1. Introduce

There are many ways that solve nonlinear and Indifferential function’s optimal problem ,We have many traditional ways such as Hooke-Jeeves algorithm, Rosenblock algorithm and so on. However these methods

are all for local extremum and can solve local extremum , not for global extremum .

Recently genetic algorithm that imitates living being’s evolution absorbed study staff in different kinds of domains and has been applied extensively all kinds of fields that include function optimize ,mode identification

,image dealing ,manual intelligence and so on. Genetic algorithm only need the value of the optimal object’s

function, towards the optimal problem for the indifferential and even uncontinuous function,Genetic algorithm can have the biggest probability to get the most optimal solution globally .Calculating time is relatively less , it possesses much better characteristic such as better stability , adaptability and collateral and so on.In the reference literature^[2] , the author suggests a method that towards continuous and differential function ,by a newton operator embedded into the genetic algorithm to solve the optimal problem ,The method got good effect but can not been applied for indifferential function problem .

The paper proposes a hybrid algorithm that put Hooke-Jeeves operator embedded into the Genetic

algorithm .In order to improve the convergence velocity and research probability of indifferential function,

Hooke-Jeeves can be called as pace accelerated algorithm, which is made up of exploration motion and mode

motion, The former reveal the object function’s changing disciplinarian and follow the avail direction (approximately gradient direction) to seek preferable point. The method can be regarded as the most fast

descending way, As a directly optimal method for indifferential function .The method is the most effective way

and combines the advances of both the genetic algorithm and the Hooke-Jeeves algorithm. The hybrid algorithm not only can have more probability to seek the most optimal solution ,but also carry on locally meticulous search.

2.problem description

 

Calculating complicate object’s optimal problem of nonlinear and indifferential function ,we can show

　In (1) a_i and b_i are the upper value and the lower value of variable x ,n is the dimension of variable

X ,f(x) is the nonlinear and indifferential function. In the most practical condition , towards a research plant

function, Giving a independent variable, we can get a corresponding function value, Except it we can not get

whether the function is continuous and differential or not.

2.1 Hooke-Jeeves algorithm

Hooke-Jeeves algorithm is a directly optimal method, which is simple and briefness , need not the optimal problem function’s differential.

Algorithm:

Step1 initialization: setting the original point x⁽¹⁾ ∈E ⁿ and n coordinate directions

E₁ ,E₂ ,E₃ ,…,E_n and original paceδ ,accelerated factor а≧1 ,reducing rate β∈(0,1),tolerate

error ε>0 ,set y⁽¹⁾=x⁽¹⁾, k=1,j=1.

Step2 if f(y^(j)+δE_j)< f(y^(j)),then put y^(j+1)=y^(j)+δE_j ,carry on step 4; else carry on step 3.

Step3 if f(y^(j)-δE_j)< f(y^(j)),then put y^(j+1)=y^(j)-δE_j ,carry on step 4; else put y^(j+1)=y^(j) ,then carry on step 3.

Step4 if j < n , then set j = j+1 ,return to step 2 ; else carry on step 5.

Step5 if f(y⁽ⁿ⁺¹⁾ )<f(x^k) , then carry on step 6 ; else carry on step 7

Step6 set x^(k+1) = y⁽ⁿ⁺¹⁾, let y⁽¹⁾=x^(k+1)+a(x^(k+1)-x^(k)) and k=k+1 , return to step 2

Step7 if δ≦ε then halt the iterate and get the point x^(k) ,else put δ=β*δ .

Hooke-Jeeves algorithm is a local directly optimal method , which can be regarded as the most fast descending way approximately. As the method need not calculate the differential of the plant function.

It is fitted for the optimal of the indifferential function.

2.2 Genetic algorithms

Genetic algorithm is a simulation of evolution where the rule of survival of the fittest is applied to a population of individuals. It transfers the aim function into gene groups ,regards adapt function as optimal

target; by the recombine ,exchange and mutation of gene , In general, the fittest individuals of any population tend to reproduce and survive to the next generation , thus improving successive generations. however ,inferior

individuals can, by chance ,survive and also reproduce. we can continually get optimal gene combination of next generation, till the final optimal result is got. Numerical calculation evinces that the method can cross the local extremum and make the group migrate to the global optimal value .The characteristic comes from internal gene diversity of Genetic algorithm, makes algorithm research in the full directions.

