REAL-CODED GENETIC ALGORITHMS CONSTRAINED …nedimtutkun.com/ders_notlari/genetik_algoritmalar/notlar11.pdf · Optimization plays a central role in operations research/management

REAL-CODED

GENETIC ALGORITHMS

CONSTRAINED OPTIMIZATION

Nedim TUTKUN

[email protected]

Outlines

• Unconstrained Optimization

• Ackley’s Function

• GA Approach for Ackley’s Function

• Nonlinear Programming

• Penalty Function

• Genetic Operators

• Numarical Examples

2

Optimization plays a central role in operations

research/management science and engineering design

problems.

It deals with problems of minimizing or maximizing a function

with several variables usually subject to equality and/or

inequality constraints.

Optimization techniques have had an increasingly great

impact on our society.

Both the number and variety of their applications continue to

grow rapidly, and no slowdown is in sight.

3

Unconstrained Optimization

However, many engineering design problems are very

complex in nature and difficult to solve with conventional

optimization techniques.

In recent years, genetic algorithms have received

considerable attention regarding their potential as a novel

optimization technique.

In this lecture we will discuss the applications of genetic

algorithms to unconstrained optimization, nonlinear

programming, stochastic programming, goal programming,

and interval programming.

4


Unconstrained optimization deals with the problem of

minimizing or maximizing a function in the absence of any

restrictions.

In general, an unconstrained optimization problem can be

mathematically represented as follows.

min 𝑓(𝑥)

Subject to 𝑥 ∈ Ω

where 𝑓 is a real-valued function and Ω, the feasible set, is a

subset of 𝐸𝑛

5


A point 𝑥∗ ∈ Ω is said to be a local minima of 𝑓 over Ω if

there is an > 0 such that f(x) f(x*) for all 𝑥 ∈ Ω within a

distance of x*.

A point 𝑥∗ ∈ Ω is said to be a global minima of 𝑓 over Ω if f(x)

f(x*) for all 𝑥 ∈ Ω.

The necessary conditions for local minima are based on the

differential calculus of 𝑓, that is, the gradient of 𝑓 defined as

follows:

6


The Hessian of f at x denoted as 𝛻2𝑓(𝑥) or F(x) is defined as

Even though most practical optimization problems have side

restrictions that must be satisfied, the study of techniques for

unconstrained optimization provides a basis for the further

studies.

In this lecture, we will discuss how to solve the

unconstrained optimization problem with genetic algorithms.

7


Ackley's function is a continuous and multimodal test

function obtained by modulating an exponential function with

a cosine wave of moderate amplitude.

Its topology, as shown in Figure 1, is characterized by an

almost flat outer region and a central hole or peak where

modulations by the cosine wave become more and more

influential. Ackley's function is as follows.

8

Ackley's Function

Ackley's function is as follows

where c1 = 20, c2 = 0.2, c3 = 2𝜋, and e = 2.71282.

The known optimal solution is (x1, x2) = 0 and f(x1,x2)= 0.

9

Ackley's Function

10

Ackley's Function

Fig. 1. Plot of Ackley’s Function

11

Ackley's Function

Fig. 2. Contour plot of Ackley’s Function

As Ackley pointed out, this function causes moderate

complications to the search.

Because a strictly local optimization algorithm that

performs hill climbing would surely get trapped in a local

optimum.

A search strategy that scans a slightly bigger

neighborhood would be able to cross intervening valleys.

Therefore, Ackley's function provides one of the

reasonable test cases for genetic search.

12

Ackley's Function

To minimize Ackley's function, we simply use the

following implementation of the genetic algorithm.

1. Real number encoding

2. Arithmetic crossover

3. Nonuniform mutation

4. Top pop_size selection

13

Minimization of Ackley's Function

The arithmetic crossover is defined as the

combination of two chromosomes v1 and v2 as

follows.

v1new= v1+(1-)v2

v2new= v2+(1-)v1

where is a uniforly distributed random number

between 0 and 1.

