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(a) Suppose we are given a linear program in standard equation form maximize cx subject to Ax=b, x ≥ 0. Given two different optimal solutions y and z to this linear program, prove that every convex combination of y and z is also an optimal solution of the program. (b) Briefly explain how to use part (a) to show that every linear program has either 0, 1, or infinitely many optimal solutions.

(a) Suppose we are given a linear program in standard equation form maximize cx subject to Ax=b, x ≥ 0. Given two different optimal solutions y and z to this linear program, prove that every convex combination of y and z is also an optimal solution of the program. (b) Briefly explain how to use part (a) to 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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(a) Suppose we are given a linear program in standard equation form maximize cx subject to Ax=b, x ≥ 0. Given two different optimal solutions y and z to this linear program, prove that every convex combination of y and z is also an optimal solution of the program. (b) Briefly explain how to use part (a) to show that every linear program has either 0, 1, or infinitely many optimal solutions.
Transcribed Image Text:(a) Suppose we are given a linear program in standard equation form maximize cx subject to Ax=b, x ≥ 0. Given two different optimal solutions y and z to this linear program, prove that every convex combination of y and z is also an optimal solution of the program. (b) Briefly explain how to use part (a) to show that every linear program has either 0, 1, or infinitely many optimal solutions.
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