A GENETIC ALGORITHM TO
SOLVE SOME SPECIAL
FUNCTIONS
BY
EMMANUEL SARKODIE ADABOR
SUPERVISED BY
MR. J. ACKORA-PRAH
1
OUTLINE

BACKGROUND
 STATEMENT OF PROBLEM
 OBJECTIVE
 METHODOLOGY
 APPLICATIONS TO SPECIAL FUNCTIONS
 CONCLUSION
 RECOMMENDATIONS
2
BACKGROUND
Darwin’s principle of survival of the fittest
was used as a starting point in introducing
evolutionary computation (EC).
 EC has four stages of development namely;
1. Genetic algorithms (Holland, 1975)
2. Genetic programming (Koza, 1992, 1994)
3. Evolutionary strategies (Rocheuberg,1973)

3
BACKROUND (CONT.)
4.

1.
2.
Evolutionary programming (Forgel et
al.1966)
The following principles in Darwin’s
principle of the survival of the fittest
inspired the genetic algorithm:
Species live in a competitive world.
Survival depends on the fitness competition
and offspring who are stronger than or
4
equally as strong as their parents.
BACKROUND (CONT.)
3. The offspring genetically take the
characteristics of their parents
4. The offspring are however unique and
there is probability of slight variations in
some of their genes.
5. In the competitive environment less fit
individuals die off and may not become
parents for breeding.
5
PROBLEM STATEMENT

Evolutionary concepts are of recent interest
since some methods may not lead to the
global optimum.
 There is therefore the need to find an
algorithm that exhausts the entire search
space for the global optima of problems
6
OBJECTIVE

A GA is applied to solve Rosenbrock’s
function, Rastrigin’s function and the
Schwefel’s function to test its ability to
search for global optima of complex,
multivariable and multimodal problems.
7
METHODOLOGY

A GA is a search technique used in
computing to find approximate solutions to
optimization and search problems through
application of the principles of evolutionary
biology.
 Terms associated with GA’s are defined as
follows:
 An individual is a solution of optimization
problem
8
DEFINITION (CONT’D)

A population is set of solutions that form
the domain search space.
 A generation is a set of solutions taken from
the population and generated at an instant of
time or iteration.
 Selection is the operation of selecting
parents from the generation to produce
offspring.
 Crossover is the process of taking two
parents and producing from them a child.
9
DEFINITION (CONT’D)

Mutation is random operation whereby the
allele of the gene in a chromosome of the
offspring is changed by a probability.
 Recombination is the operation whereby
elements of the offspring form an
intermediate generation and less fit
chromosomes are taken from the generation
 Chromosomes represent the data structure
of solutions.
10
OVERVIEW OF GA
Create initial population
Evaluate the fitness of each
individual
Evaluation
Select parent based on fitness
Create new population
Selection
Recombination
11
EXAMPLE TO ILLUSTRATE
OPERATORS AND OPERATIONS IN GA
Max
f (x) = x² for x = 0,1, … 31;
 Encoding: representing individual genes.
 Encoding
may be Binary, Octal,
Hexadecimal or Permutation encodings
 Binary encoding is used to solve the
example

12
Solution

Using a five bit (binary integer) unsigned
integer, numbers between 0(00000) and
31(11111) can be obtained.
 Population of size 4 is randomly initialized.
 Selection: choosing two parents from the
population for crossing.
 Selection Methods include
Roulette
Wheel, Random, Rank, Tournament and
13
Elitist selections
ROULETTE WHEEL

Roulette wheel selection is used.
 A roulette wheel is constructed with the
cumulative and relative fitness ( f ) of
chromosomes. Let i denote ith chromosome.
 The relative fitness of each chromosome is
P ( x )  wi 
fi
,
n

fk
k 1
14
ROULETTE WHEEL
(CONT’D)

The expected count is calculated by
E xpected count 
w here

f  xi 
 average f  x  
 average f  x  
 n

  f  xi  

  i 1
n






Where n is population size and i is the i-th chromosome
15
Roulette Wheel is formed as
s4
13.26%
s1
12.47%
54.11%
s3
2.16%
s2
16
Table of Selection
String No.
Initial
x value
Function
P(x)
value (x2)
population
Percentage
Expected
Actual
probability
count
count
1
01100
12
144
0.1247
12.47%
0.4987
1
2
11001
25
625
0.5411
54.11%
2.1645
2
3
00101
5
25
0.0216
2.61%
0.0866
0
4
10011
19
361
0.3126
31.26%
1.2502
1
Sum
1155
1.0000
100%
4.0000
4
Average
288.75
0.2500
25%
1.0000
1
Maximum
625
0.5411
54.11%
2.1645
2
17
Crossover

