4. Let C C R" be a given set and C + Ø. Let f : R" → R be a given function. Consider the following optimization problem: (Prob1) Мaximize f(x) under the constraint: x E C. Define F, := {x e R" | f(x) > r}, and consider (Prob2) Maximize under the constraint: CnF, # Ø. Here the constraint Cn F, 7 Ø is a condition on r. Assume that an optimal solution of (Prob1) and an optimal solution of (Prob2) both exist. Find and explain the relation between the optimal solution of (Prob1) and that of (Prob2). You have to justify your answer. [Hint: To get an intuition, try first n = 1 case and m = 2 case, with a linear function f and the set C given by linear constraints. You final answer should be for general n and for general f and C.] [This problem will be marked strictly: Your solution should be clearly explained, in a logical manner. Solutions that do not show clear ideas will get very low mark if not zero.]
4. Let C C R" be a given set and C + Ø. Let f : R" → R be a given function. Consider the following optimization problem: (Prob1) Мaximize f(x) under the constraint: x E C. Define F, := {x e R" | f(x) > r}, and consider (Prob2) Maximize under the constraint: CnF, # Ø. Here the constraint Cn F, 7 Ø is a condition on r. Assume that an optimal solution of (Prob1) and an optimal solution of (Prob2) both exist. Find and explain the relation between the optimal solution of (Prob1) and that of (Prob2). You have to justify your answer. [Hint: To get an intuition, try first n = 1 case and m = 2 case, with a linear function f and the set C given by linear constraints. You final answer should be for general n and for general f and C.] [This problem will be marked strictly: Your solution should be clearly explained, in a logical manner. Solutions that do not show clear ideas will get very low mark if not zero.]
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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plz solve the question with explanation I will give you upvote
![4.
Let C C R" be a given set and C + Ø. Let f : R" → R be a given function.
Consider the following optimization problem:
(Prob1)
Maximize
f(x)
under the constraint: x E C.
Define F, := {x e R" | f(x) > r}, and consider
(Prob2)
Maximize
under the constraint: Cn F, + Ø.
Here the constraint CN F, 7 Ø is a condition on r.
Assume that an optimal solution of (Prob1) and an optimal solution of (Prob2) both
exist. Find and explain the relation between the optimal solution of (Prob1) and that
of (Prob2). You have to justify your answer.
[Hint: To get an intuition, try first n = 1
2 case, with a linear function f and the set C given by linear constraints.
case and n=
You final answer should be for general n and for general f and C.] [This problem will
be marked strictly: Your solution should be clearly explained, in a logical
manner. Solutions that do not show clear ideas will get very low mark if not
zero.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc1b5c4a-c3bc-4b6f-9997-aff256c9d780%2F58573975-1771-445b-82bd-da14e868514c%2Fuh9fdie_processed.png&w=3840&q=75)
Transcribed Image Text:4.
Let C C R" be a given set and C + Ø. Let f : R" → R be a given function.
Consider the following optimization problem:
(Prob1)
Maximize
f(x)
under the constraint: x E C.
Define F, := {x e R" | f(x) > r}, and consider
(Prob2)
Maximize
under the constraint: Cn F, + Ø.
Here the constraint CN F, 7 Ø is a condition on r.
Assume that an optimal solution of (Prob1) and an optimal solution of (Prob2) both
exist. Find and explain the relation between the optimal solution of (Prob1) and that
of (Prob2). You have to justify your answer.
[Hint: To get an intuition, try first n = 1
2 case, with a linear function f and the set C given by linear constraints.
case and n=
You final answer should be for general n and for general f and C.] [This problem will
be marked strictly: Your solution should be clearly explained, in a logical
manner. Solutions that do not show clear ideas will get very low mark if not
zero.]
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