Basics of the Genetic Algorithm in Matlab

Genetic Algorithm Optimization in MATLAB

A genetic algorithm is a population-based optimization method inspired by natural selection. It repeatedly selects promising candidate solutions and creates new solutions using crossover and mutation.

Genetic algorithms can be useful when:

• The objective function is nonlinear, discontinuous or difficult to differentiate.
• The search space contains several local optima.
• Traditional derivative-based methods are unsuitable.

A genetic algorithm is not automatically faster or better than every other optimization technique. The appropriate method depends on the problem. For the linear examples below, linprog would normally be more efficient, but they provide a simple introduction to MATLAB’s ga syntax.

Main components

• Fitness function: measures the quality of a candidate solution.
• Selection: chooses candidates for reproduction.
• Crossover: combines information from selected candidates.
• Mutation: introduces controlled random changes.

MATLAB syntax

[x, fval, exitflag] = ga( ...
    fitnessFunction, ...
    numberOfVariables, ...
    A, b, ...
    Aeq, beq, ...
    lowerBounds, upperBounds);

MATLAB expresses linear inequality constraints in this form:

A × x ≤ b

Use empty brackets, [], when an optional constraint is not required.

Problem 1: Minimization with linear constraints

Minimize:

z = −x₁ + 2x₂

Subject to:

• −x₁ + 3x₂ ≤ 10
• x₁ + x₂ ≤ 6
• x₁ − x₂ ≤ 2
• x₁ ≥ 0 and x₂ ≥ 0

Fitness function: fitness1.m

function y = fitness1(x)
    y = -x(1) + 2*x(2);
end

MATLAB commands

A = [-1  3;
      1  1;
      1 -1];

b = [10;
      6;
      2];

lb = [0 0];

[x, fval, exitflag] = ga( ...
    @fitness1, 2, A, b, [], [], lb, []);

The solution should be approximately:

x ≈ [2 0]
fval ≈ -2

Genetic algorithms use randomness, so the displayed values may differ slightly between runs.

Problem 2: Converting greater-than constraints

Minimize:

z = 20x₁ + 40x₂

Subject to:

• 36x₁ + 6x₂ ≥ 108
• 3x₁ + 12x₂ ≥ 36
• 20x₁ + 12x₂ ≥ 100
• x₁ ≥ 0 and x₂ ≥ 0

MATLAB requires inequalities in the form A × x ≤ b. Therefore, multiply each greater-than constraint by −1:

• −36x₁ − 6x₂ ≤ −108
• −3x₁ − 12x₂ ≤ −36
• −20x₁ − 12x₂ ≤ −100

Fitness function: fitness2.m

function y = fitness2(x)
    y = 20*x(1) + 40*x(2);
end

MATLAB commands

A = [-36  -6;
      -3 -12;
     -20 -12];

b = [-108;
      -36;
     -100];

lb = [0 0];

[x, fval, exitflag] = ga( ...
    @fitness2, 2, A, b, [], [], lb, []);

The solution should be approximately:

x ≈ [3.7647  2.0588]
fval ≈ 157.6471

Important observation

The original third row of matrix A used [-20 -10], although the stated constraint contained 12x₂. It has been corrected to [-20 -12].

Learn With Champak: Experiment with different population sizes, mutation settings and stopping criteria to observe how they affect the quality and speed of the result.

Solving Equations using Matlab

Equations and Polynomials in MATLAB

Algebra has a long history shaped by Indian, Babylonian, Egyptian, Greek and Persian mathematicians. Brahmagupta made important contributions to arithmetic, algebra and the treatment of zero, while the word algebra comes from the title of Al-Khwarizmi’s work Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala.

Basic rule of equations

An equation remains balanced when the same valid operation is performed on both sides. If LHS = RHS, then:

• LHS + x = RHS + x
• LHS − x = RHS − x
• LHS × x = RHS × x
• LHS ÷ x = RHS ÷ x, provided x ≠ 0

Representing a polynomial in MATLAB

MATLAB represents a polynomial using a vector of coefficients arranged in descending powers.

p = [1 0 6 20];

This represents x³ + 6x + 20.

Evaluating the polynomial

Use polyval to evaluate the polynomial for a given value of x.

