Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 204

Using.Lemma.4.1,

∑ ∑

( ) ( )

i j



f

( )

q = f

( )

q ≡ C



q Q ∈ q Q ∈

. . (A.6)

Substituting . Equation.(A.6) . into . Equation.(A.5) , .one.has

KC
M

( )

P ξ , t + 1 = , ∀ξ ∈ S . (A.7)

i i

2

.

∑ ∑

KC
M

KC
M

( )

Since. P ξ , t + 1 = = × M = 1 ,.1.one.has

i

2 2

ξ ∈ S ξ ∈ S

i ξ i

M
K

. (A.8)

C = .

.

Substituting. Equation.(A.8) .into . Equation.(A.6) , .one.has

∑

M
K

( )

i

f ( ) q = .

. . (A.9)

q Q ∈

M M

( ) ( )

i i

Based.on.the.definition,. Σ Σ a = K × Σ f ( ) q = K ×

M
K

= M ..This.completes.

mn

m = 1 n = 1 q Q ∈

the.proof.

ProofofLemma4.3
Consider

L L − L L −

M M g g

∑ ∑ ∑ ∑

( ) i

[ ]

a = K δ ( f ( ξ , V f ), ( ξ , G )) × δ ( f ( ξ , V f ′ ), ( ξ , V ′ ))

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

i

i = 1 i = 1 c = 0 k = 0

L L − L L −

M

g g

∑ ∑ ∑

[ ]

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

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

i

c = 0 k = 0 i = 1

L L − L L −

M g g

∑ ∑ ∑

 

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

T i k T p c k , T i k T q k

.

c = 0 k = 0 i = 1

. . (A.10)