Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, Show that if y and z are two different optimal solutions to this program, then 1) every convex combination of y and z is also an optimal solution of the program. 2) Consider two vectors y and z with y z and consider two values 0 E [0, 1] and O e [0, 1] with 0 0. Show that dy + (1-0)z 0'y + (1 – 0')z. In other words, every distinct way of choosing the value A E [0, 1] gives a distinct convex combination Ay + (1- )z. Hint: let a = Oy + (1-0)z and b 0'y + (1 0')z, then consider a b. 3) Using the result from parts (a) and (b), show that every linear program has either 0, 1, or infinitely many optimal solutions.
Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, Show that if y and z are two different optimal solutions to this program, then 1) every convex combination of y and z is also an optimal solution of the program. 2) Consider two vectors y and z with y z and consider two values 0 E [0, 1] and O e [0, 1] with 0 0. Show that dy + (1-0)z 0'y + (1 – 0')z. In other words, every distinct way of choosing the value A E [0, 1] gives a distinct convex combination Ay + (1- )z. Hint: let a = Oy + (1-0)z and b 0'y + (1 0')z, then consider a b. 3) Using the result from parts (a) and (b), show that every linear program has either 0, 1, or infinitely many optimal solutions.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Consider an arbitrary program in standard equation form:
maximise
cTx
subject to Ax = b,
x 2 0
Show that if
1)
and z are two different optimal solutions to this program, then
every convex combination of y and z is also an optimal solution of the program.
2) Consider two vectors y and z with y z and consider two values 0 E [0, 1
and 0 e [0, 1] with 0 0. Show that Oy + (1 – 0)z 0'y + (1 – 0')z. In
other words, every distinct way of choosing the value A E [0, 1] gives a distinct
convex combination Ay + (1 – A)z.
Hint: let a = Oy + (1-0)z and b 0'y + (1- 0)z, then consider a -
3) Using the result from parts (a) and (b), show that every linear program has
either 0, 1, or infinitely many optimal solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa586075f-e62a-4dbd-99d5-0873e381613a%2Fd52218ca-f9a0-4849-af2d-2cb6ee3aa966%2Fk918nz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider an arbitrary program in standard equation form:
maximise
cTx
subject to Ax = b,
x 2 0
Show that if
1)
and z are two different optimal solutions to this program, then
every convex combination of y and z is also an optimal solution of the program.
2) Consider two vectors y and z with y z and consider two values 0 E [0, 1
and 0 e [0, 1] with 0 0. Show that Oy + (1 – 0)z 0'y + (1 – 0')z. In
other words, every distinct way of choosing the value A E [0, 1] gives a distinct
convex combination Ay + (1 – A)z.
Hint: let a = Oy + (1-0)z and b 0'y + (1- 0)z, then consider a -
3) Using the result from parts (a) and (b), show that every linear program has
either 0, 1, or infinitely many optimal solutions.
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