For x = 1

>> polyval(p, 1)

ans =

    27

For x = 2.5

>> polyval(p, 2.5)

ans =

    50.6250

For the complex value x = 1 + i

>> polyval(p, 1 + i)

ans =

    24.0000 + 8.0000i

Finding the roots

Use the roots function to calculate the roots of the polynomial.

>> roots(p)

ans =

    1.0000 + 3.0000i
    1.0000 - 3.0000i
   -2.0000

The roots can also be stored in a vector:

>> r = roots(p);

Reconstructing the polynomial

Use poly to recover the polynomial coefficients from its roots.

>> v = poly(r)

v =

    1.0000    0.0000    6.0000   20.0000

Plotting the polynomial

Create a vector of x-values, evaluate the polynomial and plot the result.

x = 0:0.01:10;
y = polyval(p, x);
plot(x, y);
grid on;
xlabel('x');
ylabel('p(x)');
title('p(x) = x^3 + 6x + 20');

Multiplying polynomials

Use conv to multiply two polynomials. To multiply the original polynomial by x − 2:

>> product = conv(p, [1 -2])

product =

     1    -2     6     8   -40

The resulting polynomial is x⁴ − 2x³ + 6x² + 8x − 40.

Dividing polynomials

Use deconv to divide polynomials. Dividing the product by x − 2 returns the original polynomial:

>> deconv(product, [1 -2])

ans =

     1     0     6    20

Learn With Champak: Practise these commands by changing the coefficients, evaluating the polynomial at different values and comparing the resulting graphs.

3 D Plots in Matlab

3 D Plots in Matlab

To create 3D plots , first of all we need to create 3 variables. x,y and z.

Create the x and y  arrays:-

>> x=-3:0.25:3;
>> y=-3:0.25:3;

Create the meshgrid:-

>> [X Y]=meshgrid(x,y);

Define:-

Z=sqrt(X.^2 + Y.^2);

mesh(X,Y,Z);

>>surf(X,Y,Z);

>>surf(X,Y,-Z);

>>waterfall(X,Y,Z):-

 >>contour(x,y,z):

 >>>> surfc(X,Y,Z):-

 Now we arrive to the view part:

View is used to give the viewing angle of the plot. the syntax of the command is :-

view(azimuth,elevation);

where  the azimuth is an angle measured in degrees  in the x-y plane and is measured relative to the negative y axis and positive as always in the anti clockwise direction.

The elevation is the angle measured from the x-y plane in the direction of the z - axis.

default values are azimuth=-37.5 and elevation =35.

Here are the results of applying different views to the last plot:-

>> view(-65,30);

 >> view(-37.5,40);

Next we shall move to interpolation using Matlab.

Matrices in Matlab

Matrices in Matlab

Matrix manipulation is probably the most important part of Matlab. It is what most people will use it for. In this post we shall learn how to create matrices, perform basic manipulations, and to solve simultaneous equations using Matlab.

Creating a Matrix:- 

>> a=[1,4,-1;1,1,-6;3,-1,-1];
a

a =

     1     4    -1
     1     1    -6
     3    -1    -1

>>

Transposing the matrix :-

>> a'

ans =

     1     1     3
     4     1    -1
    -1    -6    -1

>>

Inverse of a Matrix:-

>> inv(a)

ans =

    0.0986   -0.0704    0.3239
    0.2394   -0.0282   -0.0704
    0.0563   -0.1831    0.0423

>>

>> a*inv(a)

ans =

    1.0000   -0.0000         0
   -0.0000    1.0000         0
    0.0000    0.0000    1.0000

>> The identity matrix as you would expect.

Now let's try and solve the given equation using Matlab:-

1) x + y + z =9

2)2x - 3y + 4z =13

3)3x + 4y + 5z=40

Create two matrices a & b:-

>> a=[1,1,1;2,-3,4;3,4,5]

a =

     1     1     1
     2    -3     4
     3     4     5

>> b=[9;13;40]

b =

     9
    13
    40

>>

This equation can be represented as 

AX=B

=>inv(A) * A * X= inv(A) * B multiplying both sides by inv(A)

=> X=inv(A)* B 

>> inv(a)*b

ans =

    1.0000
    3.0000
    5.0000

>>

which means 

x=1

y=3

z=5

Now, let's try it the other way round.

