# Data Science for Engineers | Week 5

### Session: JULY-DEC 2023

Course Name: Data Science for Engineers

#### 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

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)

Q4. Find the maximum value of f(x,y)=49−x2−y2 subject to the constraints x+3y=10.
49
46
59
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)

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

#### 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

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 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)

These are NPTEL Data Science for Engineers Assignment 5 Answers

Course Name: Data Science for Engineers

#### 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 contoursII – Gradient points in the direction of greatest increase of the functionIII – Negative gradient points in the direction of the greatest decrease of the functionIV – Hessian is a non-symmetric matrixa. Ib. II and IIIc. I and IVd. 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 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)

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 NPTEL Data Science for Engineers Assignment 5 Answers

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 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

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)