Difference between revisions of "Scilab/C4/Optimization-Using-Karmarkar-Functions/English"

'Optimization of Linear Functions with Linear Constraints Using Scilab'

Objectives

• What is meant by Optimization?

And

• How to use Scilab function karmarkar for optimization.

What is Optimization ?

• Minimize or maximize a given objective function.
• Which is also called as Cost function sometimes.
• By varying the decision variables.
• The decision variables are varied subject to the pre-defined constraints.

• Optimization is extensively used in majority of the engineering as well as non-engineering fields like
  • Economics
  • Control Theory and
  • Operations & Research.

Scilab Function: karmarkar

• Optimizing the linear objective function

• subjected to linear constraints
• on the decision variables

We will solve the following example using the karmarkar function:

Minimize

minus three 'x' one minus 'x' two minus three 'x' three

for

two 'x' one plus 'x' two plus 'x' three is less than or equal to two.

'x' one plus two 'x' two plus three 'x' three is less than or equal to five.

two 'x' one plus two 'x' two plus 'x' three is less than or equal to six.

where 'x' one 'x' two 'x' three are all greater than or equal to zero

Note that all the functions objective functions as well as constraints are linear

Enter

Scilab Function: karmarkar

Output Arguments

type help karmarkar

and press Enter.

We can see the calling sequence of the argument.

The argument explaination, description and some examples in the help browser.

Close the help browser

We will summarize the input and output arguments here

Out put arguments are 'x' opt, 'f' opt, exitflag, iter,

'y' opt

• 'x' opt : is the optimum solution .
• 'f' opt : is the objective function value at optimum solution
• 'exitflag' : is the status of execution, it helps in identifying if the algorithm is converging or not.
• 'iter' : Is the number of iterations required to reach 'x' opt.
• 'y' opt : is the structure containing the dual solution
  This gives the Lagrange multipliers.

Scilab Function: karmarkar

Input Arguments:

• 'Aeq' : is the Matrix in the linear equality constraints.
• 'beq' :is the right hand side of the linear equality constraints.
• 'c' : is the Linear objective function co-efficients for 'x'.
• 'x' zero : is the Initial guess .
• 'rtolf' : is Relative tolerance on 'f' of 'x' equals to 'c' transpose multiplied by 'x'.
• 'gam' : is the Scaling factor.
• 'maxiter' : is the Maximum number of iterations after which the output is returned.
• 'outfun' : is the additional user-defined output functions .
• 'A' : is the Matrix of linear inequality constraints
• 'b' : is the right hand sideof the linear inequality constraints.
• 'lb' : is the lowerbound for 'x'.
• 'ub' : is the Upper bounds for 'x'.

Using Scilab function karmarkar to solve an example

Go to the scilab console and type

A = [2 1 1; 1 2 3; 2 2 1]

A =

2. 1. 1.

1. 2. 3.

2. 2. 1.

Enter

b = [2;5;6]

b =

2.

5.

6.

Enter

c = [-3; -1; -3]

c =

- 3.

- 1.

- 3.

Enter

lb = [0;0;0]

lb =

0.

0.

0.

Enter

clc

[xopt,fopt,exitflag,iter]=karmarkar([],[],c,[],[],[],[],[],A,b,lb)

Enter

Enter

xopt =

0.1999967

0.0000076

1.5999902

fopt =

- 5.3999683

iter =

70.

exitflag =

1.

And press Enter

similarly type,

'b' equals to open square bracket, two <semicolon>five <semicolon> six, close the square bracket.

And press Enter

'c' equals to open square bracket, minus three <semicolon> minus one <semicolon> minus three, close the square bracket.

And press Enter

Type

'lb' equals to open square bracket, zero <semicolon> zero <semicolon> zero, close the square bracket.

And press Enter

Now clear the Scilab console by clc command.

Type

open square bracket, 'x' opt <comma> 'f' opt <comma> 'exitflag' <comma> iter, close the square bracket equals to karmarkar open parenthesis, open square bracket, close the square bracket <comma> open square bracket, close the square bracket <comma> 'c' <comma> open square bracket, close the square bracket <comma> open square bracket, close the square bracket <comma> open square bracket, close the square bracket <comma> open square bracket, close the square bracket <comma> open square bracket, close the square bracket <comma> capital 'A' <comma> 'small b' <comma> 'lb', close the round bracket.

And Press enter

Press Enter to continue the Display

This will give the output as shown on the screen.

Where xopt is the optimal solution to the problem

fopt is the value of objective function calculated at optimum solution x is equal to xopt

and number of iteration required to reach optimum solution xopt is 70

Please note that: it is mandatory to specify the input arguments in same order.

In which they have been listed above, while calling the function

Summary

• What is Optimization?

And

• Use of Scilab function karmarkar in optimization to solve linear problems.

Contact

http://spoken-tutorial.org /What\_is\_a\_Spoken\_Tutorial

About the Spoken Tutorial Project

It summarises the Spoken Tutorial project

If you do not have good bandwidth, you can download and watch it

Spoken Tutorial Workshops

Conducts workshops using spoken tutorials

Gives certificates to those who pass an online test s

For more details, please write to

contact@spoken-tutorial.org

Acknowledgement

It is supported by the National Mission on Education through ICT, MHRD, Government of India

More information on this Mission is available at: http://spoken-tutorial.org\NMEICT-Intro

Thank You for joining.