Data Science for Engineers | Week 5
Session: JAN-APR 2024
Course Name: Data Science for Engineers
Course Link: Click Here
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These are NPTEL Data Science for Engineers Assignment 5 Answers
Q1. Which of the following statements is/are not TRUE with respect to the multi variate optimization?
I – The gradient of a function at a point is parallel to the contours
II – Gradient points in the direction of greatest increase of the function
III – Negative gradients points in the direction of the greatest decrease of the function
IV – Hessian is a non-symmetric matrix
I
II and III
I and IV
III and IV
Answer: I and IV
Q2. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.
True
False
Answer: False
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These are NPTEL Data Science for Engineers Assignment 5 Answers
Q3. Gradient-based algorithm methods compute
only step length at each iteration
both direction and step length at each iteration
only direction at each iteration
none of the above
Answer: both direction and step length at each iteration
Q4. For an unconstrained multivariate optimization given f(x¯), the necessary second order condition for x¯∗ to be the minimizer of f(x¯) is
∇2f(x¯∗) must be negative definite.
∇2f(x¯∗) must be positive definite.
∇f(x¯∗)=0
”(x∗)>0
Answer: ∇2f(x¯∗) must be positive definite.
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These are NPTEL Data Science for Engineers Assignment 5 Answers
Consider an optimization problem minx1,x2∈Rf(x1,x2)=x21+4×22−2×1+8×2.
Q5. Which among the following is the stationary point for f(x1,x2)?
(0, 0)
(1, −1)
(−1, −1)
(−1, 1)
Answer: (1, −1)
Q6. Find the eighen values corresponding to Hessian matrix of f.
1, −1
1, 1
2, 8
0, 2
Answer: 2, 8
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These are NPTEL Data Science for Engineers Assignment 5 Answers
Q7. Find the minimum value of f.
0
−5
−1
1
Answer: −5
Q8. Now, in order to find the minimum value of f subject to the constraint
x1+2×2=7,
what should be the first order condition for x∗¯ to be a minimizer of f(x1,x2)?
a) 2x∗1+2=λ
−8x∗2−8=2λ
x∗1+2x∗2=7
b) −2x∗1+2=λ
−8x∗2−8=2λ
x∗1+2x∗2=7
c) 2x∗1−2=−λ
8x∗2+8=−2λ
x∗1+2x∗2=7
d) −2x∗1+2=−λ
−8x∗2−8=−2λ
x∗1+2x∗2=7
Answer: b) −2x∗1+2=λ
−8x∗2−8=2λ
x∗1+2x∗2=7
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These are NPTEL Data Science for Engineers Assignment 5 Answers
Q9. What is the minimum value of f(x1,x2) subject to the above mentioned constrained?
−5
−1
27
0
Answer: 27
Q10. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraints x+3y=10.
49
46
59
39
Answer: 39
Q11. Consider an optimization problem minx1,x2x2−xy+y2 subject to the constraints
2x+y≤1
x+2y≥2
x≥−1
Find the lagrangian function for the above optimization problem.
L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(2−x−2y)+μ3(−x−1)
L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(x+2y−2)+μ3(−x−1)
L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(x+2y−2)+μ3(x+1)
L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(1−2x−y)+μ2(2−x−2y)+μ3(−x−1)
Answer: L(x,y,μ1,μ2,μ3)=x2−xy+y2+μ1(2x+y−1)+μ2(2−x−2y)+μ3(−x−1)
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These are NPTEL Data Science for Engineers Assignment 5 Answers
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Session: JULY-DEC 2023
Course Name: Data Science for Engineers
Course Link: Click Here
These are NPTEL Data Science for Engineers Assignment 5 Answers
Q1. The values of μ1,μ2 and μ3 while evaluating the Karush-Kuhn-Tucker (KKT) condition with all the constraints being inactive are
μ1=μ2=μ3=1
μ1=μ2=μ3=0
μ1=μ3=0,μ2=1
μ1=μ2=0,μ3=1
Answer: μ1=μ2=μ3=0
Q2. Gradient based algorithm methods compute
only step length at each iteration
both direction and step length at each iteration
only direction at each iteration
none of the above
Answer: both direction and step length at each iteration
These are NPTEL Data Science for Engineers Assignment 4 Answers
Q3. The point on the plane x+y−2z=6 that is closest to the origin is
(0,0,0)
(1,1,1)
(−1,1,2)
(1,1,−2)
Answer: (1,1,−2)
Q4. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraints x+3y=10.
49
46
59
39
Answer: 39
These are NPTEL Data Science for Engineers Assignment 4 Answers
Q5. The minimum value of f(x,y)=x2+4y2−2x+8y subject to the constraint x+2y=7 occurs at the below point:
(5,5)
(−5,5)
(1,5)
(5,1)
Answer: (5,1)
Q6. Which of the following statements is/are NOT TRUE with respect to the multi variate optimization?