The basic genetic algorithm is as follows:

step 1. Create an initial population (usually a randomly generated string) .

step 2. Evaluate all of the individuals (apply some function or formula to the individuals).

step 3. Select a new population from the old population based on the fitness of the individuals as given by the evaluation function.

step 4. Apply some genetic operators (mutation & crossover) to members of the population to create new solutions.

step 5. Evaluate these newly created individuals.

step 6. Repeat steps 3-6 (one generation) until the termination criteria has been satisfied (usually

perform for a certain fixed number of generations).

 

2. hybrid algorithm:

code which induces contradiction of the calculating precision and the length of the character string and operates full randomly with probability , it makes the ability of locally research of genetic algorithm is poor. However , Hooke-Jeeves algorithm possesses doughty local research ability and improve the velocity of convergence greatly. Hybrid algorithm combines the advantages of the genetic algorithm and hooke-Jeeves algorithm and make full of Hooke-Jeeves algorithm which need not the optimal problem function’ s differential and promote the ability of the genetic algorithm’ s locally meticulous research. In this paper , we embedded the a promoted Hooke-Jeeves operator into the step 3 of genetic algorithm in order to improve the　convergence velocity and probability of solution .

In the algorithm, adaptation function is defined as g(x) = f_max- f(x)+k*(f_max-f_min),where f_max and f_min is separately the maximal and minimal function value of the present group, k is the parameter which controls

the rate of the present group’s maximal and minimal adaptation value. Because each generation’s f_mac and f_min vary with the group in the algorithm, the adaptation function above makes the hybrid algorithm

In this paper, the following optimal problem of the indifferential function is considered

powell function of globally optimal problem, the global extremum point x^* of the function is (0,0,0,0), where the function value f(x^*) is zero and two order partial differential matrix is singularity. when k isn’t zero , the

when we get the initial step δ is 0.005 , reducing rate β is 0.5, permissible error is 0.0001,

when k is zero, the monotony of function is approximately good ,we can get the extremum point with Hooke-Jeeves algorithm by 100% .when x^* is (0,0,0,0) ,f(x^*) is 0.

　　when k is eight, the equation above adds many local extremum points densely into powell function,

2. genetic algorithm

when we respectively let crossover probability be 0.8 ,mutation probability be 0.05 ,the biggest propagation 　

number is 10000.

when k is zero , the global solution is got by 100% probability.

when k is eight , the global solution is got only by 90% probability.

The calculation example shows that genetic algorithm is a globally optimal algorithm that has biggish probability to get the global solution. however, under the condition of this example, full probability solution

can not be reached .As a matter of fact we can adjust the above parameter to get the global solution with the

full probability, but usually endless iterate course needs to be passed , a number of time will be consumed,

solving efficiency is very low. Considering the local research ability of genetic algorithm isn’t powerful, we can

make full use of the locally delicate research ability, improve the step 3 of the genetic algorithm by adding the

Hooke-Jeeves algorithm into it, so solving efficiency and solving probability can be heightened greatly.

3. the result of the hybrid algorithm:

Because the extremum points gather up densely in the interval of [-1,1], to discuss the local research ability of the Hooke-Jeeves operator in the hybrid algorithm ,we separately let the initial step δ be 0.005 , reducing rate β is 0.5, permissible error is 0.0004; let the initial step δ be 0.0005 , reducing rate β is 0.5, permissible error is 0.0004; let the initial step δ be 0.001 , reducing rate β is 0.5, permissible error is 0.0002;the most iterate number is 500, crossover probability P_c is 0.8, mutation probability P_m

	initial step δ	probability Pch
		0.0	0.005	0.01	0.015	0.02	0.025	0.03	0.035
K=8	δ=0.005	50	75	75	92	96	97	99	100
	probability to solve	50%	75%	75%	92%	96%	97%	99%	100%
K=8	δ=0.0005	50	78	78	94	96	98	100	100
	probability to solve	50%	78%	78%	94%	96%	98%	100%	100%
K=8	δ=0.001	50	72	87	95	98	100	100	100
	probability to solve	50%	72%	87%	95%	98%	100%	100%	100%

1).In the calculation above, we assume if f(x)≦10^-5 , the global solution is thought to be grabbled.

properly adjust the initial step δ and permissible error, the solving probability could be improved;

3).if we let the initial step δ be 0.001 and permissible error be 0.0002,when Pch is only 0.025, the global solution can be got with the full probability by means of the action of both Hooke-Jeeves operator and

genetic algorithm.

puts forwards a hybrid genetic algorithm based on Hooke-Jeeves algorithm. the algorithm synthesis

locally delicate research of Hooke-Jeeves algorithm, meanwhile the differential aspect information of

Hooke-Jeeves algorithm.