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Let p be a direction of unboundedness for the constraints Ax = b, x > 0. Prove that –p cannot be a direction of unboundedness for these constraints.

Let p be a direction of unboundedness for the constraints Ax = b, x > 0. Prove that –p cannot be a direction of unboundedness for these constraints.

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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### Unboundedness in Linear Programming **4.4. Let \( p \) be a direction of unboundedness for the constraints:** \[ Ax = b, \quad x \geq 0. \] **Prove that \( -p \) cannot be a direction of unboundedness for these constraints.** --- This problem addresses the concept of unboundedness in linear programming. We are given a set of constraints represented by a linear system, \( Ax = b \), where the vector \( x \) is non-negative, meaning each component of \( x \) is greater than or equal to zero. **Explanation:** - **Direction of Unboundedness:** A vector \( p \) is a direction of unboundedness if, starting from some feasible point, moving indefinitely in the direction of \( p \) keeps you within the feasible region and makes the objective function increase indefinitely. - **Constraints:** The system of constraints consists of a matrix \( A \), a vector \( x \), and a vector \( b \). The constraints require \( x \) to meet the equation \( Ax = b \) and to maintain non-negativity \( x \geq 0 \). - **Objective:** The task is to prove the logical assertion that \( -p \), the opposite direction of \( p \), cannot also lead to unboundedness. Intuitively, if moving in the direction of \( p \) is unbounded, moving in the opposite direction should eventually violate one of the constraint conditions or become bounded. This exercise helps in understanding the properties of feasible regions within linear programming and the behavior of solutions concerning unbounded directions.
Transcribed Image Text:### Unboundedness in Linear Programming **4.4. Let \( p \) be a direction of unboundedness for the constraints:** \[ Ax = b, \quad x \geq 0. \] **Prove that \( -p \) cannot be a direction of unboundedness for these constraints.** --- This problem addresses the concept of unboundedness in linear programming. We are given a set of constraints represented by a linear system, \( Ax = b \), where the vector \( x \) is non-negative, meaning each component of \( x \) is greater than or equal to zero. **Explanation:** - **Direction of Unboundedness:** A vector \( p \) is a direction of unboundedness if, starting from some feasible point, moving indefinitely in the direction of \( p \) keeps you within the feasible region and makes the objective function increase indefinitely. - **Constraints:** The system of constraints consists of a matrix \( A \), a vector \( x \), and a vector \( b \). The constraints require \( x \) to meet the equation \( Ax = b \) and to maintain non-negativity \( x \geq 0 \). - **Objective:** The task is to prove the logical assertion that \( -p \), the opposite direction of \( p \), cannot also lead to unboundedness. Intuitively, if moving in the direction of \( p \) is unbounded, moving in the opposite direction should eventually violate one of the constraint conditions or become bounded. This exercise helps in understanding the properties of feasible regions within linear programming and the behavior of solutions concerning unbounded directions.
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