→

∈

−∞>= ∗∗

∗≥∈∀

∈→⊂

190

978-1-4244-1734-6/08/$25.00 c© 2008 IEEE

SX ⊂ denotes any interval, thus ],[ XXX = . The width

of the interval )(Xω is defined as XXX −=)(ω , the midpoint of the interval is )(Xm

It is without loss of generality that }:)({)( XxxfXf ∈=

denote the function fitness value of f in X. The whole min

fitness value of f is ∗f in S, the set of the whole min point is M, and assuming that M is interior in the interval of S and finite. Then Mx ∈∀ ∗ its result

Sxxfxfxf ∈∀≥=′ ∗∗ ),()(,0)( 4For such optimized question (see (1)), the aim is to search

∗f and M. The thought of Moore [3] and H.Ratschek [4], who is the founder of interval Algorithms, above all, is to divide S into some subintervals iXS = , seek fitness 0fin the midpoint )( iXm of f in a interval S and exclude all intervals jX that fit )(min0 xff

jXx∈< , and save the

reserved subintervals )1(L from the algorithm program, once more divide )1(L into smaller intervals and iterative until fit the final precision. This Algorithm is trustful and safe, but the verify demand is very strict and complicated. The verify demand of the type cell Algorithms developed by Jiangshe Zhang [5] is very simple and practicable, it applied the quadratic extension norm:

))()((21

)()())(()( XmXXfXmXXfXmfXF −′′+−′+= 5

Then, a simple interval verify condition is given as follows: Assuming that )(xf ′ fits the condition of Lipschitz:

Syxyxcyfxf ∈∀−≤′−′ ,,)()( 6Assume SX ⊂ as any interval [5]. There is the lemma as follows [6]: Assume that )(xf is quadratic continuous differentiable

and fits (6), 0f is the fitness of a point within S, while X fits:

2)(81

))((0 XcXmff ω−< 7

Then, there must contain a global min point of f within X [5]. Proof. By contradiction. If there exists one point

of XMx ∈* then due to 0)( * =′ xf , given:

))()(()())(())(()( **** xXmxfxfXmfXmfxf −′−−=−

2*)(21 xXmc −≤ 8

Because there is )(21

)( * XxXm ω≤− therefore

2* )(81

))(()( XcXmfxf ω≤− 9

Thus2* )(

81

))(()( XcXmfxf ω−≥ 10

Again, due to )( *0 xff ≥ therefore due to (10) etc

deducing2

0 )(81

))(( XcXmff ω−≥ 11

This result is contrary to 7 , thus Φ=MX , namely there is not contain the global min point of the fitness of fwithin X. This contradiction completes the proof of the lemma. Namely it is contractive mapping algorithm. This lemma is a simple and convenient term that we verify a given interval iX whether there exists a whole min point. But the verify term of conventional method by Moore is too complexity to implement. Applying that CMGA, for its contractibility and if only the function is genetic reachability [8][9] in encode space, well then, after finite generation of genetic operation, the CMGA must convergence to the global optimal solution without fail. The

calculative complexity of this algorithm is )1

(logε

O for

achieving requiring precision ε that was testified in the relation references [6][10].

2.3 The Process of Contractive-Mapping-Genetic- Algorithm(CMGA)

The conventional hybrid GA flow can see the relative reference [7]. Here, we give the CMGA’s algorithm flow of 2-subdivision (divided interval into two section) as follows:

The algorithm and notation t is the genetic mutation generation, )(tL is the preserved valid interval sets of t -th generation in encode space, )(tL is divided by 2-subdivision, then { } tm

iiXL 1==′ , here )(2 tN

t Lm =

)(tL is the preserved interval number of the t -th generation, given precision 0>ε . The flow is as follows

begin

{ } ))(()(,,0 0)0( SmfftPSLt ←=←←

if εω <)( iX then exit

while not εω <)( iX do

begin

Processing the operation in valid encode space as follows:

Recombination{ })1(,,,2,1))((min)( −=← t

ti PmiXmPtP

Applying 2-section to every iX

Evaluation:

if 2)(81

))(()( ii XcXmPtP ω−< then

2008 Chinese Control and Decision Conference (CCDC 2008) 191

++←

∈=

=

=−−=Δ

=

−

−=Γ=

=

=

=

=

=

−∈−+−=

==

∈= π

= =

Table 1. Comparative of CMGA and GA Algorithm

192 2008 Chinese Control and Decision Conference (CCDC 2008)

2008 Chinese Control and Decision Conference (CCDC 2008) 193