Duality

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Julien Ugon

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Needs to be tested

9 months, 2 weeks ago

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Julien Ugon 9 months, 2 weeks ago

Published this.

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Julien Ugon 9 months, 2 weeks ago

Gave some feedback: Needs to be tested

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Julien Ugon 9 months, 4 weeks ago

Created this as a copy of Canonical form.

Name	Status	Author	Last Modified
Canonical form	Needs to be tested	Julien Ugon	06/08/2024 00:32
Duality	Needs to be tested	Julien Ugon	06/09/2024 00:38

There are 9 other versions that do you not have access to.

Question statement

Find the dual problem for the following linear program:

$\begin{gathered} \var{{optimise}}\text{ } \var{objective} \text{ subject to }\\ \var{system} \end{gathered}$

$x_i≥0, \forall j\in \var{Jp}$

$x_j≤0, \forall j\in \var{Jn}$

Give any introductory information the student needs.

			Name	Type	Generated Value

nvar	integer	3
ncons	integer	6
positivity	list	[ "≥0", "≥0", "free" ]
A	list	Nested 6×3 list
b	list	[ -4, 1, 4, 0, 3, -5 ]
direction	list	List of 6 items
C	list	[ 3, -4, 2 ]
optimise	string	maximise

			Name	Type	Generated Value

x	list	[ "x1", "x2", "x3" ]
L	list	List of 6 items
Cp	list	List of 6 items
Constraints	list	List of 6 items
system	string	\begin{aligned}4x_{1} & +5x_{2
objective_string	string	3x1+-4x2+2*x3
objective	expression	3x1 - 4x2 + 2x3
Jp	set	set(1,2)
Jn	set	set()
systema	string	\begin{aligned}4x_{1} & +5x_{2
systemb	string	\begin{gathered}4 x_{1} + 5 x_

			Name	Type	Generated Value

y	list	List of 6 items
Ad	list	Nested 3×6 list
Mv	list	Nested 6×3 list
objectived_string	string	-4y1+1y2+4y3+0y4+3y5+-5y
objectived	expression	y2 + 4y3 - 4y1 + 3y5 - 5y6
Ld	list	List of 3 items
Cpd	list	List of 3 items
Constraintsd	list	List of 3 items
Systemd	string	\begin{gathered}4 y_{1} - 5 y_
Jdn	list	[ 3, 4, 5 ]
Jdp	list	[ 6 ]
directionv	list	[ "≥", "≥", "=" ]
optimised	string	minimise
Adlt	matrix	matrix()
Adgt	matrix	Matrix of size 2×7
Adeq	matrix	matrix([-1,-2,5,1,5,-1,2])

			Name	Type	Generated Value

reorder	list	[ 4, 0, 2, 5, 1, 3 ]
I1	matrix	Matrix of size 6×6
I2	matrix	Matrix of size 7×7
A1	matrix	matrix()
A2	matrix	Matrix of size 2×7
A3	matrix	matrix([-2,5,5,-1,-1,1,2])
c1	vector	vector(1,3,4,-5,-4,0)
J1	set	set(4)
J2	set	set(3,6,2)
test	integer	2

			Name	Type	Generated Value

cstudent	list	[ -4, 1, 4, 0, 3, -5 ]
max_conversion	list	Nested 7×2 list
min_conversion	list	Nested 7×2 list
conversion	dict	Dictionary with 7 entries
nonnegativityCodes	list	[ "≤0", "free", "≥0" ]

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(3..5)

JME function reference

Description

Number of variables in the system.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

A
C
positivity
x

This variable doesn't seem to be used anywhere.

Parts

Dual Problem Gap-fill
1. Gap optimise Choose one from a list
  Add
2. Gap c Matrix entry
  Add
3. Gap lessthan Matrix entry
  Add
4. Gap equalities Matrix entry
  Add
5. Gap greatherthan Matrix entry
  Add
6. Gap Nonnegative Variables Matrix entry
  Add
7. Gap Nonpositive Variables Matrix entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

optimise c

subject to

Set the number of rows/columns to 0 manually if a constraint type does not exist in the dual.

Less than (≤) constraints

lessthan

Equality constraints

equalities

Greater than (≥) constraints

greatherthan

$y_j≥0, ∀j∈$ {Nonnegative Variables}.

$y_j≤0, ∀j∈$ {Nonpositive Variables}.

Advice

In order to convert a problem to canonical form, we need to perform the following changes

The objective should be maximised.
All variables should be positive.
All constraints should be in the form ≤

$\begin{gathered} \var{{optimise}}\text{ } \var{objective} \text{ subject to }\\ \var{system} \end{gathered}$

$x_i≥0, \forall\in \var{Jp}$

$x_j≤0, \forall j\in \var{Jn}$

The dual problem is:

$\begin{gathered} \var{optimised}\text{ } \var{objectived} \text{ subject to }\\ \var{systemd} \end{gathered}$

$y_i≥0, \forall\in \var{Jdp}$

$y_j≤0, \forall j\in \var{Jdn}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet

This question is used in the following exam:

Duality Exam by Julien Ugon in SIT316.

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Julien Ugon 9 months, 2 weeks ago

Julien Ugon 9 months, 2 weeks ago

Julien Ugon 9 months, 4 weeks ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Less than (≤) constraints

Equality constraints

Greater than (≥) constraints

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Duality

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julien Ugon 9 months, 2 weeks ago

Julien Ugon 9 months, 2 weeks ago

Julien Ugon 9 months, 4 weeks ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Less than (≤) constraints

Equality constraints

Greater than (≥) constraints

Generated value: `integer`