# Nptel Data Science for Engineers Assignment 5 Answers

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## Nptel Data Science for Engineers Assignment 4 Answers (July-Dec 2024)

1. The values of μ₁, μ₂, and μ₃ while evaluating the Karush-Kuhn-Tucker (KKT) condition with all the constraints being inactive are:

a) μ₁ = μ₂ = μ₃ = 1
b) μ₁ = μ₂ = μ₃ = 0
c) μ₁ = μ₃ = 0, μ₂ = 1
d) μ₁ = μ₂ = 0, μ₃ = 1

Answer: b) μ₁ = μ₂ = μ₃ = 0

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

3. The point on the plane (x + y – 2z = 6) that is closest to the origin is:

a) (0, 0, 0)
b) (1, 1, 1)
c) (-1, 1, 2)
d) (1, 1, -2)

4. Find the maximum value of (f(x,y) = 49 – x^2 – y^2) subject to the constraints (x + 3y = 10):

a) 49
b) 46
c) 59
d) 39

5. The minimum value of (f(x,y) = x^2 + 4y^2 – 2x + 8y) subject to the constraint (x + 2y = 7) occurs at the below point:

a) (5, 5)
b) (-5, 5)
c) (1, 5)
d) (5, 1)

6. Which of the following statements is/are NOT TRUE with respect to multivariate 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

a) I
b) II and III
c) I and IV
d) III and IV

7. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one:

a) True
b) False

8. A manufacturer incurs a monthly fixed cost of $7350 and a variable cost, (C(m) = 0.001m^3 – 2m^2 + 324m) dollars. The revenue generated by selling these units is, (R(m) = -6m^2 + 1065m). How many units produced every month (m) will generate maximum profit? a) m = 46 b) m = 90 c) m = 231 d) m = 125 Answer: b) m = 90 9. Consider an optimization problem (\min_{x,y} x^2 – xy + y^2) subject to the constraints (2x + y \leq 1), (x + 2y \geq 2), (x \geq -1). Find the Lagrangian function for the above optimization problem. a) (L(x,y,\mu_1,\mu_2,\mu_3) = x^2 – xy + y^2 + \mu_1(2x + y – 1) + \mu_2(2 – x – 2y) + \mu_3(-x – 1)) b) (L(x,y,\mu_1,\mu_2,\mu_3) = x^2 – xy + y^2 + \mu_1(2x + y – 1) + \mu_2(x + 2y – 2) + \mu_3(-x – 1)) c) (L(x,y,\mu_1,\mu_2,\mu_3) = x^2 – xy + y^2 + \mu_1(2x + y – 1) + \mu_2(x + 2y – 2) + \mu_3(x + 1)) d) (L(x,y,\mu_1,\mu_2,\mu_3) = x^2 – xy + y^2 + \mu_1(1 – 2x – y) + \mu_2(2 – x – 2y) + \mu_3(-x – 1)) Answer: d) (L(x,y,\mu_1,\mu_2,\mu_3) = x^2 – xy + y^2 + \mu_1(1 – 2x – y) + \mu_2(2 – x – 2y) + \mu_3(-x – 1)) ## Nptel Data Science for Engineers Assignment 5 Answers (JAN-APR 2024) Course Name: Data Science for Engineers Course Link: Click Here For answers or latest updates join our telegram channel: Click here to join These are Data Science for Engineers Nptel Assignment Solutions Week 5 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 For answers or latest updates join our telegram channel: Click here to join These are Data Science for Engineers Nptel Assignment Solutions Week 5 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. For answers or latest updates join our telegram channel: Click here to join These are Data Science for Engineers Nptel Assignment Solutions Week 5 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 For answers or latest updates join our telegram channel: Click here to join These are Data Science for Engineers Nptel Assignment Solutions Week 5 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 For answers or latest updates join our telegram channel: Click here to join These are Data Science for Engineers Nptel Assignment Solutions Week 5 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) For answers or latest updates join our telegram channel: Click here to join These are Data Science for Engineers Nptel Assignment Solutions Week 5 More Solutions of Data Science for Engineers: Click Here More Nptel Courses: Click here ## Nptel Data Science for Engineers Assignment 5 Answers (JULY-DEC 2023) Course Name: Data Science for Engineers Course Link: Click Here These are Data Science for Engineers Nptel Assignment Solutions Week 5 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 5 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 5 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 5 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

These are Data Science for Engineers Nptel Assignment Solutions Week 5

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 Data Science for Engineers Nptel Assignment Solutions Week 5

These are Data Science for Engineers Nptel Assignment Solutions Week 5

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

Q2. The solution to an unconstrained optimization problem is always the same as the solution to the constrained one.
a. True
b. False

These are Data Science for Engineers Nptel Assignment Solutions Week 5

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 Data Science for Engineers Nptel Assignment Solutions Week 5

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)

Q6. Find the eigen values corresponding to Hessian matrix of f.
a. 1,−1
b. 1,1
c. 2,8
d. 0,2

These are Data Science for Engineers Nptel Assignment Solutions Week 5

Q7. Find the minimum value of f.
a. 0
b. −5
c. −1
d. 1

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

These are Data Science for Engineers Nptel Assignment Solutions Week 5

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

These are Data Science for Engineers Nptel Assignment Solutions Week 5

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 Data Science for Engineers Nptel Assignment Solutions Week 5