Scilab---FOSSEE-Optimisation-Toolbox/C2/Quadratic-Optimisation-using-FOT/English

Title of the script: Quadratic Optimization

Author: Siddharth Agarwal and Mankrit Singh

Keywords: FOSSEE Optimization Toolbox, Integer Quadratic optimization, Quadratic optimization, OR, quadprog, intquadprog .

Visual Cue	Narration
Show Slide Title Slide	Hello and welcome to the Spoken tutorial on Quadratic optimization.
Show Slide Learning Objectives	In this tutorial, we will learn how to: Use fot underscore quadprog and fot underscore intquadprog functions in Scilab Solve quadratic optimization problems using, fot underscore quadprog and fot underscore intquadprog functions
Show Slide System Requirement	To record this tutorial, I am using Ubuntu 18.04 Scilab 6.1.0 and FOSSEE Optimization Toolbox version 0.4.1
Show Slide Pre-requisites https://spoken-tutorial.org	To follow this tutorial, you should Install FOSSEE Optimization Toolbox version 0.4.1 or above Have basic understanding of Scilab and optimization theory If not, for relevant tutorials please visit this site.
Show Slide Code Files	The files used in this tutorial have been provided in the Code files link. Please download and extract the files. Make a copy and then use them while practising.
Show Slide What is the Quadratic optimization problem?	What is the Quadratic optimization problem? A function is quadratic if it has a degree of two. A quadratic optimization problem is a mathematical optimization model.
Show Slide What is the Quadratic optimization problem?	It has: Quadratic objective function Bounds on the decision variables Linear constraints on decision variables
Show Slide Mathematical Formulation	A general form of the quadratic optimization problem is as shown.
Show Slide Example	We will now solve this example to illustrate the use of fot underscore quadprog In this example, we will learn how to minimize the given function. Note that the objective function is quadratic. Note that there are bounds and linear constraints on the variables.
Show Slide Example	We will use the toolbox to solve this example. I have downloaded the required files to my Downloads folder
Cursor on the Scilab console.	Now open the Scilab console.
Type editor >> press Enter. Point to the editor window.	In the Scilab console, type editor and press Enter. Editor window opens.
Click on Open button >> locate the file opt_quadprog .sce.	Click on the Open button on the toolbar and locate the file opt_quadprog .sce. Then click the Ok button. opt_quadprog.sce file opens in the editor
Show opt_quadprog .sce in scilab editor.	Now we will see the input arguments for fot underscore quadprog.
Highlight ‘nbVar’	nbVar is an integer denoting the number of decision variables.
Highlight ‘nbCon’	nbCon is an integer denoting the number of constraints.
Highlight ‘H’	H is a symmetric matrix, representing the Hessian of the quadratic problem.
Highlight ‘f’	f is a vector, representing coefficients of the linear terms in the quadratic problem.
Highlight ‘x0’	x0 is a vector containing the starting values of the decision variables.
Highlight the line with ‘A’	A is a matrix of coefficients of inequality constraints.
Highlight the line with ‘conUB’	conUB is the vector of the upper bounds of the constraints.
Highlight the line with ‘conLB’	conLB is the vector of the lower bounds of the constraints.
Highlight lb and ub	lb and ub are row vectors. They contain the lower and upper bounds of the decision variables respectively.
Point to the Output arguments.	Now we will see the output arguments. Output arguments are xopt, fopt, exitflag, output and lambda.
Highlight ‘xopt’	xopt is the optimal value of x.
Highlight ‘fopt’	fopt is the optimal objective function value.
Highlight ‘exitflag’	exitflag is the status of execution.
Highlight ‘output’	output is a structure containing detailed information about the optimization.
Highlight lambda	lambda is a structure containing the Lagrange multipliers at the optimal solution.
Highlight [xopt,fopt,exitflag,output,lambda]=fot_quadprog(nbVar,nbCon,H,f,lb,ub,A,conLB,conUB,x0)	Here we see the Scilab code to define and solve the example. We call the fot underscore quadprog function to solve the given problem.
Press CTRL + S Click on the Execute button on scilab. Select file with echo from the drop-down.	Save the file by pressing Control and ‘S’ keys simultaneously. To run the file, click on the Execute menu. Then click on the File with echo from the drop down.
Point to the Confirmation box. Click Yes to confirm.	A confirmation box to clear the Console appears. Click on the Yes button to confirm.