14


The nonuniform mutation is given as follows:

For a given parent v, if the element xk of it is

selected for mutation, the resulting offspring is vnew

= [x1, x2, x3,…..,xn], where xknew is randomly

selected from two possible choices:

xknew= xk+(t, xkU-xk)

or

xknew= xk-(t, xk-xkL)

15


The nonuniform mutation is given as follows:

For a given parent v, if the element xk of it is selected for

mutation, the resulting offspring is vnew = [x1, x2,

x3,…..,xn], where xknew is randomly selected from two

possible choices:

xknew= xk+(t, xkU-xk)

xknew= xk-(t, xk-xkL)

where xkU and xk

L are the upper and lower bounds for Xk

16


The function (t, y) returns a value in the range

[0, y] such that the value of (t, y) approaches to 0

as t increases (t is the generation number) as

follows:

(t, y)=y ×r ×(1-t / T)b

where 𝑟 is a random number from [0, 1], 𝑇 is the

maximal generation number, and 𝑏 is a parameter

determining the degree of nonuniformity. 17


Top pop_size selection produces the next

generation by selecting the best pop_size

chromosomes from parents and offspring.

For this case, we can simply use the values of

objective as fitness values and sort

chromosomes according to these values.

The parameters of the genetic algorithm are set

as follows:

18


The parameters of the genetic algorithm are set

as follows:

pop_ size: 10

Maxgen: 1000

Pm: 0.1

Pc: 0.3

19


20


Table 1. Initial Population of 10 Random Chromosomes

21


Table 2. The corresponding fitness function values

This means that the chromosomes v2 , v6 , v8,

and v9 were selected for crossover. The

offspring were generated as follows:

22


This means that the chromosomes v2 , v6 , v8,

and v9 were selected for crossover. The

offspring were generated as follows:

23


The mutation then is performed. Because there

are a total 2 x 10 = 20 genes in whole

population, we generate a sequence of random

numbers rk (k = 1, ... , 20) from the range [0, 1].

The corresponding gene to be mutated is:

24


bit_pos chrom_num variable random_num

11 6 x1 0.081393

11 6 -4.068506 -0.959655

The fitness value for each offspring is as

follows:

25


The best 10 chromosomes among parents and

offspring form a new population as follows:

26


The corresponding fitness values of variables [x1, x2] are

as follows:

27


Now we just completed one iteration of the genetic

procedure (one generation). At the 1000th generation,

we have the following chromosomes:

The fitness value is f(x1*, x2*) = -0.005456.

28


29

Ackley's Function

Fig. 3. Contour plot of Ackley’s Function

30

Contour Plot of the Cost Function

Fig. 4. Variation of fitness value with generation.

Xopt=(0.6196×10-4, -0.1231E-04), Fmin=1.7879×10-4,

31


Fig. 5. Scattering of the initial population.

32


Fig. 6. Scattering of at the 50th generation.

33



34



35



Nonlinear programming (or constrainted

optimization) deals with the problem of optimizing an

objective function in the presence of equality and/or

inequality constraints.

Nonlinear programming is an extremely important

tool used in almost every area of engineering,

operations research, and mathematics because

many practical problems cannot be successfully

modeled as a linear program.

36

Nonlinear Programming

The general nonlinear programming may be written

as follows:

where f, gi and hi are real valued functions defined

on En, X is a subset of En, and x is an n-dimensional

real vector with components x1, x2, ... ,xn.

37


The above problem must be solved for the values of

the variables x1, x2,…,xn that satisfy the restrictions

and meanwhile minimize the function f.

The function f is usually called the objective function

or criterion function.

Each of the constraints gi(x) 0 is called an

inequality constraint.

Each of the constraints hi(x) = 0 is called an equality

constraint.

38


The set X might typically include lower and upper

bounds on the variables, which is usually called

domain constraint.

A vector x ∈ X satisfying all the constraints is called a

feasible solution to the problem.

The collection of all such solutions forms the feasible

region. The nonlinear programming problem then is

to find a feasible point 𝑥 such that f(x) f(𝑥 ) for each

feasible point 𝑥.

39


Such a point is called an optimal solution. Unlike linear

programming problems, the conventional solution

methods for nonlinear programming are very complex

and not very efficient.

In the past few years, there has been a growing effort

to apply genetic algorithms to the nonlinear

programming problem. This lecture notes help us how

to solve the nonlinear programming problem with

genetic algorithms in general.

40


The central problem for applying genetic algorithms

to the constrained optimization is how to handle

constraints because genetic operators used to

manipulate the chromosomes often yield infeasible

offspring.

Recently, several techniques have been proposed to

handle constraints with genetic algorithms

Michalewicz has published a very good survey on

this problem.

