Scilab/C4/Optimization-Using-Karmarkar-Functions/English

Title of script: Optimization of Linear Functions with Linear Constraints Using Scilab

Author: Mukul R. Kulkarni

Keywords: Scilab, Optimization, karmarkar

Visual Clue	Narration
Display Slide	Welcome to the spoken tutorial on 'Optimization of Linear Functions with Linear Constraints Using Scilab'
Display Slide Objectives	In this tutorial, We will learn What is meant by Optimization? And How to use Scilab function karmarkar for optimization.
Display Slide What is Optimization ?	Optimization means 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.
	* These constraints are also in the form of some functions of the variables. Optimization is extensively used in majority of the engineering as well as non-engineering fields like Economics Control Theory and Operations & Research.
Display Slide Scilab Function: karmarkar	The Scilab function Karmarkar is used for 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
help Karmarkar Enter Scilab Function: karmarkar Output Arguments	Before we solve the given problem go to scilab console and 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.
Display Slide Scilab Function: karmarkar Input Arguments:	Input arguments are 'Aeq' 'beq' 'c' 'x' zero 'rtolf 'gam' 'maxiter' 'outfun' 'A' 'b' 'lb' and 'ub' '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'.
Display Slide Using Scilab function karmarkar to solve an example	We can now solve the given example in Scilab using karmarkar function. Go to the scilab console and type
Scilab Console 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.	'A' equals to open square bracket, two <space> one <space> one <semicolon> one <space> two <space> three <semicolon> two <space> two <space> one, close the square bracket. 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
Display Slide Summary	In this tutorial, we learned What is Optimization? And Use of Scilab function karmarkar in optimization to solve linear problems.
Display Slide Contact	To contact the scilab team, please write to contact@scilab.in
Disp1lay Slide http://spoken-tutorial.org /What\_is\_a\_Spoken\_Tutorial About the Spoken Tutorial Project	Watch the video available at the link shown below It summarises the Spoken Tutorial project If you do not have good bandwidth, you can download and watch it
Display Slide Spoken Tutorial Workshops	The Spoken Tutorial Project Team 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
Display Slide Acknowledgement	Spoken Tutorial Project is a part of the Talk to a Teacher project 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
	Anuradha Amrutkar from IIT Bombay Thank You for joining.

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Lavitha Pereira