

( ) i

( q ) f ( ) q

= = ,.then. 1 q = ..This.completes.the.proof.

1 2

2

ProofofLemma4.2
Obviously,.

( ) ( )

P ξ , t

Σ P ξ , t = Σ

1

= 1 .satisfies.the.constraint.of. ..Assuming.

i

i

S M

ξ ∈ S ξ ∈

i ξ i ξ

( )

∀ξ ∈ S

that. P ξ , t =

1

,. ,.one.has

i

i

M

L L − L L −

M M

g g

∑ ∑ ∑

1

( ) ( )

×

( ) ( )

( ) ( ) ( )

P ξ , t + 1 = K δ f ξ , V f , ξ , G × δ f ξ , V f ′ , ξ , V ′

i T i k T m c k , T i k T n k

2

M

m = 1 n = 1 c = 0 k = 0

L L − L L −

M M

g g

K
M

∑ ∑ ∑

( )

( )

( )

( ) ( ) ( )

= δ f ξ , V f , ξ , G × δ f ξ , V f ′ , ξ , V ′

T i k T m c k , T i k T n k

2

m = 1 n = 1 c = 0 k = 0

K
M

∑

( ) i

= f ( ) q

.

2

. (A.5)

q Q ∈