Nptel Data Science for Engineers Assignment 5 Answers
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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)
Answer: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
Answer: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)
Answer: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
Answer:c) I 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
Answer: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.
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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
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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.
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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
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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
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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)
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
Answer: (m)=90
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)
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
Answer: 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)
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 Data Science for Engineers Nptel Assignment Solutions Week 5
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 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
Answer: 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)