41


The existing techniques can be roughly classified as

follows:

Rejecting strategy

Repairing strategy

Modifying genetic operators strategy

Penalizing strategy

Each of these strategies have advantages and

disadvantages.

42


Rejecting strategy discards all infeasible

chromosomes created throughout the evolutionary

process.

This is a popular option in many genetic algorithms.

The method may work reasonably well when the

feasible search space is convex, and it creates a

reasonable part of the whole search space.

However, such an approach has serious limitations.

43

Rejecting Strategy

For example, for many constrained optimization

problems where the initial population consists of

infeasible chromosomes only, it might be

essential to improve them.

Moreover, quite often the system can reach the

optimum more easily if it is possible to "cross" an

infeasible region (especially in nonconvex

feasible search spaces).

44

Rejecting Strategy

Repairing a chromosome involves taking an infeasible

chromosome and generating a feasible one through

some repairing procedure.

For many combinatorial optimization problems, it is

relatively easy to create a repairing procedure.

It has been shown that through an empirical test of

genetic algorithms performance on a diverse set of

constrained combinatorial optimization problems, that the

repair strategy did indeed surpass other strategies in

both speed and performance.

45

Repairing Strategy

Repairing strategy depends on the existence of a

deterministic repair procedure to convert an infeasible

offspring into a feasible one.

The weakness of the method is in its problem

dependence.

For each particular problem, a specific repair algorithm

should be designed.

For some problems, the process of repairing infeasible

chromosomes might be as complex as solving the

original problem.

46

Repairing Strategy

One reasonable approach for dealing with the issue of

feasibility is to develop problem-specific representation and

specialized genetic operators to maintain the feasibility of

chromosomes.

Such systems are often much more reliable than any other

genetic algorithms based on the penalty approach. Many

users use problem-specific representation and specialized

operators in building very successful genetic algorithms in

many areas. However, the genetic search of this approach

is confined within the feasible region.

47

Modifying Genetic Operator Strategy

These strategies above have the advantage that

they never generate infeasible solutions but have the

disadvantage that they consider no points outside

the feasible regions.

For highly constrained problem, infeasible solutions

may take a relatively big portion of the population.

In such case, feasible solutions may be difficult to be

found if we just confine genetic search within

feasible regions.

48

Penalty Strategy

It has been suggested that constraint management

techniques allowing movement through infeasible

regions of the search space tend to yield more rapid

optimization and produce better final solutions than

do approaches limiting search trajectories only to

feasible regions of the search space.

The penalizing strategy is such kind of techniques

proposed to consider infeasible solutions in genetic

search.

49

Penalty Strategy

The penalty technique is perhaps the most common

technique used to handle infeasible solutions in the

genetic algorithms for constrained optimization

problems.

In essence, this technique transforms the

constrained problem into an unconstrained problem

by penalizing infeasible solutions, in which a penalty

term is added to the objective function for any

violation of the constraints.

50

Penalty Function

The basic idea of the penalty technique is borrowed

from conventional optimization.

It is a nature question: is there any difference when

we use the penalty method in conventional

optimization and in genetic algorithms?

In conventional optimization, the penalty technique is

used to generate a sequence of infeasible points

whose limit is an optimal solution to the original

problem.

51

Penalty Function

The major concern is how to choose a proper

value of penalty so as to speed convergence and

avoid premature termination.

In genetic algorithms, the penalty technique is

used to keep a certain amount of infeasible

solutions in each generation so as to enforce

genetic search towards an optimal solution from

both sides of feasible and infeasible regions.

52

Penalty Function

We do not simply reject the infeasible solutions in each

generation because some may provide much more

useful information about optimal solution than some

feasible solutions.

The major concern is how to determine the penalty term

so as to strike a balance between the information

protection (keeping some infeasible solutions) and the

selective pressure (rejecting some infeasible solutions),

and cancelled both under-penalty and overpenalty.

53

Penalty Function

In general, solution space contains two parts: feasible

area and infeasible area.

We do not make any assumption about these

subspaces; in particular, they need be neither convex

nor connected as shown in Figure 10.

Handling infeasible chromosomes is insignificant.

From the figure we can know that infeasible solution b is

much near to optima a than infeasible solution d and

feasiblesolution c.

54

Penalty Function

55

Penalty Function

Fig. 10. Solution space: feasible area and infeasible area.

We may hope to give less penalty to b than to d

even though it is a little farther from the feasible area

than d.