The same equation can be represented as X'*A'=B' where ' represents the Transpose.

multiplying both sides by inv(A')

=>X'*A'*inv(A')=B'*inv(A')

=>X'=B'*inv(A')

>> b'*inv(a')

ans =

    1.0000    3.0000    5.0000

which means 

x=1

y=3

z=5

Simple Plots in Matlab

Simple Plots in Matlab

This is a continuation of the earlier post Matlab Basics

Creating Plots in Matlab is probably the easiest in all possible options present in the Software Universe. We shall study only the most basic methods in this Post and then advance to comparatively complex features.

Create a Vector for representing the X - Axis

>> x=[-2*pi:.001:2*pi];
>>

>> y=sin(x);
>> plot(x,y)
>> 

>> plot(cos(x),sin(x))
>>

 

 

>> plot(cos(x),sin(x).*cos(x))
>> 

%Remember the .* represents the dot product, and is necessary because all variables in Matlab are vectors and by default it will try to multiply them as matrices.

Next, we shall advance to more complex formatting of Plots, and then 3D plots.

Creating Variables and Simple Expressions in MATLAB

Basics of Matlab

The easiest way to begin learning Matlab is to start with the Command Window under the Desktop Menu.

Executing expressions in Matlab is absolutely simple !!! Just write the expression and view the results.

1) >> 3+4/10

ans =

    3.4000

>> 

To create a Variable

Simply write the expression
>> x=10

x =

    10

>> x*x + x

ans =

   110

>>
The variable is created automatically and initialized


To view the created variables write
>> x=1,y=2

x =

     1


y =

     2

>> who

Your variables are:

ans  x    y   

Point to remember: There are no scalars in Matlab. Only vectors. Thus x would  actually be [x].
Point to remember: To suppress an output use ; at the statements end.
Point to remember: use clc to clear the display.

>> x=5;
>> x=5

x =

     5

>> 



Creating Vectors

Vectors can be created in the following manners :
>> a=[1,2,3,4,5]

a =

     1     2     3     4     5

>>


>> a=[1:1:10]

a =

     1     2     3     4     5     6     7     8     9    10

>>


The first case is a simple enumeration, the second case specifies the starting value, the increment/decrement and the last value.

>> a=[10:-1:1]

a =

    10     9     8     7     6     5     4     3     2     1

>>

We shall learn 2D and 3D plotting next.

Generic Programming in C#

Generic Programming is derived from the Template based programming that originated with C++. The syntax remains the same, however the logic has changed a little bit. In C++ the methods are generated automatically based on the type of the Input Parameters. In C# the Type Parameters are specified as a part of the Class Declaration. This Permits type verification at compile time, thus making the system Type Safe.

A template based stack in C++


#include<conio.h>
#include<string.h>
#include<iostream.h>
template<class T>
class Stack
{

private:class data
{
public:T value;
public: data* next;
};
private:data* ptr;
public:Stack()
{
ptr=NULL;
}
public:void push(T value)
{
data* temp=new data();
temp->value=value;
temp->next=ptr;
ptr=temp;
}
public:void pop(T& value)
{
if(ptr==NULL)
value=NULL;
else
value=ptr->value;
ptr=ptr->next;
}
public:int isEmpty()
{
return(ptr==NULL);
}
};
void main()
{
Stack<int> s;
for(int i=0;i<=10;i++)
s.push(i);
cout<<"\n";
while(!s.isEmpty())
{
int n;
s.pop(n);
cout<<n<<",";
}
getch();
}



A Generic Class in C#
Here we implement a class that provides a Generic Max function that compares two inputs and returns the larger of the two. Here is the Code:

Comparer.cs
public   class Comparer<T> where T:IComparable
    {
      public T max(T a, T b)
      {
          if (a.CompareTo (b)>=0)
              return a;
          else
              return b;
      }

    }




Program.cs

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

namespace Generics_Operation
{
    class Program
    {
public static int add(params int[] numbers)
        {
            int sum = 0;
            foreach (int number in numbers)
                sum += number;
            return sum;
        }
        static void Main(string[] args)
        {
            Comparer <int> intcomparer = new Comparer <int>();
            Comparer<string> stringcomparer = new Comparer<string>();
           // Comparer<Program> p = new Comparer<Program>(); This is not allowed as Program does not implement IComparable
         Console.WriteLine (intcomparer.max(1, 2));
         Console.WriteLine(stringcomparer.max ("Sachin","Rahul"));
Console.WriteLine(add(1, 2, 3, 4, 5, 6));
            Console.ReadKey();

        }
    }
}