Change the window to Scilab console	Switch to the Scilab console to see the output. The result displays “Optimal Solution Found.” We see that it prints the fopt value, xopt values, exitflag, output, and lambda in the Scilab console.
Show Slide Integer Quadratic Programming	We will now look at integer quadratic programming problems. These are problems where some decision variables are constrained to be integers.
Show Slide Mathematical Formulation	A general form of the integer constrained quadratic optimization problem is as shown.
Show Slide Example	We will now solve this example to illustrate the use of fot underscore intquadprog. In this example, we will learn how to minimize the given function. Note that the objective function is quadratic. Note that some of the decision variables are constrained to be integers.
Show opt_intquadprog.sce in scilab editor.	We will use the toolbox to solve this example. Open the Scilab console. Type editor on the Scilab console and press Enter. Open opt_intquadprog.sce in the Scilab editor.
Highlight ‘H’	H is a symmetric matrix, representing the Hessian of the quadratic problem.
Highlight ‘f’	f is a vector, representing coefficients of the linear terms in the quadratic problem.
Highlight the line with ‘A’	A is a matrix of coefficients of inequality constraints.
Highlight the line with b	b is the vector of the RHS of the inequality constraints.
Highlight the line with Aeq	Aeq is the matrix of the LHS of the equality constraints.
Highlight the line with beq	beq is the vector of the RHS of the equality constraints.
Highlight ‘intcon’	intcon is a vector of the indices of integer variables.
	Now we will see the output arguments. Output arguments are xopt, fopt, exitflag, output
Highlight ‘xopt’	xopt is the optimal value of x.
Highlight ‘fopt’	fopt is the optimal objective function value.
Highlight ‘exitflag’	exitflag is the status of execution.
Highlight output	Output is a structure containing detailed information about the optimization.
Highlight [xopt,fopt,exitflag,output]=fot_intquadprog(H,f,intcon,A,b,Aeq,beq)	This is how we call fot underscore intquadprog.
Press CTRL + S Click on the Execute button on scilab. Select file with echo from the drop-down.	Save the file by pressing Control and ‘S’ keys simultaneously. To run the file, click on the Execute menu. Then click on the File with echo from the drop down.
Point to the Confirmation box. Click Yes to confirm.	In the confirmation box, click the Yes button to clear the Console.
Change the window to Scilab console	Switch to the Scilab console to see the output. The result displays “Optimal Solution Found.” We see that it prints the fopt value, xopt values, exitflag and output in the Scilab console.
Show Slide Summary	This brings us to the end of this tutorial. Let us summarise. In this tutorial, we have learnt how to: Use fot underscore quadprog and fot underscore intquadprog functions of the FOSSEE Optimization Toolbox. Solve quadratic programming problems in Scilab.
Show Slide Assignment I	As an assignment: Minimise the following objective function. Use fot underscore quadprog function subjected to the given constraint
Show Slide Assignment I	The optimum value will be -8.22. The optimum solution will be the vector [0.66 1.33].
Show Slide Assignment II	Here is another assignment problem: Minimise the following objective function. Use fot underscore intquadprog function subjected to the given constraints.
Show Slide Assignment II	The optimum value will be -20. The optimum solution can be either vector [4 1 1 0 1] or [4 1 2 0 1].
Show Slide About Spoken Tutorial Project	The video at the following link summarises the Spoken Tutorial project. Please download and watch it.
Show Slide Spoken Tutorial Workshops	The Spoken Tutorial Project Team conducts workshops and gives certificates. For more details, please write to us
Show Slide Answers for THIS Spoken Tutorial	Please post your timed queries in this forum.
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Show Slide Textbook Companion project	The FOSSEE team coordinates the Textbook Companion project. We give Certificates and Honorarium to the contributors. For more details, please visit this site..
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Show Slide Acknowledgement	Spoken Tutorial and FOSSEE projects are funded by MoE, Government of India.
Show Slide Thank you	This is Mankrit Singh, a FOSSEE intern 2021, IIT Bombay signing off Thanks for joining.

Contributors and Content Editors

Madhurig, Mankrits, Nancyvarkey