Online Quiz # L12 Constrained optimization I

This quiz is based on the content from Lecture 12: Constrained Optimization I. Please go the lecture before attempting the quiz

1. What is the goal of a constrained optimization problem?

Minimize or maximize a function under constraints
Remove all constraints from the problem
Always minimize the given function
Always maximize the given function

2. Which of the following is an example of an equality constraint?

g(x)<= 5
h(x) = 0)
f(x) > 0)
x >= 2)

3. What does the term ‘active constraint’ mean?

A constraint that has no effect on the solution
A constraint that is not satisfied at the optimal point
A constraint that holds as an equality at the optimal point
A redundant constraint

4. What does the gradient of the Lagrangian function represent?

Direction of maximum function value
The optimality condition
The sensitivity to constraints
The infeasible solution direction

5. Which of the following is NOT a necessary condition for optimality?

Gradient of the objective function must vanish
Gradients of constraints and objective function must be proportional
Lagrange multipliers must be non-negative for inequality constraints
Feasible solutions must always be unique

6. Which method is used for optimization with inequality constraints?

Newton’s Method
Lagrange multipliers
Kuhn-Tucker conditions
Simplex Method

7. What is a feasible solution?

Any solution that does not satisfy all constraints
A solution that satisfies all the constraints
A solution outside the feasible region
A solution with the maximum function value

8. What does the Lagrange multiplier λ represent?

The slope of the objective function
The sensitivity of the objective function to changes in constraints
The infeasibility of a solution
A redundant variable

9. Which of the following represents the Lagrangian function for a constrained problem?

L(x, λ) = f(x) ⋅ g(x)
L(x, λ) = f(x) + λh(x)
L(x, λ) = f(x) − λg(x)
L(x, λ) = f(x) ⋅ h(x)

10. Which of the following conditions must be satisfied for an inequality constraint?

The constraint must be always positive
The Lagrange multiplier must be negative
The Lagrange multiplier must be non-negative
The objective function must be constant

11. What happens when all constraints are satisfied at the optimal solution?

The solution is infeasible
The problem becomes unconstrained
The optimal solution lies within the feasible region
The objective function becomes undefined

12. Which of the following statements about the Kuhn-Tucker conditions is true?

They apply only to linear problems
They apply to problems with both equality and inequality constraints
They are not necessary for optimality
They ignore inequality constraints

13. In optimization, what is meant by the ‘objective function’?

A function to minimize or maximize
A function to measure constraints
A function that has no solution
A redundant function

14. What does the constraint h(x) = 0 signify?

An inequality constraint
An equality constraint
A redundant constraint
A variable constraint

15. What is the primary assumption for applying Lagrange multipliers?

All variables are non-negative
The objective function is linear
Constraints are continuously differentiable
The problem has no solution

16. When is a solution said to be infeasible?

When all constraints are violated
When the solution is outside the feasible region
When the Lagrange multipliers are negative
When the gradient of the function is zero

17. The total derivative of a constraint ensures:

All variables remain constant
The constraint is always positive
Admissible variations satisfy the constraint
Constraints can be ignored

18. What is the effect of increasing the constraint value on the optimal solution?

It increases the optimal value proportionally
It has no effect on the solution
It makes the problem infeasible
It can be analyzed using Lagrange multipliers

19. What happens when a redundant constraint is added to the problem?

The feasible region increases
The feasible region becomes smaller
The solution remains unchanged
The problem becomes infeasible

20. What is the role of the feasible region in constrained optimization?

To minimize the Lagrange multipliers
To ensure a unique solution
To define where the optimal solution lies
To discard redundant variables