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Computer Engineering
2. Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x > 0

2. Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x > 0

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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2. Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x > 0 (a) Show that if y 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. (b) 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 – 1)z. Hint: let a = Oy + (1 – 0)z and b - O'y + (1 – 0')z, then consider a – b. (c) 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:2. Consider an arbitrary program in standard equation form: maximise cTx subject to Ax = b, x > 0 (a) Show that if y 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. (b) 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 – 1)z. Hint: let a = Oy + (1 – 0)z and b - O'y + (1 – 0')z, then consider a – b. (c) 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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