The various crossover techniques include
single point, two point, multipoint and
uniform crossovers.
 In a single point crossover, the parents are
cut at corresponding points and the sections
after the cut are exchanged.
18
Crossover cont’d
Parent 1
Parent 2
0 1 1 0 0
1 1 0 0 1
Offspring 1 0 1 1 0 1
Offspring 2 1 1 0 0 0
Similar action is performed for the next strings
resulting
19
Table of Crossover
String No.
Mating
pool
Crossover
point
Offspring
after
crossover
x value
Function
value (x2)
1
01100
4
01101
13
169
2
11001
4
11000
24
576
3
11001
3
11011
27
729
4
10011
3
10001
17
289
Sum
1763
Average
440.75
Maximum
729
20
Mutation

Mutation prevents the algorithm to be
trapped in a local minimum by maintaining
diversity in population
 Different forms of Mutation include
flipping, interchanging and reversing.
 Flipping is used.
21
Mutation cont’d
String No.
Offspring
after
crossover
Mutation
Chromosomes
for flipping
Offspring after
mutation
x value
Function
value (x2)
1
01101
10000
11101
29
841
2
11000
00000
11000
24
576
3
11011
00000
11011
27
729
4
10001
00100
10100
20
400
Sum
2564
Average
636.5
Maximum
841
22
Solution cont’d

Replacements are made by comparing
fitness values.
 The example was solved with crossover and
mutation probabilities 1.0 and 0.001.
 The procedure showed an improvement on
maximum fitness from 625 to 841 in just
one generation.
23
Convergence Criteria

Maximum generation
 Elapsed time
 No change in fitness
 Stall generations
 Stall time limit
24
APPLICATION OF GA TO
SPECIAL FUNCTIONS
The GA is used to solve Rosenbrock’s
function, Rastrigin’s function and
Schwefel’s function in order establish how
good the algorithm is.
 A MATLAB code is implemented to
minimize the functions.

25
Solution

The following parameters were used in the
simulation of all three functions:
 Probability of crossover = 0.8
 Probability of mutation = 0.2
 Initial population = 50
 Maximum generations = 100
 Stall generations = 50
26
ROSENBROCK’S FUNCTION
OR VALLEY

The function has the following definition
f  x , y   1  x   100  y  x
2
2

2

where x and y lies in [-2.048, 2.048]
 It is also called banana function because its
distinct shape in a contour plot
27
Overview of Rosenbrock’s
function
Minimum point
The global optimum lies inside a long, narrow, parabolic shaped flat valley
28


Solution cont’d
Global minimum is 0.0000496 (approximately zero (0))
It occurred at (1.0070, 1.0140) of the 51st generation
29
RASTRIGIN’S FUNCTION

Function has the following definition
n
f  x   10 n    x i2  10 cos  2 x i  
i 1



where xi lies in [-5.12, 5.12]
30
Overview of Rastrigin’s function
The function is highly multimodal. However, the
location of the minima are regularly distributed.
31


Solution cont’d
Global minimum is 0.00000000239 (approximately zero )
It occurred at the point 0.00000347 of the 51st generation
32
Solution Cont’d

The global minimum for an n = 5 is 0.0309
 It occurred at 0.0125, 0.0000, -0.0000,
0.0001, -0.0000. (which are all
approximately zero).
33
SCHWEFEL’S FUNCTION

Function has the following definition
f x 
n

i 1


  x i sin




xi 
where xi lies in [-500, 500]
34
Overview of Schwefel’s function
The global minimum is geometrically distant, over the parameter
space from the next best local minima. Therefore, algorithms are
potentially prone to convergence in the wrong direction
35
Solution


Global minimum is -418.9829
It occurred at 420.9618 at the 51st generation
The
global minimum for an n = 10 is -4.7620 ×10114
36
CONCLUSIONS

The GA has been able to produce the global
optima of complex multivariable and
multimodal functions.
 These were all obtained under 1minute.
 The GA is therefore efficient, robust and
reliable.
37
RECOMMENDATIONS

It is recommended for problems with the
properties of the three functions.
 It is recommended that further research is
conducted to establish the variants of GA
and their suitability to specific problems
38
END OF PRESENTATION
THANK
YOU
39
GA CODE








function [x,fval,exitFlag,output,population,scores] = …
ga(FUN,GenomeLength,Aineq,Bineq,Aeq,Beq,LB,UB,n
onlcon,options)
defaultopt = struct('PopulationType', 'doubleVector', ...
'PopInitRange', [0;1], ...
'PopulationSize', 20, ...
'EliteCount', 2, ...
'CrossoverFraction', 0.8, ...
'MigrationDirection','forward', ...
'MigrationInterval',20, ...
40
GA CONT’D










