Proofs of Lemmas in Chapter 4 - Multiobjective Optimization Methodology: A Jumping Gene Approach - page 211

Z

Z

flip rightmost bit

0 1 ... 1 1 0 1 0 0 1 1 0 1 ... 1 1 0 1 0 1 1 1 (a)

V k

ξ n

(b)

0 1 ... 1 1 0 1 0 0 1 1

V k

ξ n

0 1 ... 1 1 0 1 0 0 1 1

V k

ξ i

0 1 ... 1 1 0 1 0 0 1 1

W c,k = N

Figure a.4
Visualization.of.the.proof.of.Lemma.4.4.

where. N .is.the.size.of. K ′,.specifying.the.number.of.possible.pasting.positions.
in. ξ .that.make.the.pasted.region.contain.don't-care.bits.(*).only..If.

ξ ∈ ,. C row .is.
positive.or.zero;.otherwise,. C row .is.negative.or.zero..Thus,.the.proof.is.completed.

Z

ProofofLemma4.5
Based.on.the.definition.of.

( )
a mn

i

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

L L − L L −

g g

∑ ∑ ∑ ∑

( ) i ( ) i

a − a = K ∆ ξ ( , V ′ ξ ; , V ′ )

mn mn i k n k

. (A.17)

ξ ∈ Z ξ ∈ Z ′ c = 0 k = 0

m m

∑ ∑

( , V ; , G ) ( , V ; , G )

× ∆ ξ ξ − ∆ ξ ξ k m c k , k m c k ,

i i

.

ξ ∈ Z ξ ∈ Z ′

m m

and. for. any. particular.

ξ ∈ . (note:. Z. can. be. Z odd . or. Z even ),. as. shown. in.
Figure A.5 ,. three.cases.can.be.obtained.as.follows:

Z

. a.. For.any.copying.position. c ,.which.causes.some.actual.bits.of. ξ .to.be.

located.outside.

G c k ,.it.is.possible.to.lip.the.rightmost.of.those.bits.to.
find.a.corresponding.

,

ξ ∈ Z ′ .such.that

m ′

∆ ξ ( , V ; ξ , G ) = ∆ ξ ( , V ; ξ , G ).

.

i k m c k , i k m ' c k ,

.