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Quiz 6. Duality Theorems and Complimentary Slackness

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Each question has only one correct answer. The results of the quiz do not affect the final marks.

1: Which statement describes the Weak Duality Theorem?

Finite optimal solutions of the primal and dual problems have the same value of objective functions
Objective function of the minimization problem may not be smaller than that of the maximization problem
Objective function of the minimization problem may not be greater than that of the maximization problem
If a primal problem has finite optimal solution, the dual problem has also a finite optimal solution

2: A dual problem has no constraints on two non-negative variables, w1 and w2. The objective of the dual problem is to maximize the function:

z' = 2 w1 - 3 w2

What is an optimal solution of the primal (minimization) problem?

Unbounded objective function
Empty solution (infeasible region)
Unique optimal solution with the value z = 0
Alternative optimal solution with the value z = 0 and two arbitrary decision variables x1 and x2

3: The problem is to find maximum of the function

z = 3 x1 + 4 x2

subject to the constraints:

x1 - x2 >= 4
x1 <= 0, x2 >= 0

Suppose the dual variable w1 is associated to the first constraint. Find the feasible region of the dual problem.

w1 <= 0
w1 >= 0
3 <= w1 <= 4
w1 >= 3

4: A dual problem in canonical form has four variables and two constraints. The optimal solution of the dual problem is

w1 = 1, w2 = 0, w3 = 6, w4 = 0

Which statement is true for the primal problem?

At least two decision variables of the primal problem are zeros
At most two slack variables of the primal problem are non-zero
Primal problem has four variables and two constraint
Primal problem has unbounded value of the objective function

5: The problem is to maximize the function:

z = 2 x1 - 3 x2

subject to the constraints:

x1 + x2 = 4
x1 + x3 = 3
x1 >= 0, x2 >= 0, x3 >= 0

The final simplex tableau is


x1 x2 x3
x2 0 1 -1 1
x1 1 0 1 3
z 0 0 5 3

Find values of the dual variables w1 and w2 associated with the first and second constraints, respectively.

w1 = 0 and w2 = 0
w1 = 0 and w2 = -5
w1 = 3 and w2 = -5
w1 = -3 and w2 = -5

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