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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
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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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.
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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