Difference between revisions of "Scilab---FOSSEE-Optimisation-Toolbox/C2/Constrained-Optimisation-using-FOT/English"

From Script | Spoken-Tutorial
Jump to: navigation, search
(Created page with "'''Title of the script''': C'''onstrained Optimisation using fot_fmincon and fot_intfmincon functions''' '''Author: Siddharth Agarwal and Mankrit Singh''' '''Keywords: FOSSE...")
 
Line 24: Line 24:
 
<ul>
 
<ul>
 
<li><blockquote><p>Use '''fot underscore fmincon''' and '''fot underscore intfmincon''' functions in '''Scilab'''</p></blockquote></li>
 
<li><blockquote><p>Use '''fot underscore fmincon''' and '''fot underscore intfmincon''' functions in '''Scilab'''</p></blockquote></li>
−
<li><blockquote><p>Solve C'''onstrained Optimisation''' problems using '''fot underscore fmincon''' and '''fot underscore intfmincon'''</p></blockquote></li></ul>
+
<li><blockquote><p>Solve constrained '''optimisation''' problems using '''fot underscore fmincon''' and '''fot underscore intfmincon'''</p></blockquote></li></ul>
 
|-
 
|-
 
|
 
|
Line 49: Line 49:
 
<ul>
 
<ul>
 
<li><blockquote><p>Install '''FOSSEE Optimization Toolbox''' version '''0.4.1''' or above</p></blockquote></li>
 
<li><blockquote><p>Install '''FOSSEE Optimization Toolbox''' version '''0.4.1''' or above</p></blockquote></li>
−
<li><blockquote><p>Have basic understanding of optimization and Scilab</p></blockquote></li></ul>
+
<li><blockquote><p>Have basic understanding of '''optimization''' and '''Scilab'''</p></blockquote></li></ul>
  
 
If not, for relevant tutorials please visit this site.
 
If not, for relevant tutorials please visit this site.
Line 67: Line 67:
  
 
'''What is the Constrained Optimisation problem?'''
 
'''What is the Constrained Optimisation problem?'''
−
|
+
|A constrained nonlinear '''optimisation''' problem is a mathematical '''optimisation model.'''
−
'''A Constrained Nonlinear Optimisation Problem''' is a mathematical optimisation model.
+
  
 
It has:
 
It has:
  
 
<ul>
 
<ul>
−
<li><blockquote><p>'''A Linear or nonlinear''' objective function</p></blockquote></li>
+
<li><blockquote><p>A linear or nonlinear''' objective function'''</p></blockquote></li>
 
<li><blockquote><p>'''Bounds''' on the '''decision variables'''</p></blockquote></li>
 
<li><blockquote><p>'''Bounds''' on the '''decision variables'''</p></blockquote></li>
−
<li><blockquote><p>'''Linear''' and '''Nonlinear constraints'''</p></blockquote></li></ul>
+
<li><blockquote><p>Linear and nonlinear constraints on '''decision variables'''</p></blockquote></li></ul>
  
−
<blockquote>on '''decision variables'''
 
−
</blockquote>
 
 
|-
 
|-
 
|
 
|
Line 88: Line 85:
  
 
'''ceq, lb, and ub are given.'''
 
'''ceq, lb, and ub are given.'''
−
| A '''general form''' of '''Constrained Nonlinear Optimisation problem''' is as shown.
+
| A general form of constrained nonlinear '''optimisation''' problem is as shown.
 
|-
 
|-
 
|
 
|
Line 100: Line 97:
  
 
<ul>
 
<ul>
−
<li><blockquote><p>Minimize the given function</p></blockquote></li></ul>
+
<li><blockquote><p>Minimize the given '''function'''</p></blockquote></li></ul>
  
−
Note that the '''objective function''' is '''nonlinear'''
+
Note that the '''objective function''' is nonlinear.
  
−
The Example has 4 variables x1, x2, x3 and x4.
+
The example has 4 '''variables x1, x2, x3''' and '''x4'''.
  