We also can believe that b contains much more

information about optima than c even though it is

infeasible.

However, we have no a priori knowledge about

optima, so generally it is very hard for us to judge

which one is better than others.

56

Penalty Function

The main issue of penalty strategy is how to design the

penalty function 𝑝(𝑥) which can effectively guide genetic

search toward the promising area of solution space.

The relationship between infeasible chromosome and the

feasible part of the search space plays a significant role in

penalizing infeasible chromosomes:

The penalty value corresponds to the "amount" of its

infeasibility under some measurement.

There is no general guideline on designing penalty function,

and constructing an efficient penalty function is quite problem-

dependent.

57

Penalty Function

Penalty techniques transform the constrained problem into an

unconstrained problem by penalizing infeasible solutions.

In general, there are two possible ways to construct the

evaluation function with penalty term.

1) One is to take the addition form expressed as follows:

𝑒𝑣𝑎𝑙(𝑥) = 𝑓(𝑥) + 𝑝(𝑥)

where 𝑥 represents a chromosome, 𝑓(𝑥) is the objective

function of problem, and 𝑝(𝑥) is the penalty term.

58

Evaluation Function with Penalty Term

For maximization problems, we usually require that:

𝑝 𝑥 = 0 if 𝑥 is feasible

𝑝 𝑥 < 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Let l p(x) lmax and I f(x) lmin be the maximum of l p(x) l and

minimum of l f(x) l among infeasible solutions in current

population, respectively. We also require that

l p(x) lmax ≤ I f(x) lmin

to avoid negative fitness value. 59


For minimization problems, we usually require that:


𝑝 𝑥 > 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

The second way is to take the multiplication form

expressed as follows:

60


𝑒𝑣𝑎𝑙(𝑥) = 𝑓(𝑥) + 𝑝(𝑥)

In this case, for maximization problems we require that:


0≤ 𝑝 𝑥 ≤ 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

and for minimization problems we require that

61



𝑝 𝑥 > 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

Note that for the minimization problems, the fitter

chromosome has the lower value of eval(x).

For some selection methods, it is required to

transform the objective values into fitness values

in such way that the fitter one has the larger

fitness value.

62


Find the optimal value of the following

constrained function.

z=5-(x-2)2-2(y-1)2

x+4y=3

0 x, y 5

63

Example

64

Constrained Optimization (Matlab Code)

clear;clc;

% Step 1 : Initialization

a=0;b=10;n=2;rh=0.6667;

G=100;pm=0.001;pc=.6;N=1000;fmin

=[];fave=[];fmax=[];maxfit=0;

x=rand(N,n);

for k=1:n

x(:,k)=linmap(x(:,k),a,b); %

convert chromosome to real

number in a range from a to b

end

65


for g=1:G

fprintf('g:%.0f\n',g);

% Step 2 : Selection

f=fitval3(x(:,1),x(:,2),rh);

s=selpop(x,f);

% Step 3 : Crossover

c=artxover(s,pc);

% Step 4 : Mutation

x=pertmutate(c,pm,a,b);

[maxfit x]=elit(x(:,1),x(:,2),maxfit,rh);

f=fitval3(x(:,1),x(:,2),rh);

fmin=[fmin maxfit];

fave=[fave mean(f)];

fmax=[fmax max(f)];

end % end the generation

66


g=1:G;

plot(g,fmax,'r',g,fave,'b');

xlabel(‘Generation');ylabel(‘Fitness

Value');

axis([1 100 0 2.1])

legend('Max','Ave','location','best');lege

nd boxoff;

f=fun3(x(:,1),x(:,2));

[fmx ind]=max(f);

optx=x(ind(1),:)

yoptx=fun3(optx(:,1),optx(:,2))

67


function f=fitval3(x,y,rh)

f=[];n=length(x);

z=fun3(x,y);

h=x+4*y;

bh=3;

for k=1:n

if (h(k)~=bh)

f(k)=z(k)-rh*h(k);

else

f(k)=z(k);

end

end

68


function z=fun3(x,y)

z=5-(x-2).^2-2*(y-1).^2;

69

Convergence of Constrained Optimization

Documents

REAL-CODED GENETIC ALGORITHMS CONSTRAINED …nedimtutkun.com/ders_notlari/genetik_algoritmalar/notlar11.pdf · Optimization plays a central role in operations research/management