Consider the following optimization problem (P) min X1 s.t. x² +ax ≤ 1 ax² X₂ ≤ 1 where a is a parameter (i.e., it is not a variable). Answer the following questions. 1. Obtain an optimal solution of (P) graphically when a = 1. 2. Obtain an optimal solution of (P) graphically when a = -1. Find all the values of a (if any) for which 3. (1, 0) is a feasible solution to (P). 4. (0, 1) is a feasible solution to (P). 5. Problem (P) is unbounded. 6. Problem (P) is infeasible. 7. Problem (P) can be written as a linear optimization model. 8. Problem (P) has multiple optimal solutions.
Consider the following optimization problem (P) min X1 s.t. x² +ax ≤ 1 ax² X₂ ≤ 1 where a is a parameter (i.e., it is not a variable). Answer the following questions. 1. Obtain an optimal solution of (P) graphically when a = 1. 2. Obtain an optimal solution of (P) graphically when a = -1. Find all the values of a (if any) for which 3. (1, 0) is a feasible solution to (P). 4. (0, 1) is a feasible solution to (P). 5. Problem (P) is unbounded. 6. Problem (P) is infeasible. 7. Problem (P) can be written as a linear optimization model. 8. Problem (P) has multiple 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
Optimization problem
![### Optimization Problem
Consider the following optimization problem:
\[
\text{(P)} \quad \min x_1
\]
\[
\text{s.t.} \quad x_1^2 + \alpha x_2^2 \leq 1
\]
\[
x_2 \leq 1
\]
where \(\alpha\) is a parameter (i.e., it is not a variable).
#### Questions
1. **Obtain an optimal solution of \((P)\) graphically when \(\alpha = \frac{1}{4}\).**
2. **Obtain an optimal solution of \((P)\) graphically when \(\alpha = -1\).**
Find all the values of \(\alpha\) (if any) for which:
3. **(1, 0) is a feasible solution to \((P)\).**
4. **(0, 1) is a feasible solution to \((P)\).**
5. **Problem \((P)\) is unbounded.**
6. **Problem \((P)\) is infeasible.**
7. **Problem \((P)\) can be written as a linear optimization model.**
8. **Problem \((P)\) has multiple optimal solutions.**
### Notes
- **Feasibility Conditions:** These refer to whether specific values satisfy all the constraints of the optimization problem.
- **Boundedness and Unboundedness:** Explores whether the solution is limited within a certain range or can extend infinitely.
- **Linear vs Non-linear Models:** Discusses if the problem can be simplified to a linear form or remains non-linear.
- **Multiple Solutions:** Identifies conditions where more than one optimal solution exists.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff12ec146-5781-453b-8388-2e7cc9921f6c%2F114e1f3a-3a7a-4d76-9f49-ce7c643dc798%2Fjdp50ck_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Optimization Problem
Consider the following optimization problem:
\[
\text{(P)} \quad \min x_1
\]
\[
\text{s.t.} \quad x_1^2 + \alpha x_2^2 \leq 1
\]
\[
x_2 \leq 1
\]
where \(\alpha\) is a parameter (i.e., it is not a variable).
#### Questions
1. **Obtain an optimal solution of \((P)\) graphically when \(\alpha = \frac{1}{4}\).**
2. **Obtain an optimal solution of \((P)\) graphically when \(\alpha = -1\).**
Find all the values of \(\alpha\) (if any) for which:
3. **(1, 0) is a feasible solution to \((P)\).**
4. **(0, 1) is a feasible solution to \((P)\).**
5. **Problem \((P)\) is unbounded.**
6. **Problem \((P)\) is infeasible.**
7. **Problem \((P)\) can be written as a linear optimization model.**
8. **Problem \((P)\) has multiple optimal solutions.**
### Notes
- **Feasibility Conditions:** These refer to whether specific values satisfy all the constraints of the optimization problem.
- **Boundedness and Unboundedness:** Explores whether the solution is limited within a certain range or can extend infinitely.
- **Linear vs Non-linear Models:** Discusses if the problem can be simplified to a linear form or remains non-linear.
- **Multiple Solutions:** Identifies conditions where more than one optimal solution exists.
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