−
There are bounds on the variables, but no constraints otherwise.
+
There are '''bounds''' on the '''variables''', but no constraints otherwise.
 
|-
 
|-
 
|
 
|
Line 114: Line 111:
 
| I have downloaded the required files to my '''Downloads''' folder.
 
| I have downloaded the required files to my '''Downloads''' folder.
 
|-
 
|-
−
| Open the '''Scilab window''' and place the cursor on the '''Scilab console'''.
+
| Open the '''Scilab''' window >> place the cursor on the '''Scilab console'''.
 
| Now open the '''Scilab console.'''
 
| Now open the '''Scilab console.'''
 
|-
 
|-
Line 128: Line 125:
 
Video-editor: Pls put a textbox on screen. “In Windows OS ,Click on Open button”
 
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'''.
+
Click on the '''Open''' button on the '''toolbar''' and locate the file '''opt_fmincon.sce'''.
  
 
Then click the '''Ok''' button.
 
Then click the '''Ok''' button.
Line 135: Line 132:
 
|-
 
|-
 
| Show '''opt_fmincon .sce''' in scilab editor.
 
| Show '''opt_fmincon .sce''' in scilab editor.
−
| Now we will see the input arguments for '''fot underscore fmincon.'''
+
| Now we will see the '''input arguments''' for '''fot underscore fmincon.'''
 
|-
 
|-
 
| Highlight '''‘f’'''
 
| Highlight '''‘f’'''
Line 141: Line 138:
 
|-
 
|-
 
| Highlight '''‘x0’'''
 
| Highlight '''‘x0’'''
−
| '''x0''' is a vector with the initial values of the decision variables.
+
| '''x0''' is a '''vector''' with the initial values of the '''decision variables'''.
 
|-
 
|-
 
| Highlight the line with '''‘A’'''
 
| Highlight the line with '''‘A’'''
 
|
 
|
−
'''''A''''' is a '''matrix''' of '''coefficients''' of
+
'''''A''''' is a '''matrix''' of '''coefficients''' of linear inequality constraints.
−
 
+
−
'''linear inequality''' constraints.
+
 
|-
 
|-
 
| Highlight the line with '''‘b’'''
 
| Highlight the line with '''‘b’'''
 
|
 
|
−
'''''b''''' is a '''vector''' of the right-hand side
+
'''''b''''' is a '''vector''' of the right-hand side of linear inequality constraints.
−
 
+
−
of linear '''inequality''' constraints.
+
 
|-
 
|-
 
| Highlight '''‘lb’''' and '''ub'''
 
| Highlight '''‘lb’''' and '''ub'''
Line 159: Line 152:
 
'''lb''' and '''ub''' are the row vectors.
 
'''lb''' and '''ub''' are the row vectors.
  
−
They contain the lower and upper bounds of the decision variables respectively.
+
They contain the lower and upper '''bounds''' of the '''decision variables''' respectively.
 
|-
 
|-
 
| Highlight '''‘Nonlinearcon’'''
 
| Highlight '''‘Nonlinearcon’'''
 
|
 
|
−
'''‘Nonlinearcon’''' is a scilab function.
+
'''‘Nonlinearcon’''' is a '''Scilab function'''.
  
 
It represents the equality and inequality nonlinear constraints for the problem.
 
It represents the equality and inequality nonlinear constraints for the problem.
Line 169: Line 162:
 
| Highlight Output Arguments
 
| Highlight Output Arguments
 
|
 
|
−
Now we will see the output arguments.
+
Now we will see the '''output arguments'''.
  