'InitialPopulation',[], ...
'InitialScores', [], ...
'InitialPenalty', 10, ...
'PenaltyFactor', 100, ...
'PlotInterval',1, ...
'CreationFcn',@gacreationuniform, ...
'FitnessScalingFcn', @fitscalingrank, ...
'SelectionFcn', @selectionroulette, ...
'CrossoverFcn',@crossovertwopoint, ...
'MutationFcn',@mutationgaussian, ...
41
GA CONT’D








'MigrationFraction',0.2, ...
'Generations', 100, ...
'TimeLimit', inf, ...
'FitnessLimit', -inf, ...
'StallGenLimit', 50, ...
'StallTimeLimit', 20, ...
'TolFun', 1e-6, ...
'TolCon', 1e-6, ...
42
GA CONT’D









'HybridFcn',[], ...
'Display', 'final', ...
'PlotFcns', [], ...
'OutputFcns', [], ...
'Vectorized','off');
% Check number of input arguments
errmsg = nargchk(1,10,nargin);
if ~isempty(errmsg)
error('gads:ga:numberOfInputs',[errmsg,' GA requires
at least 1 input argument.']);
43
GA CONT’D






end
% If just 'defaults' passed in, return the default options
in X
if nargin == 1 && nargout <= 1 &&
isequal(FUN,'defaults')
x = defaultopt;
return
end
44
GA CONT’D


if nargin < 10, options = [];
if nargin < 9, nonlcon = [];
if nargin < 8, UB = [];
if nargin < 7, LB = [];
if nargin <6, Beq = [];
if nargin <5, Aeq = [];
if nargin < 4, Bineq = [];
if nargin < 3, Aineq = [];

end; end; end; end






45
GA CONT’D










end; end; end; end
% Is third argument a structure
if nargin == 3 && isstruct(Aineq) % Old syntax
options = Aineq; Aineq = [];
end
% Input can be a problem structure
if nargin == 1
try
options = FUN.options;
GenomeLength = FUN.nvars;
46
GA CONT’D









% If using new syntax then must have all the fields;
check one
% field
if isfield(FUN,'Aineq')
Aineq = FUN.Aineq;
Bineq = FUN.Bineq;
Aeq
= FUN.Aeq;
Beq
= FUN.Beq;
LB
= FUN.LB;
UB
= FUN.UB;
47
GA CONT’D









nonlcon = FUN.nonlcon;
else
Aineq = []; Bineq = [];
Aeq = []; Beq = [];
LB = []; UB = [];
nonlcon = [];
end
% optional fields
if isfield(FUN, 'randstate') && isfield(FUN,
'randnstate') && ...
48
GA CONT’D








isa(FUN.randstate, 'double') &&
isequal(size(FUN.randstate),[625, 1]) && ...
isa(FUN.randnstate, 'double') &&
isequal(size(FUN.randnstate),[2, 1])
rand('twister',FUN.randstate);
randn('state',FUN.randnstate);
end
FUN = FUN.fitnessfcn;
catch
error('gads:ga:invalidStructInput','The input should
49
GA CONT’D







be a structure with valid fields or provide at least two
arguments to GA.' );
end
end
% We need to check the GenomeLength here before we
call any solver
valid = isnumeric(GenomeLength) &&
isscalar(GenomeLength)&& (GenomeLength > 0) ...
&& (GenomeLength == floor(GenomeLength));
if(~valid)
50
GA CONT’D






error('gads:ga:validNumberofVariables:notValidNvars','
Number of variables (NVARS) must be a positive
integer.');
end
% Use default options if empty
if ~isempty(options) && ~isa(options,'struct')
error('gads:ga:optionsNotAStruct','Tenth input
argument must be a valid structure created with
GAOPTIMSET.');
elseif isempty(options)
51
GA CONT’D








options = defaultopt;
end
user_options = options;
% All inputs should be double
try
dataType =
superiorfloat(GenomeLength,Aineq,Bineq,Aeq,Beq,LB,
UB);
if ~isequal('double', dataType)
error('gads:ga:dataType', …
52
GA CONT’D








output.message = '';
% Determine the 'type' of the problem
if ~isempty(nonlcon)
type = 'nonlinearconstr';
% Determine the sub-problem type for the
constrained problem (used in ALPS)
if ~isempty(Aeq) || ~isempty(Beq) || ~isempty(Aineq)
|| ~isempty(Bineq)
subtype = 'linearconstraints';
elseif ~isempty(LB) || ~isempty(UB)
53
GA CONT’D








subtype = 'boundconstraints';
else
subtype = 'unconstrained';
end
% If Aeq or Aineq is not empty, then problem has
linear constraints.
elseif ~isempty(Aeq) || ~isempty(Beq) ||
~isempty(Aineq) || ~isempty(Bineq)
type = 'linearconstraints';
% This condition satisfies bound constraints
54
GA CONT’D









elseif ~isempty(LB) || ~isempty(UB)
type = 'boundconstraints';
% If all constraints fields are empty then it is
unconstrained
else
type = 'unconstrained';
end
% Initialize output structure
output.problemtype = type;
% If nonlinear constraints, then subtype is needed to
process linear
55
GA CONT’D