I – The gradient of a function at a point is parallel to the contours
II – Gradient points in the direction of greatest increase of the function
III – Negative gradients points in the direction of the greatest decrease of the function
IV – Hessian is a non-symmetric matrix
I
II and III
I and IV
III and IV
Answer: I and IV
These are NPTEL Data Science for Engineers Assignment 4 Answers
Q7. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.
True
False
Answer: False
Q8. A manufacturer incurs a monthly fixed cost of $7350 and a variable cost,C(m)=0.001m3−2m2+324m dollars. The revenue generated by selling these units is, R(m)=−6m2+1065m. How many units produced every month (m) will generate maximum profit?
(m)=46
(m)=90
(m)=231
(m)=125
Answer: (m)=90
These are NPTEL Data Science for Engineers Assignment 4 Answers
Q9. Consider an optimization problem minx1,x2 x2−xy+y2 subject to the constraints
2x+y≤1
x+2y≥2
x≥−1
Find the lagrangian function for the above optimization problem.
L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(2x + y − 1) + μ2(2 − x − 2y) + μ3( −x − 1)
L(x,y,μ1,μ2,μ3) = x2 −xy + y2 + μ1(2x + y − 1) + μ2(x + 2y − 2)) + μ3( −x − 1)
L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(2x + y − 1) + μ2(x + 2y − 2)) + μ3(x + 1)
L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(1 − 2x − y) + μ2(2 − x − 2y) + μ3( − x − 1)
Answer: L(x,y,μ1,μ2,μ3) = x2 − xy + y2 + μ1(2x + y − 1) + μ2(2 − x − 2y) + μ3( −x − 1)
These are NPTEL Data Science for Engineers Assignment 5 Answers
More Solutions of Data Science for Engineers: Click Here
More Nptel Courses: Click here
Course Name: Data Science for Engineers
Course Link: Click Here
These are NPTEL Data Science for Engineers Assignment 5 Answers
Q1. Which of the following statements is/are not TRUE with respect to the multi variate optimization?
I – The gradient of a function at a point is parallel to the contours
II – Gradient points in the direction of greatest increase of the function
III – Negative gradient points in the direction of the greatest decrease of the function
IV – Hessian is a non-symmetric matrix
a. I
b. II and III
c. I and IV
d. III and IV
Answer: c. I and IV
Q2. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.
a. True
b. False
Answer: b. False
These are NPTEL Data Science for Engineers Assignment 5 Answers
Q3. Gradient based algorithm methods compute
a. only step length at each iteration
b. both direction and step length at each iteration
c. only direction at each iteration
d. none of the above
Answer: b. both direction and step length at each iteration
Q4. For an unconstrained multivariate optimization given f(x¯¯¯), the necessary second order condition for x¯¯¯∗ to be the minimizer of f(x) is
a. ∇2f(x¯¯¯∗) must be negative definite.
b. ∇2f(x¯¯¯∗) must be positive definite.
c. ∇f(x¯¯¯∗)=0
d. f”(x¯¯¯∗)>0
Answer: b. ∇2f(x¯¯¯∗) must be positive definite.
These are NPTEL Data Science for Engineers Assignment 5 Answers
Use the following information to answer Q5, 6, 7 and 8
minx1,x2∈R f(x1,x2)=x21+4×22−2×1+8×2.
Q5. Which among the following is the stationary point for f(x1,x2)?
a. (0,0)
b. (1,−1)
c. (−1,−1)
d. (−1,1)
Answer: b. (1,−1)
Q6. Find the eigen values corresponding to Hessian matrix of f.
a. 1,−1
b. 1,1
c. 2,8
d. 0,2
Answer: c. 2,8
These are NPTEL Data Science for Engineers Assignment 5 Answers
Q7. Find the minimum value of f.
a. 0
b. −5
c. −1
d. 1
Answer: b. −5
Q8. What is the minimum value of f(x1,x2)csubject to the constraint x1+2×2=7?
a. −5
b. −1
c. 27
d. 0
Answer: c. 27
These are NPTEL Data Science for Engineers Assignment 5 Answers
Q9. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraint x+3y=10.
a. 49
b. 46
c. 59
d. 39
Answer: d. 39
These are NPTEL Data Science for Engineers Assignment 5 Answers
Q10. Consider an optimization problem minx1,x2 x2−xy+y2 subject to the constraints 2x+y≤1
x+2y≥2
x≥−1
Find the lagrangian function for the above optimization problem.
a. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(2 − x − 2y)+µ3( − x − 1)
b. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(x + 2y − 2) + µ3( − x − 1)
c. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(x + 2y − 2) + µ3(x + 1)
d. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(1 − 2x − y) + µ2(2 − x − 2y) + µ3( − x − 1)
Answer: c. L(x,y,µ1,µ2,µ3) = x2 − xy + y2 + µ1(2x + y − 1) + µ2(x + 2y − 2) + µ3(x + 1)
These are NPTEL Data Science for Engineers Assignment 5 Answers
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