−
Output arguments are '''xopt,''' '''fopt''', '''exitflag, output, lambda, gradient, hessian.'''
+
'''Output arguments''' are '''xopt, fopt, exitflag, output, lambda, gradient, hessian.'''
 
|-
 
|-
 
| Highlight '''‘xopt’'''
 
| Highlight '''‘xopt’'''
Line 177: Line 170:
 
|-
 
|-
 
| Highlight '''‘fopt’'''
 
| Highlight '''‘fopt’'''
−
| '''fopt''' is the optimal objective function value'''.'''
+
| '''fopt''' is the optimal '''objective function''' value.
 
|-
 
|-
 
| Highlight '''‘exitflag’'''
 
| Highlight '''‘exitflag’'''
−
| '''exitflag''' is the status of execution
+
| '''exitflag''' is the status of '''execution'''.
 
|-
 
|-
 
| Highlight '''‘output’'''
 
| Highlight '''‘output’'''
−
| '''Output''' is a '''structure''' containing detailed information about the optimization.
+
| '''Output''' is a '''structure''' containing detailed information about the '''optimization'''.
 
|-
 
|-
 
| Highlight '''‘lambda’'''
 
| Highlight '''‘lambda’'''
Line 189: Line 182:
 
'''Lambda''' is a structure containing the following:
 
'''Lambda''' is a structure containing the following:
  
−
'''Lagrange multipliers''' of the lower bounds and upper bounds.
+
'''Lagrange multipliers''' of the lower '''bounds''' and upper '''bounds'''.
  
−
Linear equality and inequality constraints at the optimized point.
+
Linear equality and inequality constraints at the '''optimized''' point.
 
|-
 
|-
 
| Highlight '''‘gradient’'''
 
| Highlight '''‘gradient’'''
Line 204: Line 197:
 
'''[xopt,fopt,exitflag,output, lambda,gradient, hessian] = fot_fmincon(ObjectiveFunction,x0,A,b,[],[],lb,ub,Nonlinearcon)'''
 
'''[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.
+
Here we see the '''Scilab''' code to define and solve the example.
  
 
We call the '''fot underscore fmincon function''' to solve the given problem.
 
We call the '''fot underscore fmincon function''' to solve the given problem.
Line 226: Line 219:
 
Click Yes to confirm.
 
Click Yes to confirm.
 
|
 
|
−
A confirmation box to clear the Console appears.
+
A confirmation box to clear the '''Console''' appears.
  
 
Click on the '''Yes''' button to confirm.
 
Click on the '''Yes''' button to confirm.
Line 243: Line 236:
 
'''Integer Constraints'''
 
'''Integer Constraints'''
 
|
 
|
−
We will now look at '''constrained integer nonlinear programming''' problems.
+
We will now look at constrained '''integer''' nonlinear '''programming''' problems.
  
−
These are problems in which some decision variables are constrained to be integers.
+
These are problems in which some '''decision variables''' are constrained to be '''integers'''.
 
|-
 
|-
 
|
 
|
Line 251: Line 244:
  
 
'''Mathematical Formulation'''
 
'''Mathematical Formulation'''
−
| A '''general form''' of the '''constrained integer optimisation problem''' is as shown.
+
| A general form of the constrained integer '''optimisation''' problem is as shown.
 
|-
 
|-
 
|
 
|
Line 260: Line 253:
 
We will now solve this example to illustrate the use of '''fot underscore intfmincon'''
 
We will now solve this example to illustrate the use of '''fot underscore intfmincon'''
  
−
In this example, we will demonstrate how to:
+
In this example, we will demonstrate how to minimize the given '''function'''.
−
 
+
−
<ul>
+
−
<li><blockquote><p>Minimize the given function</p></blockquote></li></ul>
+
  
−
Note that some of the '''decision variables''' are '''constrained''' to be integers.
+
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.'''
+
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.
 
We will use the toolbox to solve this example.
 
|-
 
|-
−
| Show '''opt_intfmincon.sce''' in '''scilab editor.'''
+
| Show '''opt_intfmincon.sce''' in '''Scilab editor.'''
 
|
 
|
 
Open the '''Scilab console'''.
 
Open the '''Scilab console'''.
  
