Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 209

A i ,.we.

If.we.sum.all.elements.in.the i th.column.of.the.distribution.matrix.

( )

can.get

L L − L L −

M M

g g

∑ ∑ ∑ ∑

( )

i

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

k m c k , i k i k

mi

i

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

L L − L L −

M

g g

∑ ∑ ∑

= K ∆ ξ ( , V ; ξ , G ) 1 ×

k m c k ,

i

c = 0 k = 0 m = 1

L L − L L −

g g

∑ ∑

( )

N u v

− −

= K 2

c k ,

× 2

c k ,

c = 0 k = 0

. (A.15)

L L − L L −

g g

∑ ∑

( )

N u − − v

= K 2

c k , c k ,

c = 0 k = 0

L L − L L −

g g

∑ ∑

( )

w

= K 2

k

c = 0 k = 0

L L −

g

1

∑

w

= 2

k

L L − + 1

g

.

k = 0

M

( )

i

It.is.easy.to.prove.that. Σ a .decreases.when. L g .increases.

mi

m = 1

ProofofLemma4.4
Based.on.the.definition.of.

i

a mn

( )

.given.in.Equation.(4.54),.consider

L L − L L −

g g

∑ ∑ ∑ ∑

a

( ) i

− a

( ) i

= K ∆ ξ ( , V ; ξ , G )

mn mn k m c k ,

i

ξ ∈ Z ξ ∈ Z ′ c = 0 k = 0

n n

. (A.16)

∑ ∑

× ∆ ξ ( , V ′ ξ ; , V ′ − ) ∆ ξ ( , V ′ ξ ; , V ′ )

i k n k i k n k

ξ ∈ Z ξ ∈ Z ′

.

n n