## NPTEL Data Science for Engineers Assignment 5 Answers

**Q1. An optimization problem, solved for N variables, with one equality constraint will have**

a. N equations in N variables

b. N + 1 equations in N + 1 variables

c. N equations in N + 1 variables

d. none of these

**Answer:** b. N + 1 equations in N + 1 variables

**Q2. While minimizing a function f(x,y)=6×2+4y2 with a constraint 3x+2y≤12, the unconstrained minimum solution is __________ the constrained minimum solution**

a. Equal to

b. Less than

c. Greater than

d. both (b) and (c)

**Answer:** a. Equal to

**Q3. The function min f(x,y)=3x+y subject to the given constraints x2+y2<10 is an example of**

a. Unconstrained multivariate optimisation

b. Multivariate optimisation with equality constraint

c. Multivariate optimisation with inequality constraint

d. None of the above

**Answer:** c. Multivariate optimisation with inequality constraint

**Q4.** **What is the global minimum value for the function f(x,y)=(x+y)2+5?**

a. 0

b. -∞

c. 5

d. global minimum does not exist

**Answer:** c. 5

Consider the function *f*(*x*,*y*)=5*x*2+3*y*2;+8*xy*+12*x*+6*y* as the function to be optimized and answer questions 5 and 6.

**Q5. The saddle point of the function f(x,y) exists in which of the following coordinates (x,y)**

a. (6,-9)

b. (5,3)

c. (2,-3)

d. There is no saddle point

**Answer:** a. (6,-9)

**Q6. The Hessian matrix obtained for the function f(x,y) is**

a.

b.

c.

d.

**Answer: **c.

**Q7. The eigenvalues for the Hessian matrix obtained in Q6 are**

a. -2.5, 2.5

b. 8.3584, -3.46

c. 4.5046, -5.3654

d. 16.2462, -0.2462

**Answer:** d. 16.2462, -0.2462

**Q8. Geometrically, finding the minimum of the function f(x,y)=x2+4y2 subject to the constraint 3x+7y=4 implies that**

a. The minimum value exist in the the gradient of the tangent 3x + 7y = 4

b. The minimum value is present in the contour of the ellipse x² + 4y² = k such that it’s tangent is 3x + 7y = 4

c. The minimum value is present at the point (0,0)

d. The minimum value is not dependent on the constraint

**Answer:** b. The minimum value is present in the contour of the ellipse x² + 4y² = k such that it’s tangent is 3x + 7y = 4

**Q9. State whether the following statements are True or False.**

i) The decision variables of the optimization function need not be independent of each other

ii) An optimization problem with linear objective function and linear constraints is said to be a linear optimization problem

a. i) – TRUE; ii) – FALSE

b. i) – FALSE; ii) – FALSE

c. i) – TRUE; ii) – TRUE

d. i) – FALSE; ii) – TRUE

**Answer:** d. i) – FALSE; ii) – TRUE

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