−
Type editor on the Scilab console and Press enter.
+
Type editor on the '''Scilab console''' and press '''Enter'''.
  
 
Open '''opt_intfmincon.sce''' in the '''Scilab editor.'''
 
Open '''opt_intfmincon.sce''' in the '''Scilab editor.'''
Line 281: Line 271:
 
| Show opt_intfmincon.sce in scilab editor.
 
| Show opt_intfmincon.sce in scilab editor.
 
|
 
|
−
We have the same input arguments that we had in opt_fmincon.sce.
+
We have the same '''input arguments''' that we had in '''opt_fmincon.sce'''.
  
 
'''f,x0,A,b,lb,ub''' and '''Nonlinearcon'''
 
'''f,x0,A,b,lb,ub''' and '''Nonlinearcon'''
  
−
.
 
  
−
Along with them we have an additional input argument called as '''intcon.'''
+
Along with them we have an additional '''input argument''' called as '''intcon.'''
 
|-
 
|-
 
| Highlight '''‘intcon’'''
 
| Highlight '''‘intcon’'''
−
| '''intcon''' is a '''vector''' of the '''indices''' of integer variables
+
| '''intcon''' is a '''vector''' of the '''indices''' of '''integer variables'''.
 
|-
 
|-
 
| Highlight Output Arguments
 
| Highlight Output Arguments
 
|
 
|
−
Now we will see the output arguments.
+
Now we will see the '''output arguments'''.
  
−
We have the same output arguments that we had in '''opt_fmincon.sce'''
+
We have the same '''output arguments''' that we had in '''opt_fmincon.sce'''
  
−
Output arguments are '''xopt,''' '''fopt''', '''exitflag, gradient, hessian'''
+
'''Output arguments''' are '''xopt, fopt, exitflag, gradient, hessian'''
 
|-
 
|-
 
| Highlight '''[xopt,fopt,exitflag,gradient,hessian] = fot_intfmincon(ObjectiveFunction,x0,int,A,b,[],[],lb,ub,Nonlinearcon)'''
 
| Highlight '''[xopt,fopt,exitflag,gradient,hessian] = fot_intfmincon(ObjectiveFunction,x0,int,A,b,[],[],lb,ub,Nonlinearcon)'''
Line 340: Line 329:
 
<ul>
 
<ul>
 
<li><blockquote><p>Use '''fot underscore fmincon''' and '''fot underscore intfmincon''' functions of the '''FOSSEE Optimization Toolbox'''.</p></blockquote></li>
 
<li><blockquote><p>Use '''fot underscore fmincon''' and '''fot underscore intfmincon''' functions of the '''FOSSEE Optimization Toolbox'''.</p></blockquote></li>
−
<li><blockquote><p>Solve '''constrained nonlinear''' programming examples in '''Scilab'''.</p></blockquote></li></ul>
+
<li><blockquote><p>Solve constrained nonlinear '''programming''' examples in '''Scilab'''.</p></blockquote></li></ul>
 
|-
 
|-
 
|
 
|
Line 350: Line 339:
  
 
<ul>
 
<ul>
−
<li><blockquote><p>Minimise the following objective function subject to the given constraints:</p></blockquote></li>
+
<li><blockquote><p>Minimise the following '''objective function''' subject to the given constraints</p></blockquote></li>
−
<li><blockquote><p>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</p></blockquote></li></ul>
+
<li><blockquote><p>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</p></blockquote></li></ul>
 
|-
 
|-
 
|
 
|

Revision as of 09:25, 16 November 2021

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.

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.

Show Slide

FOSSEE Forum

Please post your general and technical queries on Scilab in this forum.

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..

Show Slide

Lab Migration

The FOSSEE team coordinates the Lab Migration project.

For more details, please visit this site.

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

Retrieved from "https://script.spoken-tutorial.org/index.php?title=Scilab---FOSSEE-Optimisation-Toolbox/C2/Constrained-Optimisation-using-FOT/English&oldid=55701"