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

Title of the script: Constrained Optimisation using fot_fmincon and fot_intfmincon functions

Author: Siddharth Agarwal and Mankrit Singh

Keywords: FOSSEE Optimization Toolbox, Integer Constrained Optimisation, Constrained Optimisation, OR, fot_fmincon, fot_intfmincon .

Visual Cue	Narration
Show Slide Title Slide	Welcome to the spoken tutorial on Constrained Optimisation.
Show Slide Learning Objectives	In this tutorial, we will learn how to: Use fot underscore fmincon and fot underscore intfmincon functions in Scilab Solve constrained optimisation problems using fot underscore fmincon and fot underscore intfmincon
Show Slide System Requirements	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 optimization and Scilab If not, for relevant tutorials please visit this site.
Show Slide Code Files	The files used in this tutorial are 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 Constrained Optimisation problem?	A constrained nonlinear optimisation problem is a mathematical optimisation model. It has: A linear or nonlinear objective function Bounds on the decision variables Linear and nonlinear constraints on decision variables
Show Slide Mathematical Formulation where f , A, b, Aeq, beq, c, ceq, lb, and ub are given.	A general form of constrained nonlinear optimisation problem is as shown.
Show Slide Example	We will now solve this example to illustrate the use of fot underscore fmincon. In this example, we will learn how to: Minimize the given function Note that the objective function is nonlinear. The example has 4 variables x1, x2, x3 and x4. There are bounds on the variables, but no constraints otherwise.
Show Slide Example	I have downloaded the required files to my Downloads folder.
Open the Scilab window >> place the cursor on the Scilab console.	Now open the Scilab console.
Type editor >> press Enter.	In the Scilab console type editor and press Enter. Editor window opens.
Click on File >> Open button >> locate the file opt_fmincon .sce. Video-editor: Pls put a textbox on screen. “In Windows OS ,Click on Open button”	Click on the Open button on the toolbar and locate the file opt_fmincon.sce. Then click the Ok button. opt_fmincon.sce file opens in the editor.
Show opt_fmincon .sce in scilab editor.	Now we will see the input arguments for fot underscore fmincon.
Highlight ‘f’	f is the objective function that has to be minimized.
Highlight ‘x0’	x0 is a vector with the initial values of the decision variables.
Highlight the line with ‘A’	A is a matrix of coefficients of linear inequality constraints.
Highlight the line with ‘b’	b is a vector of the right-hand side of linear inequality constraints.
Highlight ‘lb’ and ub	lb and ub are the row vectors. They contain the lower and upper bounds of the decision variables respectively.
Highlight ‘Nonlinearcon’	‘Nonlinearcon’ is a Scilab function. It represents the equality and inequality nonlinear constraints for the problem.
Highlight Output Arguments	Now we will see the output arguments. Output arguments are xopt, fopt, exitflag, output, lambda, gradient, hessian.
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 following: Lagrange multipliers of the lower bounds and upper bounds. Linear equality and inequality constraints at the optimized point.
Highlight ‘gradient’	Gradient is a vector containing the objective's gradient of the solution.
Highlight ‘hessian’	Hessian is a matrix containing the Hessian of Lagrangian at the solution.
Highlight [xopt,fopt,exitflag,output, lambda,gradient, hessian] = fot_fmincon(ObjectiveFunction,x0,A,b,[],[],lb,ub,Nonlinearcon)	Here we see the Scilab code to define and solve the example. We call the fot underscore fmincon function to solve the given problem.
Press CTRL + s Click on execute button on scilab Click on File with Echo	Save the file by pressing Control and S keys simultaneously. To run the file, click on the Execute menu. Click on 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. We see that it prints the fopt value, xopt values, exitflag, output, lambda, gradient, and hessian on the Scilab console. We will now close the Scilab editor window.
Show Slide Integer Constraints	We will now look at constrained integer nonlinear programming problems. These are problems in which some decision variables are constrained to be integers.
Show Slide Mathematical Formulation	A general form of the constrained integer optimisation problem is as shown.
Show Slide Example	We will now solve this example to illustrate the use of fot underscore intfmincon In this example, we will demonstrate how to minimize the given function. Note that some of the decision variables are constrained to be integers. Let’s use the previous example with added integer constraints on the variables x1 and x2. We will use the toolbox to solve this example.
Show opt_intfmincon.sce in Scilab editor.	Open the Scilab console. Type editor on the Scilab console and press Enter. Open opt_intfmincon.sce in the Scilab editor.
Show opt_intfmincon.sce in scilab editor.	We have the same input arguments that we had in opt_fmincon.sce. f,x0,A,b,lb,ub and Nonlinearcon Along with them we have an additional input argument called as intcon.
Highlight ‘intcon’	intcon is a vector of the indices of integer variables.
Highlight Output Arguments	Now we will see the output arguments. We have the same output arguments that we had in opt_fmincon.sce Output arguments are xopt, fopt, exitflag, gradient, hessian
Highlight [xopt,fopt,exitflag,gradient,hessian] = fot_intfmincon(ObjectiveFunction,x0,int,A,b,[],[],lb,ub,Nonlinearcon)	This is how we call fot underscore intfmincon function.
Press CTRL + s Click on execute button on scilab	Save the file by pressing Control and S keys simultaneously. To run the file, click on the Execute menu. Click on 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.
Highlight Outputs	We see that it prints the fopt value, xopt values, exitflag, output, lambda, gradient, and hessian 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 fmincon and fot underscore intfmincon functions of the FOSSEE Optimization Toolbox. Solve constrained nonlinear programming examples in Scilab.
Show Slide Assignment	As an assignment: Minimise the following objective function subject to the given constraints The optimal value will be 75728.822 and optimal solution will be x one equal to 3.1692029 and x two equal to 6.3384058
Show Slide About Spoken Tutorial Project	The video at the following link summarises the Spoken Tutorial project. Please download and watch it.
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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

Mankrits, Nancyvarkey