'GA only accepts inputs of data type double.')
end
catch
error('gads:ga:dataType', ...
'GA only accepts inputs of data type double.')
end; % Remember the random number states used
output.randstate = rand('twister');
output.randnstate = randn('state');
output.generations = 0;
output.funccount = 0;
56
GA CONT’D









% constraints (see function preProcessLinearConstr)
if strcmp(type,'nonlinearconstr')
type = subtype;
end
% Validate options and fitness function
[options,GenomeLength,FitnessFcn,NonconFcn] =
validate(GenomeLength,FUN,nonlcon,options,type);
if ~strcmp(output.problemtype,'unconstrained')
% Determine a start point
if ~isempty(options.InitialPopulation)
57
GA CONT’D









population = [];
scores = [];
% Bound correction
[LB,UB,msg,exitFlag] =
checkbound(LB,UB,GenomeLength);
if exitFlag < 0
output.message = msg;
if options.Verbosity > 0
fprintf('%s\n',msg)
end; return;
58
GA CONT’D







end
% Linear constraints correction
[Iterate.x,Aineq,Bineq,Aeq,Beq,LB,UB,msg,exitFlag] =
...
preProcessLinearConstr(Iterate.x,Aineq,Bineq,Aeq,Beq,
LB,UB,GenomeLength,type,options.Verbosity);
if exitFlag < 0
output.message = msg;
if options.Verbosity > 0
59
GA CONT’D









fprintf('%s\n',msg)
end
return;
end
% If initial population was not empty then we replace
the first individual
% by the feasible point just found
if ~isempty(options.InitialPopulation) &&
~isempty(Iterate.x)
options.InitialPopulation(1,:) = Iterate.x';
feasible = true(pop,1);
60
GA CONT’D








try % InitialScores may not be present
options.InitialScores(1) = [];
catch
end
end
% Verify that individuals in InitialPopulation are
feasible
if ~isempty(options.InitialPopulation) &&
~strcmp(type,'unconstrained')
pop = size(options.InitialPopulation,1);
61
GA CONT’D









for i = 1:pop
feasible(i) =
isTrialFeasible(options.InitialPopulation(i,:)',Aineq,Bine
q,Aeq,Beq,LB,UB,options.TolCon);
end
options.InitialPopulation(~feasible,:) = [];
try % InitialScores may not be present
options.InitialScores(~feasible) = [];
catch
end
62
end
GA CONT’D

% Validate nonlinear constraints
 [LinearConstr, Iterate,nineqcstr,neqcstr,ncstr] =
constrValidate(NonconFcn, ...

Iterate,Aineq,Bineq,Aeq,Beq,LB,UB,type,options);
 options.LinearConstr = LinearConstr;


% Make sure that bounds and PopInitRange are
consistent
 options.PopInitRange =
checkPopulationInitRange(LB,UB,options.PopInitRang
63
e);
GA CONT’D








% Print some diagnostic information if asked for
if options.Verbosity > 2
gadiagnose(FitnessFcn,NonconFcn,GenomeLength,nine
qcstr,neqcstr,ncstr,user_options);
end
% Call appropriate single objective optimization solver
switch(output.problemtype)
case 'unconstrained'
[x,fval,exitFlag,output,population,scores] =
gaunc(FitnessFcn,GenomeLength, ...
64
GA CONT’D








options,output,Iterate);
case {'boundconstraints', 'linearconstraints'}
[x,fval,exitFlag,output,population,scores] =
galincon(FitnessFcn,GenomeLength, ...
Aineq,Bineq,Aeq,Beq,LB,UB,options,output,Iterate);
case 'nonlinearconstr'
[x,fval,exitFlag,output,population,scores] =
gacon(FitnessFcn,GenomeLength, ...
Aineq,Bineq,Aeq,Beq,LB,UB,NonconFcn,options,outpu
t,Iterate,subtype);
65
end
Function codes
Rosenbrock’s Function
 function f=rosenbrock(x)
 sumc=0;
 for i=1:length(x)-1
 sumc = sumc+100*((x(i+1)-x(i)^2)^2) + (1x(i))^2;
 end
 f = sumc

66
Codes cont’d
Rastrigin’s Function
 function scores = rastriginsfcn(x)



scores = 10.0 * size(x,2) + sum(x .^2 - 10.0
* cos(2 * pi .*x),2);
67
Codes cont’d
Schwefel’s function
 function y=schwefel(x)
 sz=size(x);
 if sz(1)==1
 x=x';
 end
 y=-sum(x.*sin(sqrt(abs(x))));

68